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Temporal causality requires that models should be sufficiently trained at time", "type": "text" }, { "bbox": [ 499, 711, 504, 720 ], "score": 0.7, "content": "t", "type": "inline_equation" } ], "index": 46 }, { "bbox": [ 105, 720, 506, 734 ], "spans": [ { "bbox": [ 105, 720, 307, 734 ], "score": 1.0, "content": "before approximating the solution at the later time", "type": "text" }, { "bbox": [ 308, 722, 336, 731 ], "score": 0.86, "content": "t + \\Delta t", "type": "inline_equation" }, { "bbox": [ 337, 720, 506, 734 ], "score": 1.0, "content": ", while continuous-time PINNs are trained", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 151, 95 ], "score": 1.0, "content": "for all time", "type": "text", "cross_page": true }, { "bbox": [ 152, 84, 157, 92 ], "score": 0.7, "content": "t", "type": "inline_equation", "cross_page": true }, { "bbox": [ 157, 82, 505, 95 ], "score": 1.0, "content": "simultaneously. To enhance the temporal causality in the training process, they proposed", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 93, 505, 107 ], "spans": [ { "bbox": [ 105, 93, 505, 107 ], "score": 1.0, "content": "a simple re-formulation of PINNs loss functions as shown in equation 1, i.e., a clever weighting", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 505, 117 ], "score": 1.0, "content": "technique that is inversely exponentially proportional to the magnitude of cumulative residual losses", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 106, 115, 505, 127 ], "spans": [ { "bbox": [ 106, 115, 505, 127 ], "score": 1.0, "content": "from prior times. This casual PINN method has been demonstrated to be effective for some difficult", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 105, 126, 505, 140 ], "spans": [ { "bbox": [ 105, 126, 432, 140 ], "score": 1.0, "content": "problems. However their method is sensitive to the new causality hyper-parameter", "type": "text", "cross_page": true }, { "bbox": [ 432, 129, 437, 137 ], "score": 0.7, "content": "\\epsilon", "type": "inline_equation", "cross_page": true }, { "bbox": [ 438, 126, 505, 140 ], "score": 1.0, "content": ", and the training", "type": "text", "cross_page": true } ], "index": 4 }, { "bbox": [ 105, 137, 298, 150 ], "spans": [ { "bbox": [ 105, 137, 298, 150 ], "score": 1.0, "content": "time is substantially longer than vanilla PINNs.", "type": "text", "cross_page": true } ], "index": 5 } ], "index": 43, "bbox_fs": [ 105, 632, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 108, 82, 505, 149 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 151, 95 ], "score": 1.0, "content": "for all time", "type": "text" }, { "bbox": [ 152, 84, 157, 92 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 157, 82, 505, 95 ], "score": 1.0, "content": "simultaneously. To enhance the temporal causality in the training process, they proposed", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 107 ], "spans": [ { "bbox": [ 105, 93, 505, 107 ], "score": 1.0, "content": "a simple re-formulation of PINNs loss functions as shown in equation 1, i.e., a clever weighting", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 505, 117 ], "spans": [ { "bbox": [ 105, 104, 505, 117 ], "score": 1.0, "content": "technique that is inversely exponentially proportional to the magnitude of cumulative residual losses", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 115, 505, 127 ], "spans": [ { "bbox": [ 106, 115, 505, 127 ], "score": 1.0, "content": "from prior times. This casual PINN method has been demonstrated to be effective for some difficult", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 505, 140 ], "spans": [ { "bbox": [ 105, 126, 432, 140 ], "score": 1.0, "content": "problems. However their method is sensitive to the new causality hyper-parameter", "type": "text" }, { "bbox": [ 432, 129, 437, 137 ], "score": 0.7, "content": "\\epsilon", "type": "inline_equation" }, { "bbox": [ 438, 126, 505, 140 ], "score": 1.0, "content": ", and the training", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 137, 298, 150 ], "spans": [ { "bbox": [ 105, 137, 298, 150 ], "score": 1.0, "content": "time is substantially longer than vanilla PINNs.", "type": "text" } ], "index": 5 } ], "index": 2.5 }, { "type": "interline_equation", "bbox": [ 169, 154, 443, 189 ], "lines": [ { "bbox": [ 169, 154, 443, 189 ], "spans": [ { "bbox": [ 169, 154, 443, 189 ], "score": 0.94, "content": "\\mathcal { L } ( \\boldsymbol { \\theta } ) = \\frac { 1 } { N _ { t } } \\sum _ { i = 1 } ^ { N _ { t } } w _ { i } \\mathcal { L } ( t _ { i } , \\boldsymbol { \\theta } ) , \\quad \\mathrm { w i t h } \\quad w _ { i } = \\exp \\left( - \\epsilon \\sum _ { k = 1 } ^ { i - 1 } \\mathcal { L } ( t _ { k } , \\boldsymbol { \\theta } ) \\right) .", "type": "interline_equation", "image_path": "1bde2ec7e51813312ff42d93bbf1e10ed5565835c804cc1f8bceffae187b5f12.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 169, 154, 443, 165.66666666666666 ], "spans": [], "index": 6 }, { "bbox": [ 169, 165.66666666666666, 443, 177.33333333333331 ], "spans": [], "index": 7 }, { "bbox": [ 169, 177.33333333333331, 443, 188.99999999999997 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 106, 199, 505, 331 ], "lines": [ { "bbox": [ 105, 198, 506, 213 ], "spans": [ { "bbox": [ 105, 198, 506, 213 ], "score": 1.0, "content": "In this paper, we introduce a new PINN implementation technique for efficiently and precisely solving", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 211, 506, 223 ], "spans": [ { "bbox": [ 106, 211, 506, 223 ], "score": 1.0, "content": "evolutionary PDEs. Our technique relies on two key elements: (a) using discrete-time PINNs instead", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 220, 506, 235 ], "spans": [ { "bbox": [ 105, 220, 506, 235 ], "score": 1.0, "content": "of continuous-time PINNs to satisfy the principle of temporal causality, thereby making the training", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 233, 506, 245 ], "spans": [ { "bbox": [ 105, 233, 506, 245 ], "score": 1.0, "content": "process stable and accurate; and (b) utilizing transfer learning to accelerate PINN training in later", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 243, 506, 255 ], "spans": [ { "bbox": [ 106, 243, 506, 255 ], "score": 1.0, "content": "time frames. The time-differencing schemes such as forward/backward Euler, Crank-Nicolson, and", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 253, 506, 268 ], "spans": [ { "bbox": [ 105, 253, 506, 268 ], "score": 1.0, "content": "Runge-Kutta enable solutions to be learned from earlier times to later times, therefore satisfying", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 266, 505, 278 ], "spans": [ { "bbox": [ 106, 266, 505, 278 ], "score": 1.0, "content": "the temporal causality principle. Moreover, the errors from time differencing can be theoretically", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 275, 506, 289 ], "spans": [ { "bbox": [ 105, 275, 506, 289 ], "score": 1.0, "content": "controlled Ascher (2008), making the training procedure stable and accurate. We accelerate PINN", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 287, 505, 299 ], "spans": [ { "bbox": [ 105, 287, 505, 299 ], "score": 1.0, "content": "training naturally by initializing the PINN parameters at the next time frame with the trained PINN", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 298, 506, 310 ], "spans": [ { "bbox": [ 105, 298, 506, 310 ], "score": 1.0, "content": "parameters at the current time frame. In the following sections, we will show that our transfer", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 309, 505, 322 ], "spans": [ { "bbox": [ 106, 309, 505, 322 ], "score": 1.0, "content": "learning enhanced discrete physics-informed neural networks (TL-DPINN) method is theoretically", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 321, 294, 332 ], "spans": [ { "bbox": [ 106, 321, 294, 332 ], "score": 1.0, "content": "and numerically stable, accurate, and efficient.", "type": "text" } ], "index": 20 } ], "index": 14.5 }, { "type": "text", "bbox": [ 107, 337, 333, 348 ], "lines": [ { "bbox": [ 105, 335, 334, 351 ], "spans": [ { "bbox": [ 105, 335, 334, 351 ], "score": 1.0, "content": "Following is a summary of the contribution of the paper.", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 133, 357, 505, 455 ], "lines": [ { "bbox": [ 133, 357, 506, 371 ], "spans": [ { "bbox": [ 133, 357, 506, 371 ], "score": 1.0, "content": "• Implicit time differencing with the transfer-learning tuned PINN provides more accurate and", "type": "text" } ], "index": 22 }, { "bbox": [ 141, 370, 506, 381 ], "spans": [ { "bbox": [ 141, 370, 506, 381 ], "score": 1.0, "content": "robust predictions of evolutionary PDEs’ solutions while retaining a low computational cost.", "type": "text" } ], "index": 23 }, { "bbox": [ 132, 383, 506, 396 ], "spans": [ { "bbox": [ 132, 383, 506, 396 ], "score": 1.0, "content": "• We prove theoretically the error estimation result of our TL-DPINN method, indicating that", "type": "text" } ], "index": 24 }, { "bbox": [ 141, 394, 506, 408 ], "spans": [ { "bbox": [ 141, 394, 506, 408 ], "score": 1.0, "content": "TL-DPINN solutions converge as long as the time step is small and each PINN in different", "type": "text" } ], "index": 25 }, { "bbox": [ 142, 406, 253, 417 ], "spans": [ { "bbox": [ 142, 406, 253, 417 ], "score": 1.0, "content": "time frames is well trained.", "type": "text" } ], "index": 26 }, { "bbox": [ 132, 420, 507, 433 ], "spans": [ { "bbox": [ 132, 420, 507, 433 ], "score": 1.0, "content": "• Through extensive numerical results, we demonstrate that our method can attain state-of-the-", "type": "text" } ], "index": 27 }, { "bbox": [ 141, 432, 505, 445 ], "spans": [ { "bbox": [ 141, 432, 505, 445 ], "score": 1.0, "content": "art (SOTA) performance among various PINN frameworks in a trade-off between accuracy", "type": "text" } ], "index": 28 }, { "bbox": [ 142, 442, 202, 456 ], "spans": [ { "bbox": [ 142, 442, 202, 456 ], "score": 1.0, "content": "and efficiency.", "type": "text" } ], "index": 29 } ], "index": 25.5 }, { "type": "title", "bbox": [ 108, 470, 215, 483 ], "lines": [ { "bbox": [ 105, 469, 216, 485 ], "spans": [ { "bbox": [ 105, 469, 216, 485 ], "score": 1.0, "content": "2 RELATED WORKS", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 106, 495, 505, 627 ], "lines": [ { "bbox": [ 106, 495, 505, 508 ], "spans": [ { "bbox": [ 106, 495, 505, 508 ], "score": 1.0, "content": "Discrete PINN. Raissi et al. Raissi et al. (2019) have applied the general form of Runge–Kutta", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 506, 506, 519 ], "spans": [ { "bbox": [ 105, 506, 201, 519 ], "score": 1.0, "content": "methods with arbitrary", "type": "text" }, { "bbox": [ 201, 509, 208, 518 ], "score": 0.76, "content": "q", "type": "inline_equation" }, { "bbox": [ 208, 506, 506, 519 ], "score": 1.0, "content": "stages to the evolutionary PDEs. However, only an implicit Runge-Kutta", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 517, 505, 529 ], "spans": [ { "bbox": [ 106, 517, 158, 529 ], "score": 1.0, "content": "scheme with", "type": "text" }, { "bbox": [ 158, 518, 192, 529 ], "score": 0.91, "content": "q = 1 0 0", "type": "inline_equation" }, { "bbox": [ 193, 517, 327, 529 ], "score": 1.0, "content": "stages and a single large time step", "type": "text" }, { "bbox": [ 327, 518, 367, 528 ], "score": 0.9, "content": "\\Delta t = 0 . 8", "type": "inline_equation" }, { "bbox": [ 367, 517, 505, 529 ], "score": 1.0, "content": "are computed. Low-order methods", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 529, 505, 541 ], "spans": [ { "bbox": [ 106, 529, 505, 541 ], "score": 1.0, "content": "cannot retain their predictive accuracy for large time steps. In our research, we demonstrate the", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 540, 505, 551 ], "spans": [ { "bbox": [ 106, 540, 505, 551 ], "score": 1.0, "content": "capability of discrete PINNs both theoretically and experimentally, indicating that robust low-order", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 550, 506, 563 ], "spans": [ { "bbox": [ 106, 550, 506, 563 ], "score": 1.0, "content": "implicit Runge-Kutta combined with PINN can obtain high-precision solutions with multiple small-", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 561, 505, 574 ], "spans": [ { "bbox": [ 106, 561, 505, 574 ], "score": 1.0, "content": "sized time steps. Jagtap and Karniadakis Jagtap & Karniadakis (2021) propose a generalized domain", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 572, 505, 584 ], "spans": [ { "bbox": [ 105, 572, 505, 584 ], "score": 1.0, "content": "decomposition framework that allows for multiple sub-networks over different subdomains to be", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 583, 506, 595 ], "spans": [ { "bbox": [ 105, 583, 506, 595 ], "score": 1.0, "content": "stitched together and trained in parallel. However, it is not causal and has the same training issues as", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 594, 506, 606 ], "spans": [ { "bbox": [ 105, 594, 506, 606 ], "score": 1.0, "content": "conventional PINNs. The implicit Runge-Kutta scheme combined with PINN has been used to solve", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 604, 505, 618 ], "spans": [ { "bbox": [ 106, 604, 505, 618 ], "score": 1.0, "content": "simple ODE systems Stiasny et al. (2021); Moya & Lin (2023), but not dynamic PDE systems with", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 616, 281, 628 ], "spans": [ { "bbox": [ 106, 616, 281, 628 ], "score": 1.0, "content": "multi-scale or turbulent behavior over time.", "type": "text" } ], "index": 42 } ], "index": 36.5 }, { "type": "text", "bbox": [ 106, 632, 506, 732 ], "lines": [ { "bbox": [ 105, 632, 506, 646 ], "spans": [ { "bbox": [ 105, 632, 506, 646 ], "score": 1.0, "content": "Temporal decomposition. Diverse strategies have been studied for enhancing PINN training by", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 642, 506, 657 ], "spans": [ { "bbox": [ 105, 642, 506, 657 ], "score": 1.0, "content": "splitting the domain into numerous small “time-slab”. Wight and Zhao L. Wight & Zhao (2021)", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 655, 506, 668 ], "spans": [ { "bbox": [ 105, 655, 506, 668 ], "score": 1.0, "content": "propose an adaptive time-sampling strategy to learn solutions from the previous small time domain to", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 666, 506, 679 ], "spans": [ { "bbox": [ 106, 666, 506, 679 ], "score": 1.0, "content": "the whole time domain. However, collocation points are costly to add, and the computational cost rises.", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 677, 506, 690 ], "spans": [ { "bbox": [ 106, 677, 506, 690 ], "score": 1.0, "content": "This time marching strategy has been enhanced further in Krishnapriyan et al. (2021); Mattey & Ghosh", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 505, 700 ], "score": 1.0, "content": "(2022); McClenny & Braga-Neto (2023). Nevertheless, causality is only enforced on the scale of the", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 699, 506, 712 ], "spans": [ { "bbox": [ 106, 699, 506, 712 ], "score": 1.0, "content": "time slabs and not inside each time slab, thus the convergence can not be theoretically guaranteed.", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 709, 507, 723 ], "spans": [ { "bbox": [ 105, 709, 507, 723 ], "score": 1.0, "content": "A unified framework for causal sweeping strategies for PINNs is summarized in Penwarden et al.", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 721, 506, 733 ], "spans": [ { "bbox": [ 106, 721, 506, 733 ], "score": 1.0, "content": "(2023). Wang et al. Wang et al. 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Our technique relies on two key elements: (a) using discrete-time PINNs instead", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 220, 506, 235 ], "spans": [ { "bbox": [ 105, 220, 506, 235 ], "score": 1.0, "content": "of continuous-time PINNs to satisfy the principle of temporal causality, thereby making the training", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 233, 506, 245 ], "spans": [ { "bbox": [ 105, 233, 506, 245 ], "score": 1.0, "content": "process stable and accurate; and (b) utilizing transfer learning to accelerate PINN training in later", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 243, 506, 255 ], "spans": [ { "bbox": [ 106, 243, 506, 255 ], "score": 1.0, "content": "time frames. The time-differencing schemes such as forward/backward Euler, Crank-Nicolson, and", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 253, 506, 268 ], "spans": [ { "bbox": [ 105, 253, 506, 268 ], "score": 1.0, "content": "Runge-Kutta enable solutions to be learned from earlier times to later times, therefore satisfying", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 266, 505, 278 ], "spans": [ { "bbox": [ 106, 266, 505, 278 ], "score": 1.0, "content": "the temporal causality principle. Moreover, the errors from time differencing can be theoretically", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 275, 506, 289 ], "spans": [ { "bbox": [ 105, 275, 506, 289 ], "score": 1.0, "content": "controlled Ascher (2008), making the training procedure stable and accurate. We accelerate PINN", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 287, 505, 299 ], "spans": [ { "bbox": [ 105, 287, 505, 299 ], "score": 1.0, "content": "training naturally by initializing the PINN parameters at the next time frame with the trained PINN", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 298, 506, 310 ], "spans": [ { "bbox": [ 105, 298, 506, 310 ], "score": 1.0, "content": "parameters at the current time frame. In the following sections, we will show that our transfer", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 309, 505, 322 ], "spans": [ { "bbox": [ 106, 309, 505, 322 ], "score": 1.0, "content": "learning enhanced discrete physics-informed neural networks (TL-DPINN) method is theoretically", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 321, 294, 332 ], "spans": [ { "bbox": [ 106, 321, 294, 332 ], "score": 1.0, "content": "and numerically stable, accurate, and efficient.", "type": "text" } ], "index": 20 } ], "index": 14.5, "bbox_fs": [ 105, 198, 506, 332 ] }, { "type": "text", "bbox": [ 107, 337, 333, 348 ], "lines": [ { "bbox": [ 105, 335, 334, 351 ], "spans": [ { "bbox": [ 105, 335, 334, 351 ], "score": 1.0, "content": "Following is a summary of the contribution of the paper.", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 105, 335, 334, 351 ] }, { "type": "list", "bbox": [ 133, 357, 505, 455 ], "lines": [ { "bbox": [ 133, 357, 506, 371 ], "spans": [ { "bbox": [ 133, 357, 506, 371 ], "score": 1.0, "content": "• Implicit time differencing with the transfer-learning tuned PINN provides more accurate and", "type": "text" } ], "index": 22, "is_list_start_line": true }, { "bbox": [ 141, 370, 506, 381 ], "spans": [ { "bbox": [ 141, 370, 506, 381 ], "score": 1.0, "content": "robust predictions of evolutionary PDEs’ solutions while retaining a low computational cost.", "type": "text" } ], "index": 23 }, { "bbox": [ 132, 383, 506, 396 ], "spans": [ { "bbox": [ 132, 383, 506, 396 ], "score": 1.0, "content": "• We prove theoretically the error estimation result of our TL-DPINN method, indicating that", "type": "text" } ], "index": 24, "is_list_start_line": true }, { "bbox": [ 141, 394, 506, 408 ], "spans": [ { "bbox": [ 141, 394, 506, 408 ], "score": 1.0, "content": "TL-DPINN solutions converge as long as the time step is small and each PINN in different", "type": "text" } ], "index": 25 }, { "bbox": [ 142, 406, 253, 417 ], "spans": [ { "bbox": [ 142, 406, 253, 417 ], "score": 1.0, "content": "time frames is well trained.", "type": "text" } ], "index": 26, "is_list_end_line": true }, { "bbox": [ 132, 420, 507, 433 ], "spans": [ { "bbox": [ 132, 420, 507, 433 ], "score": 1.0, "content": "• Through extensive numerical results, we demonstrate that our method can attain state-of-the-", "type": "text" } ], "index": 27, "is_list_start_line": true }, { "bbox": [ 141, 432, 505, 445 ], "spans": [ { "bbox": [ 141, 432, 505, 445 ], "score": 1.0, "content": "art (SOTA) performance among various PINN frameworks in a trade-off between accuracy", "type": "text" } ], "index": 28 }, { "bbox": [ 142, 442, 202, 456 ], "spans": [ { "bbox": [ 142, 442, 202, 456 ], "score": 1.0, "content": "and efficiency.", "type": "text" } ], "index": 29, "is_list_end_line": true } ], "index": 25.5, "bbox_fs": [ 132, 357, 507, 456 ] }, { "type": "title", "bbox": [ 108, 470, 215, 483 ], "lines": [ { "bbox": [ 105, 469, 216, 485 ], "spans": [ { "bbox": [ 105, 469, 216, 485 ], "score": 1.0, "content": "2 RELATED WORKS", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 106, 495, 505, 627 ], "lines": [ { "bbox": [ 106, 495, 505, 508 ], "spans": [ { "bbox": [ 106, 495, 505, 508 ], "score": 1.0, "content": "Discrete PINN. Raissi et al. Raissi et al. (2019) have applied the general form of Runge–Kutta", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 506, 506, 519 ], "spans": [ { "bbox": [ 105, 506, 201, 519 ], "score": 1.0, "content": "methods with arbitrary", "type": "text" }, { "bbox": [ 201, 509, 208, 518 ], "score": 0.76, "content": "q", "type": "inline_equation" }, { "bbox": [ 208, 506, 506, 519 ], "score": 1.0, "content": "stages to the evolutionary PDEs. However, only an implicit Runge-Kutta", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 517, 505, 529 ], "spans": [ { "bbox": [ 106, 517, 158, 529 ], "score": 1.0, "content": "scheme with", "type": "text" }, { "bbox": [ 158, 518, 192, 529 ], "score": 0.91, "content": "q = 1 0 0", "type": "inline_equation" }, { "bbox": [ 193, 517, 327, 529 ], "score": 1.0, "content": "stages and a single large time step", "type": "text" }, { "bbox": [ 327, 518, 367, 528 ], "score": 0.9, "content": "\\Delta t = 0 . 8", "type": "inline_equation" }, { "bbox": [ 367, 517, 505, 529 ], "score": 1.0, "content": "are computed. Low-order methods", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 529, 505, 541 ], "spans": [ { "bbox": [ 106, 529, 505, 541 ], "score": 1.0, "content": "cannot retain their predictive accuracy for large time steps. In our research, we demonstrate the", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 540, 505, 551 ], "spans": [ { "bbox": [ 106, 540, 505, 551 ], "score": 1.0, "content": "capability of discrete PINNs both theoretically and experimentally, indicating that robust low-order", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 550, 506, 563 ], "spans": [ { "bbox": [ 106, 550, 506, 563 ], "score": 1.0, "content": "implicit Runge-Kutta combined with PINN can obtain high-precision solutions with multiple small-", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 561, 505, 574 ], "spans": [ { "bbox": [ 106, 561, 505, 574 ], "score": 1.0, "content": "sized time steps. Jagtap and Karniadakis Jagtap & Karniadakis (2021) propose a generalized domain", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 572, 505, 584 ], "spans": [ { "bbox": [ 105, 572, 505, 584 ], "score": 1.0, "content": "decomposition framework that allows for multiple sub-networks over different subdomains to be", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 583, 506, 595 ], "spans": [ { "bbox": [ 105, 583, 506, 595 ], "score": 1.0, "content": "stitched together and trained in parallel. However, it is not causal and has the same training issues as", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 594, 506, 606 ], "spans": [ { "bbox": [ 105, 594, 506, 606 ], "score": 1.0, "content": "conventional PINNs. The implicit Runge-Kutta scheme combined with PINN has been used to solve", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 604, 505, 618 ], "spans": [ { "bbox": [ 106, 604, 505, 618 ], "score": 1.0, "content": "simple ODE systems Stiasny et al. (2021); Moya & Lin (2023), but not dynamic PDE systems with", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 616, 281, 628 ], "spans": [ { "bbox": [ 106, 616, 281, 628 ], "score": 1.0, "content": "multi-scale or turbulent behavior over time.", "type": "text" } ], "index": 42 } ], "index": 36.5, "bbox_fs": [ 105, 495, 506, 628 ] }, { "type": "text", "bbox": [ 106, 632, 506, 732 ], "lines": [ { "bbox": [ 105, 632, 506, 646 ], "spans": [ { "bbox": [ 105, 632, 506, 646 ], "score": 1.0, "content": "Temporal decomposition. Diverse strategies have been studied for enhancing PINN training by", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 642, 506, 657 ], "spans": [ { "bbox": [ 105, 642, 506, 657 ], "score": 1.0, "content": "splitting the domain into numerous small “time-slab”. Wight and Zhao L. Wight & Zhao (2021)", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 655, 506, 668 ], "spans": [ { "bbox": [ 105, 655, 506, 668 ], "score": 1.0, "content": "propose an adaptive time-sampling strategy to learn solutions from the previous small time domain to", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 666, 506, 679 ], "spans": [ { "bbox": [ 106, 666, 506, 679 ], "score": 1.0, "content": "the whole time domain. However, collocation points are costly to add, and the computational cost rises.", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 677, 506, 690 ], "spans": [ { "bbox": [ 106, 677, 506, 690 ], "score": 1.0, "content": "This time marching strategy has been enhanced further in Krishnapriyan et al. (2021); Mattey & Ghosh", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 505, 700 ], "score": 1.0, "content": "(2022); McClenny & Braga-Neto (2023). Nevertheless, causality is only enforced on the scale of the", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 699, 506, 712 ], "spans": [ { "bbox": [ 106, 699, 506, 712 ], "score": 1.0, "content": "time slabs and not inside each time slab, thus the convergence can not be theoretically guaranteed.", "type": "text" } ], "index": 49 }, { "bbox": [ 105, 709, 507, 723 ], "spans": [ { "bbox": [ 105, 709, 507, 723 ], "score": 1.0, "content": "A unified framework for causal sweeping strategies for PINNs is summarized in Penwarden et al.", "type": "text" } ], "index": 50 }, { "bbox": [ 106, 721, 506, 733 ], "spans": [ { "bbox": [ 106, 721, 506, 733 ], "score": 1.0, "content": "(2023). Wang et al. Wang et al. (2022a) introduced a simple causal weight in the form of equation 1", "type": "text" } ], "index": 51 }, { "bbox": [ 105, 82, 506, 95 ], "spans": [ { "bbox": [ 105, 82, 506, 95 ], "score": 1.0, "content": "to naturally match the principle of temporal causality with high precision. However, this significantly", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 93, 505, 106 ], "spans": [ { "bbox": [ 105, 93, 505, 106 ], "score": 1.0, "content": "increased computational costs and did not guarantee convergence Penwarden et al. (2023). Our", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 104, 506, 117 ], "spans": [ { "bbox": [ 105, 104, 506, 117 ], "score": 1.0, "content": "methods can attain the same level of precision, are theoretically convergent, and are 4 to 40 times", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 104, 115, 141, 128 ], "spans": [ { "bbox": [ 104, 115, 141, 128 ], "score": 1.0, "content": "quicker.", "type": "text", "cross_page": true } ], "index": 3 } ], "index": 47, "bbox_fs": [ 105, 632, 507, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 127 ], "lines": [ { "bbox": [ 105, 82, 506, 95 ], "spans": [ { "bbox": [ 105, 82, 506, 95 ], "score": 1.0, "content": "to naturally match the principle of temporal causality with high precision. However, this significantly", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 106 ], "spans": [ { "bbox": [ 105, 93, 505, 106 ], "score": 1.0, "content": "increased computational costs and did not guarantee convergence Penwarden et al. (2023). Our", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 104, 506, 117 ], "spans": [ { "bbox": [ 105, 104, 506, 117 ], "score": 1.0, "content": "methods can attain the same level of precision, are theoretically convergent, and are 4 to 40 times", "type": "text" } ], "index": 2 }, { "bbox": [ 104, 115, 141, 128 ], "spans": [ { "bbox": [ 104, 115, 141, 128 ], "score": 1.0, "content": "quicker.", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "text", "bbox": [ 106, 132, 506, 232 ], "lines": [ { "bbox": [ 105, 130, 506, 146 ], "spans": [ { "bbox": [ 105, 130, 506, 146 ], "score": 1.0, "content": "Transfer learning. Transfer-learning has been previously combined with various deep-learning", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 144, 506, 155 ], "spans": [ { "bbox": [ 105, 144, 506, 155 ], "score": 1.0, "content": "models for solving PDEs problems, such as PINN for phase-field modeling of fracture Goswami et al.", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 506, 167 ], "spans": [ { "bbox": [ 105, 154, 506, 167 ], "score": 1.0, "content": "(2020), DeepONet for PDEs under conditional shift Goswami et al. (2022), DNN-based PDE solvers", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 165, 506, 178 ], "spans": [ { "bbox": [ 106, 165, 506, 178 ], "score": 1.0, "content": "Chen et al. (2021), PINN for inverse problems Xu et al. (2023), one-shot transfer learning of PINN", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 175, 507, 189 ], "spans": [ { "bbox": [ 105, 175, 507, 189 ], "score": 1.0, "content": "Desai et al. (2022), and training of CNNs on multi-fidelity data Song & Tartakovsky (2022). Xu et al.", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 506, 200 ], "spans": [ { "bbox": [ 105, 187, 506, 200 ], "score": 1.0, "content": "Xu et al. (2022) proposed a transfer learning enhanced DeepONet for the long-term prediction of", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 199, 505, 210 ], "spans": [ { "bbox": [ 106, 199, 505, 210 ], "score": 1.0, "content": "evolution equations. However, their method necessitates a substantial amount of training data from", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 208, 505, 222 ], "spans": [ { "bbox": [ 105, 208, 505, 222 ], "score": 1.0, "content": "traditional numerical methods. In contrast, our methods are physics-informed and do not require", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 220, 204, 233 ], "spans": [ { "bbox": [ 106, 220, 204, 233 ], "score": 1.0, "content": "additional training data.", "type": "text" } ], "index": 12 } ], "index": 8 }, { "type": "title", "bbox": [ 108, 247, 238, 260 ], "lines": [ { "bbox": [ 105, 246, 239, 262 ], "spans": [ { "bbox": [ 105, 246, 239, 262 ], "score": 1.0, "content": "3 NUMERICAL METHOD", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 105, 272, 504, 294 ], "lines": [ { "bbox": [ 105, 271, 505, 285 ], "spans": [ { "bbox": [ 105, 271, 505, 285 ], "score": 1.0, "content": "Problem set-up Here we consider the initial-boundary value problem for a general evolutionary", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 282, 468, 296 ], "spans": [ { "bbox": [ 105, 282, 468, 296 ], "score": 1.0, "content": "parabolic differential equation. The extension to hyperbolic equations are straightforward.", "type": "text" } ], "index": 15 } ], "index": 14.5 }, { "type": "interline_equation", "bbox": [ 221, 297, 389, 341 ], "lines": [ { "bbox": [ 221, 297, 389, 341 ], "spans": [ { "bbox": [ 221, 297, 389, 341 ], "score": 0.93, "content": "\\left\\{ \\begin{array} { l l } { u _ { t } = \\mathcal { N } ( u ) , \\quad x \\in \\Omega , t \\in [ 0 , T ] , } \\\\ { u ( 0 , x ) = u _ { 0 } ( x ) , \\quad x \\in \\Omega , } \\\\ { u ( t , x ) = g ( t , x ) , \\quad t \\in [ 0 , T ] , x \\in \\partial \\Omega , } \\end{array} \\right.", "type": "interline_equation", "image_path": "62b7255b5b8c64010c48fd71d29c743efa329f796be6517877b5f7b5728e0064.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 221, 297, 389, 311.6666666666667 ], "spans": [], "index": 16 }, { "bbox": [ 221, 311.6666666666667, 389, 326.33333333333337 ], "spans": [], "index": 17 }, { "bbox": [ 221, 326.33333333333337, 389, 341.00000000000006 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 107, 344, 505, 390 ], "lines": [ { "bbox": [ 106, 344, 506, 357 ], "spans": [ { "bbox": [ 106, 344, 134, 357 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 344, 162, 356 ], "score": 0.92, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 162, 344, 282, 357 ], "score": 1.0, "content": "denotes the hidden solution,", "type": "text" }, { "bbox": [ 283, 346, 288, 354 ], "score": 0.75, "content": "t", "type": "inline_equation" }, { "bbox": [ 289, 344, 307, 357 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 308, 347, 315, 354 ], "score": 0.76, "content": "x", "type": "inline_equation" }, { "bbox": [ 315, 344, 506, 357 ], "score": 1.0, "content": "represent temporal and spatial coordinates re-", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 354, 506, 369 ], "spans": [ { "bbox": [ 105, 354, 152, 369 ], "score": 1.0, "content": "spectively,", "type": "text" }, { "bbox": [ 152, 356, 176, 367 ], "score": 0.9, "content": "\\dot { \\mathcal { N } } ( u )", "type": "inline_equation" }, { "bbox": [ 177, 354, 361, 369 ], "score": 1.0, "content": "denotes a differential operator (for example,", "type": "text" }, { "bbox": [ 361, 355, 415, 367 ], "score": 0.92, "content": "\\mathcal { N } \\bar { ( } u ) = u _ { x x }", "type": "inline_equation" }, { "bbox": [ 415, 354, 506, 369 ], "score": 1.0, "content": "for the simplest Heat", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 365, 506, 380 ], "spans": [ { "bbox": [ 105, 365, 165, 380 ], "score": 1.0, "content": "equation), and", "type": "text" }, { "bbox": [ 165, 366, 201, 377 ], "score": 0.91, "content": "\\Omega \\subset \\mathbb { R } ^ { D }", "type": "inline_equation" }, { "bbox": [ 201, 365, 407, 380 ], "score": 1.0, "content": "is an open, bounded domain with smooth boundary", "type": "text" }, { "bbox": [ 407, 367, 421, 376 ], "score": 0.8, "content": "\\partial \\Omega", "type": "inline_equation" }, { "bbox": [ 422, 365, 506, 380 ], "score": 1.0, "content": ". This study assumes", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 377, 454, 391 ], "spans": [ { "bbox": [ 105, 377, 306, 391 ], "score": 1.0, "content": "that the equations are dissipative in the sense that", "type": "text" }, { "bbox": [ 306, 377, 390, 390 ], "score": 0.89, "content": "\\begin{array} { r } { \\int _ { \\Omega } u \\cdot \\mathcal { N } ( u ) d x \\leq 0 \\mathrm { ~ } \\mathrm { X } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 391, 377, 454, 391 ], "score": 1.0, "content": "u et al. (2022).", "type": "text" } ], "index": 22 } ], "index": 20.5 }, { "type": "text", "bbox": [ 107, 395, 505, 418 ], "lines": [ { "bbox": [ 106, 395, 505, 408 ], "spans": [ { "bbox": [ 106, 395, 181, 408 ], "score": 1.0, "content": "Our goal is to learn", "type": "text" }, { "bbox": [ 182, 396, 210, 407 ], "score": 0.92, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 210, 395, 505, 408 ], "score": 1.0, "content": "by neural network approximation. We briefly mention the basic background", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 405, 430, 419 ], "spans": [ { "bbox": [ 105, 405, 430, 419 ], "score": 1.0, "content": "of PINN in Section 3.1 and then describe our TL-DPINN method in Section 3.2.", "type": "text" } ], "index": 24 } ], "index": 23.5 }, { "type": "title", "bbox": [ 106, 431, 308, 442 ], "lines": [ { "bbox": [ 106, 430, 308, 443 ], "spans": [ { "bbox": [ 106, 430, 308, 443 ], "score": 1.0, "content": "3.1 PHYSICS-INFORMED NEURAL NETWORKS", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 106, 451, 505, 496 ], "lines": [ { "bbox": [ 105, 451, 506, 464 ], "spans": [ { "bbox": [ 105, 451, 385, 464 ], "score": 1.0, "content": "In the original study of PINNs Raissi et al. (2019), it approximates", "type": "text" }, { "bbox": [ 386, 451, 414, 464 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 414, 451, 506, 464 ], "score": 1.0, "content": "to equation 2 using a", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 462, 505, 475 ], "spans": [ { "bbox": [ 106, 463, 194, 475 ], "score": 1.0, "content": "deep neural network", "type": "text" }, { "bbox": [ 194, 462, 226, 474 ], "score": 0.93, "content": "u _ { \\theta } ( t , x )", "type": "inline_equation" }, { "bbox": [ 227, 463, 259, 475 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 259, 463, 266, 473 ], "score": 0.79, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 266, 463, 505, 475 ], "score": 1.0, "content": "represents the neural network’s parameters (e.g., weights", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 473, 505, 486 ], "spans": [ { "bbox": [ 105, 473, 419, 486 ], "score": 1.0, "content": "and biases). Consequently, the objective of a vanilla PINN is to discover the", "type": "text" }, { "bbox": [ 420, 474, 426, 483 ], "score": 0.8, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 426, 473, 505, 486 ], "score": 1.0, "content": "that minimizes the", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 485, 221, 497 ], "spans": [ { "bbox": [ 106, 485, 221, 497 ], "score": 1.0, "content": "physics-based loss function:", "type": "text" } ], "index": 29 } ], "index": 27.5 }, { "type": "interline_equation", "bbox": [ 222, 500, 388, 514 ], "lines": [ { "bbox": [ 222, 500, 388, 514 ], "spans": [ { "bbox": [ 222, 500, 388, 514 ], "score": 0.93, "content": "\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) = \\lambda _ { b } \\mathcal { L } _ { b } ( \\boldsymbol { \\theta } ) + \\lambda _ { u } \\mathcal { L } _ { u } ( \\boldsymbol { \\theta } ) + \\lambda _ { r } \\mathcal { L } _ { r } ( \\boldsymbol { \\theta } ) , } \\end{array}", "type": "interline_equation", "image_path": "f38ec2510a988be83cc88d9a0cf70df16ce0572ceb3572f39cbfdc8c2d0a0a28.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 222, 500, 388, 514 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 106, 517, 505, 604 ], "lines": [ { "bbox": [ 106, 511, 510, 538 ], "spans": [ { "bbox": [ 106, 511, 132, 538 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 517, 487, 534 ], "score": 0.64, "content": "\\begin{array} { r } { \\mathcal { L } _ { b } ( \\theta ) = \\frac { 1 } { N _ { b } } \\sum _ { i = 1 } ^ { N _ { b } } \\| u _ { \\theta } ( t _ { b } ^ { i } , x _ { b } ^ { i } ) - g ( t _ { b } ^ { i } , x _ { b } ^ { i } ) \\| ^ { 2 } , \\mathcal { L } _ { u } ( \\theta ) = \\frac { 1 } { N _ { u } } \\sum _ { i = 1 } ^ { N _ { u } } \\| u _ { \\theta } ( 0 , x _ { t } ^ { i } ) - u _ { 0 } ( x _ { t } ^ { i } ) \\| ^ { 2 } } \\end{array}", "type": "inline_equation", "image_path": "154fa693dd519652863b490df1e1792b44c386556bbe760c7239fd081f449130.jpg" }, { "bbox": [ 487, 511, 510, 538 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 527, 506, 552 ], "spans": [ { "bbox": [ 106, 533, 252, 549 ], "score": 0.89, "content": "\\begin{array} { r } { \\mathcal { L } _ { r } ( \\theta ) = \\frac { 1 } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } \\| \\mathcal { R } ( u _ { \\theta } ( t _ { r } ^ { i } , x _ { r } ^ { i } ) \\| ^ { 2 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 252, 527, 274, 552 ], "score": 1.0, "content": ". The", "type": "text" }, { "bbox": [ 274, 534, 311, 547 ], "score": 0.92, "content": "t _ { b } ^ { i } , x _ { b } ^ { i } , x _ { t } ^ { i }", "type": "inline_equation" }, { "bbox": [ 311, 527, 506, 552 ], "score": 1.0, "content": "u represent the boundary and initial sampling data", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 546, 506, 564 ], "spans": [ { "bbox": [ 104, 546, 120, 564 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 549, 153, 560 ], "score": 0.9, "content": "u _ { \\theta } ( t , x )", "type": "inline_equation" }, { "bbox": [ 154, 546, 192, 564 ], "score": 1.0, "content": ", whereas", "type": "text" }, { "bbox": [ 193, 548, 216, 561 ], "score": 0.86, "content": "t _ { r } ^ { i } , x _ { r } ^ { i }", "type": "inline_equation" }, { "bbox": [ 217, 546, 470, 564 ], "score": 1.0, "content": "represent the data points utilized to calculate the residual term", "type": "text" }, { "bbox": [ 470, 548, 506, 561 ], "score": 0.9, "content": "\\mathcal { R } ( u ) =", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 107, 559, 506, 573 ], "spans": [ { "bbox": [ 107, 560, 153, 572 ], "score": 0.91, "content": "u _ { t } - \\mathcal { N } ( u )", "type": "inline_equation" }, { "bbox": [ 153, 559, 227, 573 ], "score": 1.0, "content": ". The coefficients", "type": "text" }, { "bbox": [ 227, 560, 238, 571 ], "score": 0.69, "content": "\\lambda _ { b }", "type": "inline_equation" }, { "bbox": [ 239, 559, 242, 573 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 243, 560, 255, 571 ], "score": 0.68, "content": "\\lambda _ { u }", "type": "inline_equation" }, { "bbox": [ 255, 559, 276, 573 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 277, 560, 289, 571 ], "score": 0.88, "content": "\\lambda _ { r }", "type": "inline_equation" }, { "bbox": [ 289, 559, 506, 573 ], "score": 1.0, "content": "in the loss function are utilized to assign a different", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 570, 506, 584 ], "spans": [ { "bbox": [ 105, 570, 506, 584 ], "score": 1.0, "content": "learning rate, which can be specified by humans or automatically adjusted during trainingWang et al.", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 581, 505, 594 ], "spans": [ { "bbox": [ 105, 581, 242, 594 ], "score": 1.0, "content": "(2021a; 2022b). We note that the", "type": "text" }, { "bbox": [ 242, 582, 254, 593 ], "score": 0.89, "content": "\\mathcal { L } _ { b }", "type": "inline_equation" }, { "bbox": [ 255, 581, 505, 594 ], "score": 1.0, "content": "term can be further omitted if we apply hard constraint in the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 592, 437, 605 ], "spans": [ { "bbox": [ 105, 592, 437, 605 ], "score": 1.0, "content": "PINN’s design Lu et al. (2021b); Liu et al. (2022a); Sukumar & Srivastava (2022).", "type": "text" } ], "index": 37 } ], "index": 34 }, { "type": "text", "bbox": [ 107, 609, 505, 676 ], "lines": [ { "bbox": [ 106, 610, 505, 621 ], "spans": [ { "bbox": [ 106, 610, 505, 621 ], "score": 1.0, "content": "As demonstrated in Wang et al. (2022a), the vanilla PINN may violate the principle of temporal", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 620, 505, 633 ], "spans": [ { "bbox": [ 105, 620, 505, 633 ], "score": 1.0, "content": "causality, as the residual loss at the later time may be minimized even if the predictions at previous", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 631, 506, 644 ], "spans": [ { "bbox": [ 106, 631, 506, 644 ], "score": 1.0, "content": "times are incorrect. Figure 1 demonstrates the training result for solving the Allen-Chan equation,", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 642, 505, 655 ], "spans": [ { "bbox": [ 106, 642, 399, 655 ], "score": 1.0, "content": "confirming this phenomenon. For conventional PINN, the residual loss", "type": "text" }, { "bbox": [ 400, 643, 412, 653 ], "score": 0.89, "content": "\\mathcal { L } _ { r }", "type": "inline_equation" }, { "bbox": [ 413, 642, 505, 655 ], "score": 1.0, "content": "is quite large near the", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 653, 506, 666 ], "spans": [ { "bbox": [ 105, 653, 506, 666 ], "score": 1.0, "content": "initial state and decays quickly to a small value when the learned solution is incorrect. Comparatively,", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 664, 503, 677 ], "spans": [ { "bbox": [ 106, 664, 280, 677 ], "score": 1.0, "content": "our method’s residual remains small for all", "type": "text" }, { "bbox": [ 280, 664, 317, 676 ], "score": 0.92, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 317, 664, 503, 677 ], "score": 1.0, "content": "and captures the solution with high precision.", "type": "text" } ], "index": 43 } ], "index": 40.5 }, { "type": "title", "bbox": [ 107, 689, 347, 700 ], "lines": [ { "bbox": [ 106, 689, 348, 702 ], "spans": [ { "bbox": [ 106, 689, 348, 702 ], "score": 1.0, "content": "3.2 TRANSFER LEARNING ENHANCED DISCRETE PINN", "type": "text" } ], "index": 44 } ], "index": 44 }, { "type": "text", "bbox": [ 107, 709, 504, 732 ], "lines": [ { "bbox": [ 105, 708, 505, 722 ], "spans": [ { "bbox": [ 105, 708, 505, 722 ], "score": 1.0, "content": "Discrete PINN Since the continuous-time PINN violates temporal causality, we shift to numerical", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 457, 733 ], "score": 1.0, "content": "temporal differencing schemes that naturally respect temporal causality. Given a time step", "type": "text" }, { "bbox": [ 457, 721, 470, 731 ], "score": 0.83, "content": "\\Delta t", "type": "inline_equation" }, { "bbox": [ 471, 720, 505, 733 ], "score": 1.0, "content": ", assume", "type": "text" } ], "index": 46 } ], "index": 45.5 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 25, 308, 38 ], "spans": [ { "bbox": [ 107, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 127 ], "lines": [], "index": 1.5, "bbox_fs": [ 104, 82, 506, 128 ], "lines_deleted": true }, { "type": "text", "bbox": [ 106, 132, 506, 232 ], "lines": [ { "bbox": [ 105, 130, 506, 146 ], "spans": [ { "bbox": [ 105, 130, 506, 146 ], "score": 1.0, "content": "Transfer learning. Transfer-learning has been previously combined with various deep-learning", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 144, 506, 155 ], "spans": [ { "bbox": [ 105, 144, 506, 155 ], "score": 1.0, "content": "models for solving PDEs problems, such as PINN for phase-field modeling of fracture Goswami et al.", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 154, 506, 167 ], "spans": [ { "bbox": [ 105, 154, 506, 167 ], "score": 1.0, "content": "(2020), DeepONet for PDEs under conditional shift Goswami et al. (2022), DNN-based PDE solvers", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 165, 506, 178 ], "spans": [ { "bbox": [ 106, 165, 506, 178 ], "score": 1.0, "content": "Chen et al. (2021), PINN for inverse problems Xu et al. (2023), one-shot transfer learning of PINN", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 175, 507, 189 ], "spans": [ { "bbox": [ 105, 175, 507, 189 ], "score": 1.0, "content": "Desai et al. (2022), and training of CNNs on multi-fidelity data Song & Tartakovsky (2022). Xu et al.", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 187, 506, 200 ], "spans": [ { "bbox": [ 105, 187, 506, 200 ], "score": 1.0, "content": "Xu et al. (2022) proposed a transfer learning enhanced DeepONet for the long-term prediction of", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 199, 505, 210 ], "spans": [ { "bbox": [ 106, 199, 505, 210 ], "score": 1.0, "content": "evolution equations. However, their method necessitates a substantial amount of training data from", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 208, 505, 222 ], "spans": [ { "bbox": [ 105, 208, 505, 222 ], "score": 1.0, "content": "traditional numerical methods. In contrast, our methods are physics-informed and do not require", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 220, 204, 233 ], "spans": [ { "bbox": [ 106, 220, 204, 233 ], "score": 1.0, "content": "additional training data.", "type": "text" } ], "index": 12 } ], "index": 8, "bbox_fs": [ 105, 130, 507, 233 ] }, { "type": "title", "bbox": [ 108, 247, 238, 260 ], "lines": [ { "bbox": [ 105, 246, 239, 262 ], "spans": [ { "bbox": [ 105, 246, 239, 262 ], "score": 1.0, "content": "3 NUMERICAL METHOD", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 105, 272, 504, 294 ], "lines": [ { "bbox": [ 105, 271, 505, 285 ], "spans": [ { "bbox": [ 105, 271, 505, 285 ], "score": 1.0, "content": "Problem set-up Here we consider the initial-boundary value problem for a general evolutionary", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 282, 468, 296 ], "spans": [ { "bbox": [ 105, 282, 468, 296 ], "score": 1.0, "content": "parabolic differential equation. The extension to hyperbolic equations are straightforward.", "type": "text" } ], "index": 15 } ], "index": 14.5, "bbox_fs": [ 105, 271, 505, 296 ] }, { "type": "interline_equation", "bbox": [ 221, 297, 389, 341 ], "lines": [ { "bbox": [ 221, 297, 389, 341 ], "spans": [ { "bbox": [ 221, 297, 389, 341 ], "score": 0.93, "content": "\\left\\{ \\begin{array} { l l } { u _ { t } = \\mathcal { N } ( u ) , \\quad x \\in \\Omega , t \\in [ 0 , T ] , } \\\\ { u ( 0 , x ) = u _ { 0 } ( x ) , \\quad x \\in \\Omega , } \\\\ { u ( t , x ) = g ( t , x ) , \\quad t \\in [ 0 , T ] , x \\in \\partial \\Omega , } \\end{array} \\right.", "type": "interline_equation", "image_path": "62b7255b5b8c64010c48fd71d29c743efa329f796be6517877b5f7b5728e0064.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 221, 297, 389, 311.6666666666667 ], "spans": [], "index": 16 }, { "bbox": [ 221, 311.6666666666667, 389, 326.33333333333337 ], "spans": [], "index": 17 }, { "bbox": [ 221, 326.33333333333337, 389, 341.00000000000006 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 107, 344, 505, 390 ], "lines": [ { "bbox": [ 106, 344, 506, 357 ], "spans": [ { "bbox": [ 106, 344, 134, 357 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 344, 162, 356 ], "score": 0.92, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 162, 344, 282, 357 ], "score": 1.0, "content": "denotes the hidden solution,", "type": "text" }, { "bbox": [ 283, 346, 288, 354 ], "score": 0.75, "content": "t", "type": "inline_equation" }, { "bbox": [ 289, 344, 307, 357 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 308, 347, 315, 354 ], "score": 0.76, "content": "x", "type": "inline_equation" }, { "bbox": [ 315, 344, 506, 357 ], "score": 1.0, "content": "represent temporal and spatial coordinates re-", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 354, 506, 369 ], "spans": [ { "bbox": [ 105, 354, 152, 369 ], "score": 1.0, "content": "spectively,", "type": "text" }, { "bbox": [ 152, 356, 176, 367 ], "score": 0.9, "content": "\\dot { \\mathcal { N } } ( u )", "type": "inline_equation" }, { "bbox": [ 177, 354, 361, 369 ], "score": 1.0, "content": "denotes a differential operator (for example,", "type": "text" }, { "bbox": [ 361, 355, 415, 367 ], "score": 0.92, "content": "\\mathcal { N } \\bar { ( } u ) = u _ { x x }", "type": "inline_equation" }, { "bbox": [ 415, 354, 506, 369 ], "score": 1.0, "content": "for the simplest Heat", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 365, 506, 380 ], "spans": [ { "bbox": [ 105, 365, 165, 380 ], "score": 1.0, "content": "equation), and", "type": "text" }, { "bbox": [ 165, 366, 201, 377 ], "score": 0.91, "content": "\\Omega \\subset \\mathbb { R } ^ { D }", "type": "inline_equation" }, { "bbox": [ 201, 365, 407, 380 ], "score": 1.0, "content": "is an open, bounded domain with smooth boundary", "type": "text" }, { "bbox": [ 407, 367, 421, 376 ], "score": 0.8, "content": "\\partial \\Omega", "type": "inline_equation" }, { "bbox": [ 422, 365, 506, 380 ], "score": 1.0, "content": ". This study assumes", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 377, 454, 391 ], "spans": [ { "bbox": [ 105, 377, 306, 391 ], "score": 1.0, "content": "that the equations are dissipative in the sense that", "type": "text" }, { "bbox": [ 306, 377, 390, 390 ], "score": 0.89, "content": "\\begin{array} { r } { \\int _ { \\Omega } u \\cdot \\mathcal { N } ( u ) d x \\leq 0 \\mathrm { ~ } \\mathrm { X } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 391, 377, 454, 391 ], "score": 1.0, "content": "u et al. (2022).", "type": "text" } ], "index": 22 } ], "index": 20.5, "bbox_fs": [ 105, 344, 506, 391 ] }, { "type": "text", "bbox": [ 107, 395, 505, 418 ], "lines": [ { "bbox": [ 106, 395, 505, 408 ], "spans": [ { "bbox": [ 106, 395, 181, 408 ], "score": 1.0, "content": "Our goal is to learn", "type": "text" }, { "bbox": [ 182, 396, 210, 407 ], "score": 0.92, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 210, 395, 505, 408 ], "score": 1.0, "content": "by neural network approximation. We briefly mention the basic background", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 405, 430, 419 ], "spans": [ { "bbox": [ 105, 405, 430, 419 ], "score": 1.0, "content": "of PINN in Section 3.1 and then describe our TL-DPINN method in Section 3.2.", "type": "text" } ], "index": 24 } ], "index": 23.5, "bbox_fs": [ 105, 395, 505, 419 ] }, { "type": "title", "bbox": [ 106, 431, 308, 442 ], "lines": [ { "bbox": [ 106, 430, 308, 443 ], "spans": [ { "bbox": [ 106, 430, 308, 443 ], "score": 1.0, "content": "3.1 PHYSICS-INFORMED NEURAL NETWORKS", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "text", "bbox": [ 106, 451, 505, 496 ], "lines": [ { "bbox": [ 105, 451, 506, 464 ], "spans": [ { "bbox": [ 105, 451, 385, 464 ], "score": 1.0, "content": "In the original study of PINNs Raissi et al. (2019), it approximates", "type": "text" }, { "bbox": [ 386, 451, 414, 464 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 414, 451, 506, 464 ], "score": 1.0, "content": "to equation 2 using a", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 462, 505, 475 ], "spans": [ { "bbox": [ 106, 463, 194, 475 ], "score": 1.0, "content": "deep neural network", "type": "text" }, { "bbox": [ 194, 462, 226, 474 ], "score": 0.93, "content": "u _ { \\theta } ( t , x )", "type": "inline_equation" }, { "bbox": [ 227, 463, 259, 475 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 259, 463, 266, 473 ], "score": 0.79, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 266, 463, 505, 475 ], "score": 1.0, "content": "represents the neural network’s parameters (e.g., weights", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 473, 505, 486 ], "spans": [ { "bbox": [ 105, 473, 419, 486 ], "score": 1.0, "content": "and biases). Consequently, the objective of a vanilla PINN is to discover the", "type": "text" }, { "bbox": [ 420, 474, 426, 483 ], "score": 0.8, "content": "\\theta", "type": "inline_equation" }, { "bbox": [ 426, 473, 505, 486 ], "score": 1.0, "content": "that minimizes the", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 485, 221, 497 ], "spans": [ { "bbox": [ 106, 485, 221, 497 ], "score": 1.0, "content": "physics-based loss function:", "type": "text" } ], "index": 29 } ], "index": 27.5, "bbox_fs": [ 105, 451, 506, 497 ] }, { "type": "interline_equation", "bbox": [ 222, 500, 388, 514 ], "lines": [ { "bbox": [ 222, 500, 388, 514 ], "spans": [ { "bbox": [ 222, 500, 388, 514 ], "score": 0.93, "content": "\\begin{array} { r } { \\mathcal { L } ( \\boldsymbol { \\theta } ) = \\lambda _ { b } \\mathcal { L } _ { b } ( \\boldsymbol { \\theta } ) + \\lambda _ { u } \\mathcal { L } _ { u } ( \\boldsymbol { \\theta } ) + \\lambda _ { r } \\mathcal { L } _ { r } ( \\boldsymbol { \\theta } ) , } \\end{array}", "type": "interline_equation", "image_path": "f38ec2510a988be83cc88d9a0cf70df16ce0572ceb3572f39cbfdc8c2d0a0a28.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 222, 500, 388, 514 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 106, 517, 505, 604 ], "lines": [ { "bbox": [ 106, 511, 510, 538 ], "spans": [ { "bbox": [ 106, 511, 132, 538 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 517, 487, 534 ], "score": 0.64, "content": "\\begin{array} { r } { \\mathcal { L } _ { b } ( \\theta ) = \\frac { 1 } { N _ { b } } \\sum _ { i = 1 } ^ { N _ { b } } \\| u _ { \\theta } ( t _ { b } ^ { i } , x _ { b } ^ { i } ) - g ( t _ { b } ^ { i } , x _ { b } ^ { i } ) \\| ^ { 2 } , \\mathcal { L } _ { u } ( \\theta ) = \\frac { 1 } { N _ { u } } \\sum _ { i = 1 } ^ { N _ { u } } \\| u _ { \\theta } ( 0 , x _ { t } ^ { i } ) - u _ { 0 } ( x _ { t } ^ { i } ) \\| ^ { 2 } } \\end{array}", "type": "inline_equation", "image_path": "154fa693dd519652863b490df1e1792b44c386556bbe760c7239fd081f449130.jpg" }, { "bbox": [ 487, 511, 510, 538 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 527, 506, 552 ], "spans": [ { "bbox": [ 106, 533, 252, 549 ], "score": 0.89, "content": "\\begin{array} { r } { \\mathcal { L } _ { r } ( \\theta ) = \\frac { 1 } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } \\| \\mathcal { R } ( u _ { \\theta } ( t _ { r } ^ { i } , x _ { r } ^ { i } ) \\| ^ { 2 } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 252, 527, 274, 552 ], "score": 1.0, "content": ". The", "type": "text" }, { "bbox": [ 274, 534, 311, 547 ], "score": 0.92, "content": "t _ { b } ^ { i } , x _ { b } ^ { i } , x _ { t } ^ { i }", "type": "inline_equation" }, { "bbox": [ 311, 527, 506, 552 ], "score": 1.0, "content": "u represent the boundary and initial sampling data", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 546, 506, 564 ], "spans": [ { "bbox": [ 104, 546, 120, 564 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 549, 153, 560 ], "score": 0.9, "content": "u _ { \\theta } ( t , x )", "type": "inline_equation" }, { "bbox": [ 154, 546, 192, 564 ], "score": 1.0, "content": ", whereas", "type": "text" }, { "bbox": [ 193, 548, 216, 561 ], "score": 0.86, "content": "t _ { r } ^ { i } , x _ { r } ^ { i }", "type": "inline_equation" }, { "bbox": [ 217, 546, 470, 564 ], "score": 1.0, "content": "represent the data points utilized to calculate the residual term", "type": "text" }, { "bbox": [ 470, 548, 506, 561 ], "score": 0.9, "content": "\\mathcal { R } ( u ) =", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 107, 559, 506, 573 ], "spans": [ { "bbox": [ 107, 560, 153, 572 ], "score": 0.91, "content": "u _ { t } - \\mathcal { N } ( u )", "type": "inline_equation" }, { "bbox": [ 153, 559, 227, 573 ], "score": 1.0, "content": ". The coefficients", "type": "text" }, { "bbox": [ 227, 560, 238, 571 ], "score": 0.69, "content": "\\lambda _ { b }", "type": "inline_equation" }, { "bbox": [ 239, 559, 242, 573 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 243, 560, 255, 571 ], "score": 0.68, "content": "\\lambda _ { u }", "type": "inline_equation" }, { "bbox": [ 255, 559, 276, 573 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 277, 560, 289, 571 ], "score": 0.88, "content": "\\lambda _ { r }", "type": "inline_equation" }, { "bbox": [ 289, 559, 506, 573 ], "score": 1.0, "content": "in the loss function are utilized to assign a different", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 570, 506, 584 ], "spans": [ { "bbox": [ 105, 570, 506, 584 ], "score": 1.0, "content": "learning rate, which can be specified by humans or automatically adjusted during trainingWang et al.", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 581, 505, 594 ], "spans": [ { "bbox": [ 105, 581, 242, 594 ], "score": 1.0, "content": "(2021a; 2022b). We note that the", "type": "text" }, { "bbox": [ 242, 582, 254, 593 ], "score": 0.89, "content": "\\mathcal { L } _ { b }", "type": "inline_equation" }, { "bbox": [ 255, 581, 505, 594 ], "score": 1.0, "content": "term can be further omitted if we apply hard constraint in the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 592, 437, 605 ], "spans": [ { "bbox": [ 105, 592, 437, 605 ], "score": 1.0, "content": "PINN’s design Lu et al. (2021b); Liu et al. (2022a); Sukumar & Srivastava (2022).", "type": "text" } ], "index": 37 } ], "index": 34, "bbox_fs": [ 104, 511, 510, 605 ] }, { "type": "text", "bbox": [ 107, 609, 505, 676 ], "lines": [ { "bbox": [ 106, 610, 505, 621 ], "spans": [ { "bbox": [ 106, 610, 505, 621 ], "score": 1.0, "content": "As demonstrated in Wang et al. (2022a), the vanilla PINN may violate the principle of temporal", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 620, 505, 633 ], "spans": [ { "bbox": [ 105, 620, 505, 633 ], "score": 1.0, "content": "causality, as the residual loss at the later time may be minimized even if the predictions at previous", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 631, 506, 644 ], "spans": [ { "bbox": [ 106, 631, 506, 644 ], "score": 1.0, "content": "times are incorrect. Figure 1 demonstrates the training result for solving the Allen-Chan equation,", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 642, 505, 655 ], "spans": [ { "bbox": [ 106, 642, 399, 655 ], "score": 1.0, "content": "confirming this phenomenon. For conventional PINN, the residual loss", "type": "text" }, { "bbox": [ 400, 643, 412, 653 ], "score": 0.89, "content": "\\mathcal { L } _ { r }", "type": "inline_equation" }, { "bbox": [ 413, 642, 505, 655 ], "score": 1.0, "content": "is quite large near the", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 653, 506, 666 ], "spans": [ { "bbox": [ 105, 653, 506, 666 ], "score": 1.0, "content": "initial state and decays quickly to a small value when the learned solution is incorrect. Comparatively,", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 664, 503, 677 ], "spans": [ { "bbox": [ 106, 664, 280, 677 ], "score": 1.0, "content": "our method’s residual remains small for all", "type": "text" }, { "bbox": [ 280, 664, 317, 676 ], "score": 0.92, "content": "t \\in [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 317, 664, 503, 677 ], "score": 1.0, "content": "and captures the solution with high precision.", "type": "text" } ], "index": 43 } ], "index": 40.5, "bbox_fs": [ 105, 610, 506, 677 ] }, { "type": "title", "bbox": [ 107, 689, 347, 700 ], "lines": [ { "bbox": [ 106, 689, 348, 702 ], "spans": [ { "bbox": [ 106, 689, 348, 702 ], "score": 1.0, "content": "3.2 TRANSFER LEARNING ENHANCED DISCRETE PINN", "type": "text" } ], "index": 44 } ], "index": 44 }, { "type": "text", "bbox": [ 107, 709, 504, 732 ], "lines": [ { "bbox": [ 105, 708, 505, 722 ], "spans": [ { "bbox": [ 105, 708, 505, 722 ], "score": 1.0, "content": "Discrete PINN Since the continuous-time PINN violates temporal causality, we shift to numerical", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 457, 733 ], "score": 1.0, "content": "temporal differencing schemes that naturally respect temporal causality. Given a time step", "type": "text" }, { "bbox": [ 457, 721, 470, 731 ], "score": 0.83, "content": "\\Delta t", "type": "inline_equation" }, { "bbox": [ 471, 720, 505, 733 ], "score": 1.0, "content": ", assume", "type": "text" } ], "index": 46 } ], "index": 45.5, "bbox_fs": [ 105, 708, 505, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 119, 80, 497, 306 ], "blocks": [ { "type": "image_body", "bbox": [ 119, 80, 497, 306 ], "group_id": 0, "lines": [ { "bbox": [ 119, 80, 497, 306 ], "spans": [ { "bbox": [ 119, 80, 497, 306 ], "score": 0.972, "type": "image", "image_path": "0d7babde3a700eba8ba3e0322bcdeee9e7155677dc375223f4216307790e97ca.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 119, 80, 497, 155.33333333333331 ], "spans": [], "index": 0 }, { "bbox": [ 119, 155.33333333333331, 497, 230.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 119, 230.66666666666663, 497, 305.99999999999994 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 315, 505, 339 ], "group_id": 0, "lines": [ { "bbox": [ 106, 316, 506, 329 ], "spans": [ { "bbox": [ 106, 316, 506, 329 ], "score": 1.0, "content": "Figure 1: Allen-Cahn equation: (a)reference solution. (b)PINN solution. (c)TL-DPINN solution.", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 326, 481, 341 ], "spans": [ { "bbox": [ 105, 326, 240, 341 ], "score": 1.0, "content": "(d)PINN’s temporal residual loss", "type": "text" }, { "bbox": [ 240, 327, 279, 339 ], "score": 0.93, "content": "\\mathcal { L } _ { r } ( t _ { n } , \\theta )", "type": "inline_equation" }, { "bbox": [ 279, 326, 439, 341 ], "score": 1.0, "content": ". (e)TL-DPINN’s temporal residual loss", "type": "text" }, { "bbox": [ 439, 327, 477, 339 ], "score": 0.93, "content": "\\mathcal { L } _ { r } ( t _ { n } , \\theta )", "type": "inline_equation" }, { "bbox": [ 478, 326, 481, 341 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 106, 359, 504, 383 ], "lines": [ { "bbox": [ 106, 359, 505, 372 ], "spans": [ { "bbox": [ 106, 359, 186, 372 ], "score": 1.0, "content": "we have computed", "type": "text" }, { "bbox": [ 186, 360, 212, 371 ], "score": 0.92, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 212, 359, 329, 372 ], "score": 1.0, "content": "to approximate the solution", "type": "text" }, { "bbox": [ 329, 360, 371, 372 ], "score": 0.93, "content": "u ( n \\Delta t , x )", "type": "inline_equation" }, { "bbox": [ 372, 359, 505, 372 ], "score": 1.0, "content": "to equation 2, then we consider", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 370, 374, 383 ], "spans": [ { "bbox": [ 106, 370, 137, 383 ], "score": 1.0, "content": "finding", "type": "text" }, { "bbox": [ 137, 370, 173, 383 ], "score": 0.93, "content": "u ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 173, 370, 374, 383 ], "score": 1.0, "content": "by the Crank-Nicolson time differencing scheme:", "type": "text" } ], "index": 6 } ], "index": 5.5 }, { "type": "interline_equation", "bbox": [ 211, 387, 400, 416 ], "lines": [ { "bbox": [ 211, 387, 400, 416 ], "spans": [ { "bbox": [ 211, 387, 400, 416 ], "score": 0.94, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\ N \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n } ( x ) } { 2 } \\right] .", "type": "interline_equation", "image_path": "f9c0159728fd55fe213e80bab8603c137129b05495328bde588876c5efc774ff.jpg" } ] } ], "index": 7.5, "virtual_lines": [ { "bbox": [ 211, 387, 400, 401.5 ], "spans": [], "index": 7 }, { "bbox": [ 211, 401.5, 400, 416.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 108, 419, 505, 453 ], "lines": [ { "bbox": [ 105, 419, 505, 433 ], "spans": [ { "bbox": [ 105, 419, 505, 433 ], "score": 1.0, "content": "Instead of solving equation 2 in the whole space-temporal domain directly, our goal is to solve", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 430, 505, 444 ], "spans": [ { "bbox": [ 106, 430, 351, 444 ], "score": 1.0, "content": "equation 4 from one step to the next in the space domain:", "type": "text" }, { "bbox": [ 351, 430, 505, 443 ], "score": 0.91, "content": "u _ { 0 } ( x ) \\mapsto u ^ { 1 } ( \\bar { x ) } \\mapsto \\bar { \\cdot \\cdot \\cdot } \\mapsto u ^ { n } ( x ) \\mapsto", "type": "inline_equation" } ], "index": 10 }, { "bbox": [ 107, 439, 480, 456 ], "spans": [ { "bbox": [ 107, 441, 171, 454 ], "score": 0.91, "content": "u ^ { \\bar { n } + 1 } ( x ) \\mapsto \\cdot \\cdot \\cdot", "type": "inline_equation" }, { "bbox": [ 171, 439, 480, 456 ], "score": 1.0, "content": ", so that the evolutionary dynamics can be captured over a long time horizon.", "type": "text" } ], "index": 11 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 458, 505, 504 ], "lines": [ { "bbox": [ 105, 459, 505, 470 ], "spans": [ { "bbox": [ 105, 459, 505, 470 ], "score": 1.0, "content": "Next, we apply PINN to solve equation 4. It is also called discrete PINN in Raissi et al. (2019) when", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 469, 505, 483 ], "spans": [ { "bbox": [ 105, 469, 505, 483 ], "score": 1.0, "content": "the Crank-Nicolson scheme is replaced by implicit high-order Runge-Kutta schemes. Assuming", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 480, 505, 493 ], "spans": [ { "bbox": [ 105, 480, 253, 493 ], "score": 1.0, "content": "we have obtained a neural network", "type": "text" }, { "bbox": [ 253, 481, 282, 493 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 282, 480, 349, 493 ], "score": 1.0, "content": "to approximate", "type": "text" }, { "bbox": [ 349, 481, 391, 493 ], "score": 0.92, "content": "u ( n \\Delta t , x )", "type": "inline_equation" }, { "bbox": [ 392, 480, 505, 493 ], "score": 1.0, "content": "in equation 2, we compute", "type": "text" } ], "index": 14 }, { "bbox": [ 107, 491, 395, 504 ], "spans": [ { "bbox": [ 107, 492, 145, 504 ], "score": 0.93, "content": "u _ { \\theta ^ { n + 1 } } ( x )", "type": "inline_equation" }, { "bbox": [ 145, 491, 240, 504 ], "score": 1.0, "content": "by finding another new", "type": "text" }, { "bbox": [ 241, 492, 262, 502 ], "score": 0.9, "content": "\\theta ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 262, 491, 395, 504 ], "score": 1.0, "content": "that minimize the loss functions", "type": "text" } ], "index": 15 } ], "index": 13.5 }, { "type": "interline_equation", "bbox": [ 137, 508, 474, 579 ], "lines": [ { "bbox": [ 137, 508, 474, 579 ], "spans": [ { "bbox": [ 137, 508, 474, 579 ], "score": 0.93, "content": "\\begin{array} { r l r } { { \\mathcal { L } ^ { n + 1 } ( \\theta ^ { n + 1 } ) = \\frac { \\lambda _ { b } } { N _ { b } } \\sum _ { i = 1 } ^ { N _ { b } } | u _ { \\theta ^ { n + 1 } } ( x _ { b } ^ { i } ) - g ( x _ { b } ^ { i } ) | ^ { 2 } } } \\\\ & { } & { \\quad + \\frac { \\lambda _ { r } } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } | \\frac { u _ { \\theta ^ { n + 1 } } ( x _ { r } ^ { i } ) - u _ { \\theta ^ { n } } ( x _ { r } ^ { i } ) } { \\Delta t } - \\mathcal { N } [ \\frac { u _ { \\theta ^ { n + 1 } } ( x _ { r } ^ { i } ) + u _ { \\theta ^ { n } } ( x _ { r } ^ { i } ) } { 2 } ] | ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "eee27ab40451a23ee464b607f3c2bb65a9cd480c44d196891b34f094546eaf81.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 137, 508, 474, 531.6666666666666 ], "spans": [], "index": 16 }, { "bbox": [ 137, 531.6666666666666, 474, 555.3333333333333 ], "spans": [], "index": 17 }, { "bbox": [ 137, 555.3333333333333, 474, 578.9999999999999 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 105, 583, 504, 595 ], "lines": [ { "bbox": [ 105, 581, 506, 597 ], "spans": [ { "bbox": [ 105, 581, 197, 597 ], "score": 1.0, "content": "These multiple PINNs", "type": "text" }, { "bbox": [ 198, 583, 227, 595 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 227, 581, 247, 597 ], "score": 1.0, "content": "take", "type": "text" }, { "bbox": [ 248, 585, 254, 593 ], "score": 0.77, "content": "x", "type": "inline_equation" }, { "bbox": [ 254, 581, 506, 597 ], "score": 1.0, "content": "as input and output the solution values at different timestamps.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 106, 598, 505, 665 ], "lines": [ { "bbox": [ 105, 597, 506, 611 ], "spans": [ { "bbox": [ 105, 597, 506, 611 ], "score": 1.0, "content": "Remark 3.1. We remark that there exist alternative options for time differencing beyond the second-", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 610, 506, 621 ], "spans": [ { "bbox": [ 106, 610, 506, 621 ], "score": 1.0, "content": "order Crank-Nicolson scheme. Several implicit Runge-Kutta schemes, including the first-order", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 619, 506, 633 ], "spans": [ { "bbox": [ 105, 619, 506, 633 ], "score": 1.0, "content": "backward Euler scheme and the fourth-order Gauss-Legendre scheme, have been found to be ef-", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 631, 505, 644 ], "spans": [ { "bbox": [ 105, 631, 505, 644 ], "score": 1.0, "content": "fective. The second-order Crank-Nicolson scheme is favored due to its optimal trade-off between", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 642, 506, 655 ], "spans": [ { "bbox": [ 105, 642, 506, 655 ], "score": 1.0, "content": "computational efficiency and numerical accuracy. A comprehensive exposition of these techniques is", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 653, 214, 666 ], "spans": [ { "bbox": [ 105, 653, 214, 666 ], "score": 1.0, "content": "available in Appendix A.2.", "type": "text" } ], "index": 25 } ], "index": 22.5 }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 105, 675, 505, 690 ], "spans": [ { "bbox": [ 105, 675, 505, 690 ], "score": 1.0, "content": "Transfer learning The transfer learning methodology is utilized to expedite the training procedure", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 293, 700 ], "score": 1.0, "content": "between two adjacent PINNs. All the PINNs", "type": "text" }, { "bbox": [ 293, 689, 322, 700 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 322, 688, 505, 700 ], "score": 1.0, "content": "share the same neural network architectures", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 211, 711 ], "score": 1.0, "content": "with different parameters", "type": "text" }, { "bbox": [ 212, 699, 223, 709 ], "score": 0.86, "content": "\\theta ^ { n }", "type": "inline_equation" }, { "bbox": [ 223, 698, 316, 711 ], "score": 1.0, "content": ". 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So the parameters", "type": "text" }, { "bbox": [ 343, 709, 363, 720 ], "score": 0.9, "content": "\\theta ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 364, 708, 506, 722 ], "score": 1.0, "content": "to be trained are very close to the", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 719, 505, 734 ], "spans": [ { "bbox": [ 105, 719, 181, 734 ], "score": 1.0, "content": "trained parameters", "type": "text" }, { "bbox": [ 181, 721, 192, 730 ], "score": 0.85, "content": "\\theta ^ { n }", "type": "inline_equation" }, { "bbox": [ 192, 719, 475, 734 ], "score": 1.0, "content": ". The approach involves freezing a significant portion of the well-trained", "type": "text" }, { "bbox": [ 475, 721, 505, 732 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" } ], "index": 30 } ], "index": 28 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 301, 750, 309, 762 ], "spans": [ { "bbox": [ 301, 750, 309, 762 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 8 } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 119, 80, 497, 306 ], "blocks": [ { "type": "image_body", "bbox": [ 119, 80, 497, 306 ], "group_id": 0, "lines": [ { "bbox": [ 119, 80, 497, 306 ], "spans": [ { "bbox": [ 119, 80, 497, 306 ], "score": 0.972, "type": "image", "image_path": "0d7babde3a700eba8ba3e0322bcdeee9e7155677dc375223f4216307790e97ca.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 119, 80, 497, 155.33333333333331 ], "spans": [], "index": 0 }, { "bbox": [ 119, 155.33333333333331, 497, 230.66666666666663 ], "spans": [], "index": 1 }, { "bbox": [ 119, 230.66666666666663, 497, 305.99999999999994 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 315, 505, 339 ], "group_id": 0, "lines": [ { "bbox": [ 106, 316, 506, 329 ], "spans": [ { "bbox": [ 106, 316, 506, 329 ], "score": 1.0, "content": "Figure 1: Allen-Cahn equation: (a)reference solution. (b)PINN solution. (c)TL-DPINN solution.", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 326, 481, 341 ], "spans": [ { "bbox": [ 105, 326, 240, 341 ], "score": 1.0, "content": "(d)PINN’s temporal residual loss", "type": "text" }, { "bbox": [ 240, 327, 279, 339 ], "score": 0.93, "content": "\\mathcal { L } _ { r } ( t _ { n } , \\theta )", "type": "inline_equation" }, { "bbox": [ 279, 326, 439, 341 ], "score": 1.0, "content": ". (e)TL-DPINN’s temporal residual loss", "type": "text" }, { "bbox": [ 439, 327, 477, 339 ], "score": 0.93, "content": "\\mathcal { L } _ { r } ( t _ { n } , \\theta )", "type": "inline_equation" }, { "bbox": [ 478, 326, 481, 341 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 106, 359, 504, 383 ], "lines": [ { "bbox": [ 106, 359, 505, 372 ], "spans": [ { "bbox": [ 106, 359, 186, 372 ], "score": 1.0, "content": "we have computed", "type": "text" }, { "bbox": [ 186, 360, 212, 371 ], "score": 0.92, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 212, 359, 329, 372 ], "score": 1.0, "content": "to approximate the solution", "type": "text" }, { "bbox": [ 329, 360, 371, 372 ], "score": 0.93, "content": "u ( n \\Delta t , x )", "type": "inline_equation" }, { "bbox": [ 372, 359, 505, 372 ], "score": 1.0, "content": "to equation 2, then we consider", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 370, 374, 383 ], "spans": [ { "bbox": [ 106, 370, 137, 383 ], "score": 1.0, "content": "finding", "type": "text" }, { "bbox": [ 137, 370, 173, 383 ], "score": 0.93, "content": "u ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 173, 370, 374, 383 ], "score": 1.0, "content": "by the Crank-Nicolson time differencing scheme:", "type": "text" } ], "index": 6 } ], "index": 5.5, "bbox_fs": [ 106, 359, 505, 383 ] }, { "type": "interline_equation", "bbox": [ 211, 387, 400, 416 ], "lines": [ { "bbox": [ 211, 387, 400, 416 ], "spans": [ { "bbox": [ 211, 387, 400, 416 ], "score": 0.94, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\ N \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n } ( x ) } { 2 } \\right] .", "type": "interline_equation", "image_path": "f9c0159728fd55fe213e80bab8603c137129b05495328bde588876c5efc774ff.jpg" } ] } ], "index": 7.5, "virtual_lines": [ { "bbox": [ 211, 387, 400, 401.5 ], "spans": [], "index": 7 }, { "bbox": [ 211, 401.5, 400, 416.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 108, 419, 505, 453 ], "lines": [ { "bbox": [ 105, 419, 505, 433 ], "spans": [ { "bbox": [ 105, 419, 505, 433 ], "score": 1.0, "content": "Instead of solving equation 2 in the whole space-temporal domain directly, our goal is to solve", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 430, 505, 444 ], "spans": [ { "bbox": [ 106, 430, 351, 444 ], "score": 1.0, "content": "equation 4 from one step to the next in the space domain:", "type": "text" }, { "bbox": [ 351, 430, 505, 443 ], "score": 0.91, "content": "u _ { 0 } ( x ) \\mapsto u ^ { 1 } ( \\bar { x ) } \\mapsto \\bar { \\cdot \\cdot \\cdot } \\mapsto u ^ { n } ( x ) \\mapsto", "type": "inline_equation" } ], "index": 10 }, { "bbox": [ 107, 439, 480, 456 ], "spans": [ { "bbox": [ 107, 441, 171, 454 ], "score": 0.91, "content": "u ^ { \\bar { n } + 1 } ( x ) \\mapsto \\cdot \\cdot \\cdot", "type": "inline_equation" }, { "bbox": [ 171, 439, 480, 456 ], "score": 1.0, "content": ", so that the evolutionary dynamics can be captured over a long time horizon.", "type": "text" } ], "index": 11 } ], "index": 10, "bbox_fs": [ 105, 419, 505, 456 ] }, { "type": "text", "bbox": [ 107, 458, 505, 504 ], "lines": [ { "bbox": [ 105, 459, 505, 470 ], "spans": [ { "bbox": [ 105, 459, 505, 470 ], "score": 1.0, "content": "Next, we apply PINN to solve equation 4. It is also called discrete PINN in Raissi et al. (2019) when", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 469, 505, 483 ], "spans": [ { "bbox": [ 105, 469, 505, 483 ], "score": 1.0, "content": "the Crank-Nicolson scheme is replaced by implicit high-order Runge-Kutta schemes. Assuming", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 480, 505, 493 ], "spans": [ { "bbox": [ 105, 480, 253, 493 ], "score": 1.0, "content": "we have obtained a neural network", "type": "text" }, { "bbox": [ 253, 481, 282, 493 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 282, 480, 349, 493 ], "score": 1.0, "content": "to approximate", "type": "text" }, { "bbox": [ 349, 481, 391, 493 ], "score": 0.92, "content": "u ( n \\Delta t , x )", "type": "inline_equation" }, { "bbox": [ 392, 480, 505, 493 ], "score": 1.0, "content": "in equation 2, we compute", "type": "text" } ], "index": 14 }, { "bbox": [ 107, 491, 395, 504 ], "spans": [ { "bbox": [ 107, 492, 145, 504 ], "score": 0.93, "content": "u _ { \\theta ^ { n + 1 } } ( x )", "type": "inline_equation" }, { "bbox": [ 145, 491, 240, 504 ], "score": 1.0, "content": "by finding another new", "type": "text" }, { "bbox": [ 241, 492, 262, 502 ], "score": 0.9, "content": "\\theta ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 262, 491, 395, 504 ], "score": 1.0, "content": "that minimize the loss functions", "type": "text" } ], "index": 15 } ], "index": 13.5, "bbox_fs": [ 105, 459, 505, 504 ] }, { "type": "interline_equation", "bbox": [ 137, 508, 474, 579 ], "lines": [ { "bbox": [ 137, 508, 474, 579 ], "spans": [ { "bbox": [ 137, 508, 474, 579 ], "score": 0.93, "content": "\\begin{array} { r l r } { { \\mathcal { L } ^ { n + 1 } ( \\theta ^ { n + 1 } ) = \\frac { \\lambda _ { b } } { N _ { b } } \\sum _ { i = 1 } ^ { N _ { b } } | u _ { \\theta ^ { n + 1 } } ( x _ { b } ^ { i } ) - g ( x _ { b } ^ { i } ) | ^ { 2 } } } \\\\ & { } & { \\quad + \\frac { \\lambda _ { r } } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } | \\frac { u _ { \\theta ^ { n + 1 } } ( x _ { r } ^ { i } ) - u _ { \\theta ^ { n } } ( x _ { r } ^ { i } ) } { \\Delta t } - \\mathcal { N } [ \\frac { u _ { \\theta ^ { n + 1 } } ( x _ { r } ^ { i } ) + u _ { \\theta ^ { n } } ( x _ { r } ^ { i } ) } { 2 } ] | ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "eee27ab40451a23ee464b607f3c2bb65a9cd480c44d196891b34f094546eaf81.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 137, 508, 474, 531.6666666666666 ], "spans": [], "index": 16 }, { "bbox": [ 137, 531.6666666666666, 474, 555.3333333333333 ], "spans": [], "index": 17 }, { "bbox": [ 137, 555.3333333333333, 474, 578.9999999999999 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 105, 583, 504, 595 ], "lines": [ { "bbox": [ 105, 581, 506, 597 ], "spans": [ { "bbox": [ 105, 581, 197, 597 ], "score": 1.0, "content": "These multiple PINNs", "type": "text" }, { "bbox": [ 198, 583, 227, 595 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 227, 581, 247, 597 ], "score": 1.0, "content": "take", "type": "text" }, { "bbox": [ 248, 585, 254, 593 ], "score": 0.77, "content": "x", "type": "inline_equation" }, { "bbox": [ 254, 581, 506, 597 ], "score": 1.0, "content": "as input and output the solution values at different timestamps.", "type": "text" } ], "index": 19 } ], "index": 19, "bbox_fs": [ 105, 581, 506, 597 ] }, { "type": "text", "bbox": [ 106, 598, 505, 665 ], "lines": [ { "bbox": [ 105, 597, 506, 611 ], "spans": [ { "bbox": [ 105, 597, 506, 611 ], "score": 1.0, "content": "Remark 3.1. We remark that there exist alternative options for time differencing beyond the second-", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 610, 506, 621 ], "spans": [ { "bbox": [ 106, 610, 506, 621 ], "score": 1.0, "content": "order Crank-Nicolson scheme. Several implicit Runge-Kutta schemes, including the first-order", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 619, 506, 633 ], "spans": [ { "bbox": [ 105, 619, 506, 633 ], "score": 1.0, "content": "backward Euler scheme and the fourth-order Gauss-Legendre scheme, have been found to be ef-", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 631, 505, 644 ], "spans": [ { "bbox": [ 105, 631, 505, 644 ], "score": 1.0, "content": "fective. The second-order Crank-Nicolson scheme is favored due to its optimal trade-off between", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 642, 506, 655 ], "spans": [ { "bbox": [ 105, 642, 506, 655 ], "score": 1.0, "content": "computational efficiency and numerical accuracy. A comprehensive exposition of these techniques is", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 653, 214, 666 ], "spans": [ { "bbox": [ 105, 653, 214, 666 ], "score": 1.0, "content": "available in Appendix A.2.", "type": "text" } ], "index": 25 } ], "index": 22.5, "bbox_fs": [ 105, 597, 506, 666 ] }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 105, 675, 505, 690 ], "spans": [ { "bbox": [ 105, 675, 505, 690 ], "score": 1.0, "content": "Transfer learning The transfer learning methodology is utilized to expedite the training procedure", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 293, 700 ], "score": 1.0, "content": "between two adjacent PINNs. All the PINNs", "type": "text" }, { "bbox": [ 293, 689, 322, 700 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 322, 688, 505, 700 ], "score": 1.0, "content": "share the same neural network architectures", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 698, 505, 711 ], "spans": [ { "bbox": [ 105, 698, 211, 711 ], "score": 1.0, "content": "with different parameters", "type": "text" }, { "bbox": [ 212, 699, 223, 709 ], "score": 0.86, "content": "\\theta ^ { n }", "type": "inline_equation" }, { "bbox": [ 223, 698, 316, 711 ], "score": 1.0, "content": ". For a small time step", "type": "text" }, { "bbox": [ 317, 700, 329, 709 ], "score": 0.85, "content": "\\Delta t", "type": "inline_equation" }, { "bbox": [ 329, 698, 505, 711 ], "score": 1.0, "content": ", there are little difference between the two", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 708, 506, 722 ], "spans": [ { "bbox": [ 105, 708, 173, 722 ], "score": 1.0, "content": "adjacent PINNs", "type": "text" }, { "bbox": [ 174, 709, 203, 721 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 204, 708, 222, 722 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 223, 710, 261, 722 ], "score": 0.92, "content": "u _ { \\theta ^ { n + 1 } } ( x )", "type": "inline_equation" }, { "bbox": [ 261, 708, 342, 722 ], "score": 1.0, "content": ". So the parameters", "type": "text" }, { "bbox": [ 343, 709, 363, 720 ], "score": 0.9, "content": "\\theta ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 364, 708, 506, 722 ], "score": 1.0, "content": "to be trained are very close to the", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 719, 505, 734 ], "spans": [ { "bbox": [ 105, 719, 181, 734 ], "score": 1.0, "content": "trained parameters", "type": "text" }, { "bbox": [ 181, 721, 192, 730 ], "score": 0.85, "content": "\\theta ^ { n }", "type": "inline_equation" }, { "bbox": [ 192, 719, 475, 734 ], "score": 1.0, "content": ". The approach involves freezing a significant portion of the well-trained", "type": "text" }, { "bbox": [ 475, 721, 505, 732 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" } ], "index": 30 } ], "index": 28, "bbox_fs": [ 105, 675, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 82, 505, 105 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "and solely updating the weights in the last hidden layer through the application of a comparable", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 94, 279, 106 ], "spans": [ { "bbox": [ 105, 94, 279, 106 ], "score": 1.0, "content": "physics-informed loss function equation 5.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 107, 110, 504, 144 ], "lines": [ { "bbox": [ 105, 109, 505, 123 ], "spans": [ { "bbox": [ 105, 109, 342, 123 ], "score": 1.0, "content": "To be more precise, we first approximate the initial condition", "type": "text" }, { "bbox": [ 342, 110, 367, 123 ], "score": 0.92, "content": "u _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 367, 109, 453, 123 ], "score": 1.0, "content": "by the neural network", "type": "text" }, { "bbox": [ 454, 110, 482, 122 ], "score": 0.92, "content": "u _ { \\theta ^ { 0 } } ( x )", "type": "inline_equation" }, { "bbox": [ 483, 109, 505, 123 ], "score": 1.0, "content": ", then", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 121, 506, 134 ], "spans": [ { "bbox": [ 105, 121, 129, 134 ], "score": 1.0, "content": "learn", "type": "text" }, { "bbox": [ 129, 121, 205, 133 ], "score": 0.48, "content": "u _ { \\theta ^ { 1 } } ( \\bar { x } ) , u _ { \\theta ^ { 2 } } ( x ) , . . .", "type": "inline_equation" }, { "bbox": [ 205, 121, 506, 134 ], "score": 1.0, "content": "sequentially by transfer learning. The general structure of our TL-DPINN", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 132, 255, 144 ], "spans": [ { "bbox": [ 106, 132, 255, 144 ], "score": 1.0, "content": "method is illustrated in Algorithm 1.", "type": "text" } ], "index": 4 } ], "index": 3 }, { "type": "title", "bbox": [ 106, 159, 365, 171 ], "lines": [ { "bbox": [ 106, 158, 366, 173 ], "spans": [ { "bbox": [ 106, 158, 366, 173 ], "score": 1.0, "content": "Algorithm 1: The training procedure of our TL-DPINN method", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 97, 172, 486, 363 ], "lines": [ { "bbox": [ 105, 172, 414, 185 ], "spans": [ { "bbox": [ 105, 172, 352, 185 ], "score": 1.0, "content": "Input :Target evolutionary PDE equation 2; initial network", "type": "text" }, { "bbox": [ 352, 175, 363, 184 ], "score": 0.79, "content": "u _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 364, 172, 405, 185 ], "score": 1.0, "content": "; end time", "type": "text" }, { "bbox": [ 405, 173, 414, 183 ], "score": 0.64, "content": "T", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 105, 182, 345, 198 ], "spans": [ { "bbox": [ 105, 182, 228, 198 ], "score": 1.0, "content": "Output :The predicted model", "type": "text" }, { "bbox": [ 228, 184, 258, 196 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 258, 182, 333, 198 ], "score": 1.0, "content": "at each timestamp", "type": "text" }, { "bbox": [ 334, 185, 344, 195 ], "score": 0.86, "content": "t _ { n }", "type": "inline_equation" }, { "bbox": [ 344, 182, 345, 198 ], "score": 0.0, "content": "", "type": "text" } ], "index": 7 }, { "bbox": [ 98, 194, 462, 208 ], "spans": [ { "bbox": [ 98, 194, 286, 208 ], "score": 1.0, "content": "1 Set hyper-parameters: timestamps number", "type": "text" }, { "bbox": [ 286, 195, 298, 206 ], "score": 0.89, "content": "N _ { t }", "type": "inline_equation" }, { "bbox": [ 299, 194, 462, 208 ], "score": 1.0, "content": ", number of maximum training iterations", "type": "text" } ], "index": 8 }, { "bbox": [ 111, 205, 288, 218 ], "spans": [ { "bbox": [ 111, 206, 145, 217 ], "score": 0.8, "content": "M _ { 0 } , M _ { 1 }", "type": "inline_equation" }, { "bbox": [ 145, 205, 201, 218 ], "score": 1.0, "content": ", learning rate", "type": "text" }, { "bbox": [ 202, 207, 208, 217 ], "score": 0.78, "content": "\\eta", "type": "inline_equation" }, { "bbox": [ 209, 205, 275, 218 ], "score": 1.0, "content": ", threshold value", "type": "text" }, { "bbox": [ 275, 208, 281, 216 ], "score": 0.75, "content": "\\epsilon", "type": "inline_equation" }, { "bbox": [ 282, 205, 288, 218 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 9 }, { "bbox": [ 97, 215, 246, 230 ], "spans": [ { "bbox": [ 97, 215, 171, 230 ], "score": 1.0, "content": "2 Step (a): learn", "type": "text" }, { "bbox": [ 172, 217, 200, 228 ], "score": 0.91, "content": "u _ { \\theta ^ { 0 } } ( x )", "type": "inline_equation" }, { "bbox": [ 200, 215, 246, 230 ], "score": 1.0, "content": "by PINN ;", "type": "text" } ], "index": 10 }, { "bbox": [ 97, 226, 200, 240 ], "spans": [ { "bbox": [ 97, 226, 121, 240 ], "score": 1.0, "content": "3 for", "type": "text" }, { "bbox": [ 122, 228, 185, 239 ], "score": 0.89, "content": "i = 1 , 2 , . . . , M _ { 0 }", "type": "inline_equation" }, { "bbox": [ 185, 226, 200, 240 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 11 }, { "bbox": [ 100, 238, 302, 252 ], "spans": [ { "bbox": [ 100, 243, 103, 248 ], "score": 1.0, "content": "4", "type": "text" }, { "bbox": [ 120, 238, 268, 252 ], "score": 1.0, "content": "Compute the mean square error loss", "type": "text" }, { "bbox": [ 268, 238, 297, 251 ], "score": 0.92, "content": "{ \\mathcal { L } } ^ { 0 } ( \\theta ^ { 0 } )", "type": "inline_equation" }, { "bbox": [ 298, 238, 302, 252 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 12 }, { "bbox": [ 99, 248, 408, 266 ], "spans": [ { "bbox": [ 99, 255, 104, 261 ], "score": 1.0, "content": "5", "type": "text" }, { "bbox": [ 118, 248, 210, 266 ], "score": 1.0, "content": "Update the parameter", "type": "text" }, { "bbox": [ 210, 251, 221, 261 ], "score": 0.86, "content": "\\theta ^ { 0 }", "type": "inline_equation" }, { "bbox": [ 221, 248, 304, 266 ], "score": 1.0, "content": "via gradient descent", "type": "text" }, { "bbox": [ 304, 250, 402, 264 ], "score": 0.91, "content": "\\theta _ { i + 1 } ^ { 0 } = \\theta _ { i } ^ { 0 } - \\eta \\nabla \\mathcal { L } ^ { 0 } ( \\theta _ { i } ^ { 0 } )", "type": "inline_equation" }, { "bbox": [ 402, 248, 408, 266 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 13 }, { "bbox": [ 96, 265, 485, 281 ], "spans": [ { "bbox": [ 96, 265, 178, 281 ], "score": 1.0, "content": "6 Step (b): denote", "type": "text" }, { "bbox": [ 178, 266, 218, 280 ], "score": 0.92, "content": "\\theta _ { * } ^ { 0 } = \\theta _ { M _ { 0 } } ^ { 0 }", "type": "inline_equation" }, { "bbox": [ 219, 265, 259, 281 ], "score": 1.0, "content": "and learn", "type": "text" }, { "bbox": [ 260, 267, 346, 279 ], "score": 0.92, "content": "u _ { \\theta ^ { 1 } } ( x ) , . . . , u _ { \\theta ^ { n } } ( x ) , . . .", "type": "inline_equation" }, { "bbox": [ 347, 265, 485, 281 ], "score": 1.0, "content": "sequentially by transfer learning ;", "type": "text" } ], "index": 14 }, { "bbox": [ 98, 278, 226, 292 ], "spans": [ { "bbox": [ 98, 282, 104, 289 ], "score": 0.44, "content": "^ { 7 }", "type": "inline_equation" }, { "bbox": [ 104, 278, 121, 292 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 280, 211, 291 ], "score": 0.88, "content": "n = 0 , 1 , 2 , . . . , N _ { t } - 1", "type": "inline_equation" }, { "bbox": [ 211, 278, 226, 292 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 15 }, { "bbox": [ 98, 290, 216, 302 ], "spans": [ { "bbox": [ 98, 293, 105, 301 ], "score": 1.0, "content": "8", "type": "text" }, { "bbox": [ 120, 290, 137, 302 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 137, 291, 200, 302 ], "score": 0.85, "content": "i = 1 , 2 , . . . , M _ { 1 }", "type": "inline_equation" }, { "bbox": [ 201, 290, 216, 302 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 16 }, { "bbox": [ 96, 299, 470, 335 ], "spans": [ { "bbox": [ 96, 305, 105, 328 ], "score": 1.0, "content": "9 10", "type": "text" }, { "bbox": [ 132, 299, 194, 335 ], "score": 1.0, "content": "Compute loss Update the pa", "type": "text" }, { "bbox": [ 194, 301, 246, 315 ], "score": 0.93, "content": "\\mathcal { L } _ { i } ^ { n + 1 } ( \\boldsymbol { \\theta } _ { i } ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 226, 315, 247, 325 ], "score": 0.89, "content": "\\theta ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 247, 299, 330, 335 ], "score": 1.0, "content": "by equation 5 ;via gradient descent", "type": "text" }, { "bbox": [ 331, 314, 464, 329 ], "score": 0.92, "content": "\\theta _ { i + 1 } ^ { n + 1 } = \\theta _ { i } ^ { n + 1 } - \\eta \\nabla \\mathcal { L } ^ { n + 1 } ( \\theta _ { i } ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 464, 299, 470, 335 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 17 }, { "bbox": [ 96, 324, 309, 346 ], "spans": [ { "bbox": [ 96, 331, 105, 340 ], "score": 1.0, "content": "11", "type": "text" }, { "bbox": [ 133, 324, 146, 346 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 146, 327, 284, 341 ], "score": 0.89, "content": "| { \\mathcal { L } } ^ { n + 1 } ( \\theta _ { i + 1 } ^ { n + 1 } ) - { \\mathcal { L } } ^ { n + 1 } ( \\theta _ { i } ^ { n + 1 } ) | < \\epsilon", "type": "inline_equation" }, { "bbox": [ 284, 324, 309, 346 ], "score": 1.0, "content": "then", "type": "text" } ], "index": 18 }, { "bbox": [ 96, 338, 291, 356 ], "spans": [ { "bbox": [ 96, 344, 106, 353 ], "score": 1.0, "content": "12", "type": "text" }, { "bbox": [ 149, 338, 182, 356 ], "score": 1.0, "content": "denote", "type": "text" }, { "bbox": [ 182, 341, 237, 353 ], "score": 0.91, "content": "\\dot { \\theta } _ { * } ^ { n + 1 } = \\theta _ { i } ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 237, 338, 291, 356 ], "score": 1.0, "content": "and break ;", "type": "text" } ], "index": 19 } ], "index": 12.5 }, { "type": "text", "bbox": [ 96, 364, 364, 377 ], "lines": [ { "bbox": [ 93, 361, 361, 380 ], "spans": [ { "bbox": [ 93, 361, 305, 380 ], "score": 1.0, "content": "13 Return the optimized neural network parameters", "type": "text" }, { "bbox": [ 305, 364, 361, 377 ], "score": 0.91, "content": "\\theta _ { * } ^ { 1 } , \\theta _ { * } ^ { 2 } , . . . , \\theta _ { * } ^ { N _ { t } }", "type": "inline_equation" } ], "index": 20 } ], "index": 20 }, { "type": "title", "bbox": [ 107, 411, 242, 424 ], "lines": [ { "bbox": [ 105, 410, 243, 425 ], "spans": [ { "bbox": [ 105, 410, 243, 425 ], "score": 1.0, "content": "4 THEORETICAL RESULT", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 436, 504, 459 ], "lines": [ { "bbox": [ 105, 436, 505, 449 ], "spans": [ { "bbox": [ 105, 436, 505, 449 ], "score": 1.0, "content": "In this section, we analyze the TL-DPINN method and give an error estimate result to approximate", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 447, 464, 460 ], "spans": [ { "bbox": [ 106, 447, 464, 460 ], "score": 1.0, "content": "the evolutionary differential equation 2. We have two reasonable assumptions as follows.", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "text", "bbox": [ 106, 461, 506, 514 ], "lines": [ { "bbox": [ 105, 461, 506, 475 ], "spans": [ { "bbox": [ 105, 461, 362, 475 ], "score": 1.0, "content": "Assumption 4.1. The equation equation 2 is dissipative, i.e.", "type": "text" }, { "bbox": [ 362, 461, 444, 475 ], "score": 0.91, "content": "\\textstyle \\int _ { \\Omega } u \\cdot { \\mathcal { N } } ( u ) d x \\leq 0", "type": "inline_equation" }, { "bbox": [ 444, 461, 474, 475 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 475, 462, 502, 474 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 503, 461, 506, 475 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 474, 506, 488 ], "spans": [ { "bbox": [ 105, 474, 237, 488 ], "score": 1.0, "content": "Moreover, if N is nonlinear, then", "type": "text" }, { "bbox": [ 237, 474, 393, 488 ], "score": 0.91, "content": "\\begin{array} { r } { \\int _ { \\Omega } ( u _ { 1 } - u _ { 2 } ) \\cdot ( \\mathcal { N } ( u _ { 1 } ) - \\mathcal { N } ( u _ { 2 } ) ) \\ddot { d x } \\le 0 } \\end{array}", "type": "inline_equation" }, { "bbox": [ 393, 474, 419, 488 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 420, 475, 452, 487 ], "score": 0.92, "content": "u _ { 1 } ( t , x )", "type": "inline_equation" }, { "bbox": [ 452, 474, 470, 488 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 470, 475, 502, 487 ], "score": 0.93, "content": "u _ { 2 } ( t , x )", "type": "inline_equation" }, { "bbox": [ 503, 474, 506, 488 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 490, 505, 504 ], "spans": [ { "bbox": [ 105, 490, 237, 504 ], "score": 1.0, "content": "Assumption 4.2. The solution", "type": "text" }, { "bbox": [ 237, 491, 265, 503 ], "score": 0.92, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 266, 490, 462, 504 ], "score": 1.0, "content": "to equation 2 and the neural network solution", "type": "text" }, { "bbox": [ 463, 491, 492, 503 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 493, 490, 505, 504 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 502, 424, 515 ], "spans": [ { "bbox": [ 106, 502, 424, 515 ], "score": 1.0, "content": "equation 5 are all smooth and bounded, as well as their high order derivatives.", "type": "text" } ], "index": 27 } ], "index": 25.5 }, { "type": "text", "bbox": [ 106, 522, 506, 574 ], "lines": [ { "bbox": [ 105, 521, 505, 535 ], "spans": [ { "bbox": [ 105, 521, 468, 535 ], "score": 1.0, "content": "The first assumption is to guarantee that the solution is not increasing over time. Consider the", "type": "text" }, { "bbox": [ 469, 522, 481, 533 ], "score": 0.86, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 481, 521, 505, 535 ], "score": 1.0, "content": "norm", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 530, 505, 553 ], "spans": [ { "bbox": [ 107, 533, 216, 548 ], "score": 0.92, "content": "\\begin{array} { r } { \\left\\| u ( t , \\cdot ) \\right\\| ^ { 2 } = \\bar { \\int _ { \\Omega } u } ( t , x ) ^ { 2 } \\bar { d } x } \\end{array}", "type": "inline_equation" }, { "bbox": [ 217, 530, 331, 553 ], "score": 1.0, "content": ", we multiply equation 2 by", "type": "text" }, { "bbox": [ 331, 537, 338, 545 ], "score": 0.7, "content": "u", "type": "inline_equation" }, { "bbox": [ 339, 530, 406, 553 ], "score": 1.0, "content": "and integrate in", "type": "text" }, { "bbox": [ 406, 538, 413, 545 ], "score": 0.73, "content": "x", "type": "inline_equation" }, { "bbox": [ 414, 530, 441, 553 ], "score": 1.0, "content": "to get", "type": "text" }, { "bbox": [ 441, 533, 505, 548 ], "score": 0.9, "content": "\\begin{array} { r l } { \\frac { 1 } { 2 } \\frac { d } { d t } \\left\\| u \\right\\| ^ { 2 } ( t ) = } & { { } } \\end{array}", "type": "inline_equation" } ], "index": 29 }, { "bbox": [ 106, 546, 508, 563 ], "spans": [ { "bbox": [ 106, 549, 258, 561 ], "score": 0.56, "content": "\\begin{array} { r } { \\int _ { \\Omega } u \\cdot \\mathcal { N } u d x \\leq 0 , \\mathrm { s o } \\left\\| u ( t , \\cdot ) \\right\\| \\leq \\left\\| u _ { 0 } \\right\\| } \\end{array}", "type": "inline_equation" }, { "bbox": [ 258, 546, 283, 563 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 283, 548, 306, 558 ], "score": 0.87, "content": "t > 0", "type": "inline_equation" }, { "bbox": [ 307, 546, 450, 563 ], "score": 1.0, "content": ". For the simplest Heat equation with", "type": "text" }, { "bbox": [ 450, 548, 502, 560 ], "score": 0.9, "content": "\\mathcal { N } ( u ) = u _ { x x }", "type": "inline_equation" }, { "bbox": [ 503, 546, 508, 563 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 30 }, { "bbox": [ 104, 559, 457, 575 ], "spans": [ { "bbox": [ 104, 559, 198, 575 ], "score": 1.0, "content": "it is easy to verify that", "type": "text" }, { "bbox": [ 198, 560, 344, 574 ], "score": 0.94, "content": "\\begin{array} { r } { \\int _ { \\Omega } u \\cdot \\mathcal { N } ( u ) d x = - \\int _ { \\Omega } | u _ { x } | ^ { 2 } d x \\leq 0 } \\end{array}", "type": "inline_equation" }, { "bbox": [ 344, 559, 457, 575 ], "score": 1.0, "content": ", satisfying Assumption 4.1.", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 107, 577, 505, 600 ], "lines": [ { "bbox": [ 105, 577, 505, 590 ], "spans": [ { "bbox": [ 105, 577, 505, 590 ], "score": 1.0, "content": "The second assumption can be verified by the standard regularity estimate result of PDEs Evans", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 588, 263, 600 ], "spans": [ { "bbox": [ 106, 588, 263, 600 ], "score": 1.0, "content": "(2022), and we omit it here for brevity.", "type": "text" } ], "index": 33 } ], "index": 32.5 }, { "type": "text", "bbox": [ 105, 605, 504, 628 ], "lines": [ { "bbox": [ 105, 603, 505, 619 ], "spans": [ { "bbox": [ 105, 603, 182, 619 ], "score": 1.0, "content": "Denote the symbol", "type": "text" }, { "bbox": [ 183, 606, 214, 615 ], "score": 0.91, "content": "\\tau = \\Delta t", "type": "inline_equation" }, { "bbox": [ 215, 603, 232, 619 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 232, 606, 267, 617 ], "score": 0.91, "content": "t _ { n } = n \\tau", "type": "inline_equation" }, { "bbox": [ 267, 603, 505, 619 ], "score": 1.0, "content": ", we show that the error can be strictly controlled by the time", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 616, 395, 629 ], "spans": [ { "bbox": [ 105, 616, 124, 629 ], "score": 1.0, "content": "step", "type": "text" }, { "bbox": [ 125, 619, 132, 627 ], "score": 0.73, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 132, 616, 225, 629 ], "score": 1.0, "content": ", the training loss value", "type": "text" }, { "bbox": [ 226, 617, 239, 626 ], "score": 0.88, "content": "{ \\mathcal { L } } ^ { n }", "type": "inline_equation" }, { "bbox": [ 239, 616, 378, 629 ], "score": 1.0, "content": "and the collocation points number", "type": "text" }, { "bbox": [ 379, 617, 392, 627 ], "score": 0.88, "content": "N _ { r }", "type": "inline_equation" }, { "bbox": [ 392, 616, 395, 629 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 35 } ], "index": 34.5 }, { "type": "text", "bbox": [ 107, 631, 505, 665 ], "lines": [ { "bbox": [ 105, 630, 505, 644 ], "spans": [ { "bbox": [ 105, 630, 505, 644 ], "score": 1.0, "content": "Theorem 4.1. With the assumptions equation 4.1 and equation 4.2 hold, then the error between", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 641, 507, 656 ], "spans": [ { "bbox": [ 105, 641, 159, 656 ], "score": 1.0, "content": "the solution", "type": "text" }, { "bbox": [ 159, 642, 193, 654 ], "score": 0.92, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 193, 641, 394, 656 ], "score": 1.0, "content": "to equation 2 and the neural network solution", "type": "text" }, { "bbox": [ 394, 642, 424, 654 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 424, 641, 507, 656 ], "score": 1.0, "content": "to equation 5, i.e.,", "type": "text" } ], "index": 37 }, { "bbox": [ 107, 652, 366, 666 ], "spans": [ { "bbox": [ 107, 653, 218, 665 ], "score": 0.91, "content": "e ^ { n } ( x ) = u ( t _ { n } , x ) - u _ { \\theta ^ { n } } ( x ) ,", "type": "inline_equation" }, { "bbox": [ 218, 652, 316, 666 ], "score": 1.0, "content": ", can be estimated in the", "type": "text" }, { "bbox": [ 316, 653, 328, 663 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 329, 652, 366, 666 ], "score": 1.0, "content": "norm by", "type": "text" } ], "index": 38 } ], "index": 37 }, { "type": "interline_equation", "bbox": [ 180, 670, 429, 693 ], "lines": [ { "bbox": [ 180, 670, 429, 693 ], "spans": [ { "bbox": [ 180, 670, 429, 693 ], "score": 0.92, "content": "\\| e ^ { n } \\| \\leq C \\sqrt { 1 + t _ { n } } ( \\tau ^ { 2 } + \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } ) , \\quad n = 1 , . . . , N _ { t } ,", "type": "interline_equation", "image_path": "d02a32cfd56d92270f897654e9d9746cac878d5e910a8e52b7eb6068b5b0fe33.jpg" } ] } ], "index": 39, "virtual_lines": [ { "bbox": [ 180, 670, 429, 693 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 107, 699, 362, 712 ], "lines": [ { "bbox": [ 105, 697, 362, 713 ], "spans": [ { "bbox": [ 105, 697, 133, 713 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 700, 142, 709 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 142, 697, 277, 713 ], "score": 1.0, "content": "is a bounded constant depend on", "type": "text" }, { "bbox": [ 277, 701, 310, 712 ], "score": 0.93, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 311, 697, 330, 713 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 330, 701, 358, 712 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 359, 697, 362, 713 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "text", "bbox": [ 106, 720, 337, 732 ], "lines": [ { "bbox": [ 105, 719, 338, 734 ], "spans": [ { "bbox": [ 105, 719, 338, 734 ], "score": 1.0, "content": "The proof of Theorem 4.1 can be found in Appendix A.3.", "type": "text" } ], "index": 41 } ], "index": 41 } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 82, 505, 105 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "and solely updating the weights in the last hidden layer through the application of a comparable", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 94, 279, 106 ], "spans": [ { "bbox": [ 105, 94, 279, 106 ], "score": 1.0, "content": "physics-informed loss function equation 5.", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 82, 505, 106 ] }, { "type": "text", "bbox": [ 107, 110, 504, 144 ], "lines": [ { "bbox": [ 105, 109, 505, 123 ], "spans": [ { "bbox": [ 105, 109, 342, 123 ], "score": 1.0, "content": "To be more precise, we first approximate the initial condition", "type": "text" }, { "bbox": [ 342, 110, 367, 123 ], "score": 0.92, "content": "u _ { 0 } ( x )", "type": "inline_equation" }, { "bbox": [ 367, 109, 453, 123 ], "score": 1.0, "content": "by the neural network", "type": "text" }, { "bbox": [ 454, 110, 482, 122 ], "score": 0.92, "content": "u _ { \\theta ^ { 0 } } ( x )", "type": "inline_equation" }, { "bbox": [ 483, 109, 505, 123 ], "score": 1.0, "content": ", then", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 121, 506, 134 ], "spans": [ { "bbox": [ 105, 121, 129, 134 ], "score": 1.0, "content": "learn", "type": "text" }, { "bbox": [ 129, 121, 205, 133 ], "score": 0.48, "content": "u _ { \\theta ^ { 1 } } ( \\bar { x } ) , u _ { \\theta 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The general structure of our TL-DPINN", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 132, 255, 144 ], "spans": [ { "bbox": [ 106, 132, 255, 144 ], "score": 1.0, "content": "method is illustrated in Algorithm 1.", "type": "text" } ], "index": 4 } ], "index": 3, "bbox_fs": [ 105, 109, 506, 144 ] }, { "type": "title", "bbox": [ 106, 159, 365, 171 ], "lines": [ { "bbox": [ 106, 158, 366, 173 ], "spans": [ { "bbox": [ 106, 158, 366, 173 ], "score": 1.0, "content": "Algorithm 1: The training procedure of our TL-DPINN method", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 97, 172, 486, 363 ], "lines": [ { "bbox": [ 105, 172, 414, 185 ], "spans": [ { "bbox": [ 105, 172, 352, 185 ], "score": 1.0, "content": "Input :Target evolutionary PDE equation 2; initial network", "type": "text" }, { "bbox": [ 352, 175, 363, 184 ], "score": 0.79, "content": "u _ { \\theta }", "type": "inline_equation" }, { "bbox": [ 364, 172, 405, 185 ], "score": 1.0, "content": "; end time", "type": "text" }, { "bbox": [ 405, 173, 414, 183 ], "score": 0.64, "content": "T", "type": "inline_equation" } ], "index": 6 }, { "bbox": [ 105, 182, 345, 198 ], "spans": [ { "bbox": [ 105, 182, 228, 198 ], "score": 1.0, "content": "Output :The predicted model", "type": "text" }, { "bbox": [ 228, 184, 258, 196 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 258, 182, 333, 198 ], "score": 1.0, "content": "at each timestamp", "type": "text" }, { "bbox": [ 334, 185, 344, 195 ], "score": 0.86, "content": "t _ { n }", "type": "inline_equation" }, { "bbox": [ 344, 182, 345, 198 ], "score": 0.0, "content": "", "type": "text" } ], "index": 7 }, { "bbox": [ 98, 194, 462, 208 ], "spans": [ { "bbox": [ 98, 194, 286, 208 ], "score": 1.0, "content": "1 Set hyper-parameters: timestamps number", "type": "text" }, { "bbox": [ 286, 195, 298, 206 ], "score": 0.89, "content": "N _ { t }", "type": "inline_equation" }, { "bbox": [ 299, 194, 462, 208 ], "score": 1.0, "content": ", number of maximum training iterations", "type": "text" } ], "index": 8 }, { "bbox": [ 111, 205, 288, 218 ], "spans": [ { "bbox": [ 111, 206, 145, 217 ], "score": 0.8, "content": "M _ { 0 } , M _ { 1 }", "type": "inline_equation" }, { "bbox": [ 145, 205, 201, 218 ], "score": 1.0, "content": ", learning rate", "type": "text" }, { "bbox": [ 202, 207, 208, 217 ], "score": 0.78, "content": "\\eta", "type": "inline_equation" }, { "bbox": [ 209, 205, 275, 218 ], "score": 1.0, "content": ", threshold value", "type": "text" }, { "bbox": [ 275, 208, 281, 216 ], "score": 0.75, "content": "\\epsilon", "type": "inline_equation" }, { "bbox": [ 282, 205, 288, 218 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 9 }, { "bbox": [ 97, 215, 246, 230 ], "spans": [ { "bbox": [ 97, 215, 171, 230 ], "score": 1.0, "content": "2 Step (a): learn", "type": "text" }, { "bbox": [ 172, 217, 200, 228 ], "score": 0.91, "content": "u _ { \\theta ^ { 0 } } ( x )", "type": "inline_equation" }, { "bbox": [ 200, 215, 246, 230 ], "score": 1.0, "content": "by PINN ;", "type": "text" } ], "index": 10 }, { "bbox": [ 97, 226, 200, 240 ], "spans": [ { "bbox": [ 97, 226, 121, 240 ], "score": 1.0, "content": "3 for", "type": "text" }, { "bbox": [ 122, 228, 185, 239 ], "score": 0.89, "content": "i = 1 , 2 , . . . , M _ { 0 }", "type": "inline_equation" }, { "bbox": [ 185, 226, 200, 240 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 11 }, { "bbox": [ 100, 238, 302, 252 ], "spans": [ { "bbox": [ 100, 243, 103, 248 ], "score": 1.0, "content": "4", "type": "text" }, { "bbox": [ 120, 238, 268, 252 ], "score": 1.0, "content": "Compute the mean square error loss", "type": "text" }, { "bbox": [ 268, 238, 297, 251 ], "score": 0.92, "content": "{ \\mathcal { L } } ^ { 0 } ( \\theta ^ { 0 } )", "type": "inline_equation" }, { "bbox": [ 298, 238, 302, 252 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 12 }, { "bbox": [ 99, 248, 408, 266 ], "spans": [ { "bbox": [ 99, 255, 104, 261 ], "score": 1.0, "content": "5", "type": "text" }, { "bbox": [ 118, 248, 210, 266 ], "score": 1.0, "content": "Update the parameter", "type": "text" }, { "bbox": [ 210, 251, 221, 261 ], "score": 0.86, "content": "\\theta ^ { 0 }", "type": "inline_equation" }, { "bbox": [ 221, 248, 304, 266 ], "score": 1.0, "content": "via gradient descent", "type": "text" }, { "bbox": [ 304, 250, 402, 264 ], "score": 0.91, "content": "\\theta _ { i + 1 } ^ { 0 } = \\theta _ { i } ^ { 0 } - \\eta \\nabla \\mathcal { L } ^ { 0 } ( \\theta _ { i } ^ { 0 } )", "type": "inline_equation" }, { "bbox": [ 402, 248, 408, 266 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 13 }, { "bbox": [ 96, 265, 485, 281 ], "spans": [ { "bbox": [ 96, 265, 178, 281 ], "score": 1.0, "content": "6 Step (b): denote", "type": "text" }, { "bbox": [ 178, 266, 218, 280 ], "score": 0.92, "content": "\\theta _ { * } ^ { 0 } = \\theta _ { M _ { 0 } } ^ { 0 }", "type": "inline_equation" }, { "bbox": [ 219, 265, 259, 281 ], "score": 1.0, "content": "and learn", "type": "text" }, { "bbox": [ 260, 267, 346, 279 ], "score": 0.92, "content": "u _ { \\theta ^ { 1 } } ( x ) , . . . , u _ { \\theta ^ { n } } ( x ) , . . .", "type": "inline_equation" }, { "bbox": [ 347, 265, 485, 281 ], "score": 1.0, "content": "sequentially by transfer learning ;", "type": "text" } ], "index": 14 }, { "bbox": [ 98, 278, 226, 292 ], "spans": [ { "bbox": [ 98, 282, 104, 289 ], "score": 0.44, "content": "^ { 7 }", "type": "inline_equation" }, { "bbox": [ 104, 278, 121, 292 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 280, 211, 291 ], "score": 0.88, "content": "n = 0 , 1 , 2 , . . . , N _ { t } - 1", "type": "inline_equation" }, { "bbox": [ 211, 278, 226, 292 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 15 }, { "bbox": [ 98, 290, 216, 302 ], "spans": [ { "bbox": [ 98, 293, 105, 301 ], "score": 1.0, "content": "8", "type": "text" }, { "bbox": [ 120, 290, 137, 302 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 137, 291, 200, 302 ], "score": 0.85, "content": "i = 1 , 2 , . . . , M _ { 1 }", "type": "inline_equation" }, { "bbox": [ 201, 290, 216, 302 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 16 }, { "bbox": [ 96, 299, 470, 335 ], "spans": [ { "bbox": [ 96, 305, 105, 328 ], "score": 1.0, "content": "9 10", "type": "text" }, { "bbox": [ 132, 299, 194, 335 ], "score": 1.0, "content": "Compute loss Update the pa", "type": "text" }, { "bbox": [ 194, 301, 246, 315 ], "score": 0.93, "content": "\\mathcal { L } _ { i } ^ { n + 1 } ( \\boldsymbol { \\theta } _ { i } ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 226, 315, 247, 325 ], "score": 0.89, "content": "\\theta ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 247, 299, 330, 335 ], "score": 1.0, "content": "by equation 5 ;via gradient descent", "type": "text" }, { "bbox": [ 331, 314, 464, 329 ], "score": 0.92, "content": "\\theta _ { i + 1 } ^ { n + 1 } = \\theta _ { i } ^ { n + 1 } - \\eta \\nabla \\mathcal { L } ^ { n + 1 } ( \\theta _ { i } ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 464, 299, 470, 335 ], "score": 1.0, "content": ";", "type": "text" } ], "index": 17 }, { "bbox": [ 96, 324, 309, 346 ], "spans": [ { "bbox": [ 96, 331, 105, 340 ], "score": 1.0, "content": "11", "type": "text" }, { "bbox": [ 133, 324, 146, 346 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 146, 327, 284, 341 ], "score": 0.89, "content": "| { \\mathcal { L } } ^ { n + 1 } ( \\theta _ { i + 1 } ^ { n + 1 } ) - { \\mathcal { L } } ^ { n + 1 } ( \\theta _ { i } ^ { n + 1 } ) | < \\epsilon", "type": "inline_equation" }, { "bbox": [ 284, 324, 309, 346 ], "score": 1.0, "content": "then", "type": "text" } ], "index": 18 }, { "bbox": [ 96, 338, 291, 356 ], "spans": [ { "bbox": [ 96, 344, 106, 353 ], "score": 1.0, "content": "12", "type": "text" }, { "bbox": [ 149, 338, 182, 356 ], "score": 1.0, "content": "denote", "type": "text" }, { "bbox": [ 182, 341, 237, 353 ], "score": 0.91, "content": "\\dot { \\theta } _ { * } ^ { n + 1 } = \\theta _ { i } ^ { n + 1 }", "type": "inline_equation" }, { "bbox": [ 237, 338, 291, 356 ], "score": 1.0, "content": "and break ;", "type": "text" } ], "index": 19 } ], "index": 12.5, "bbox_fs": [ 96, 172, 485, 356 ] }, { "type": "text", "bbox": [ 96, 364, 364, 377 ], "lines": [ { "bbox": [ 93, 361, 361, 380 ], "spans": [ { "bbox": [ 93, 361, 305, 380 ], "score": 1.0, "content": "13 Return the optimized neural network parameters", "type": "text" }, { "bbox": [ 305, 364, 361, 377 ], "score": 0.91, "content": "\\theta _ { * } ^ { 1 } , \\theta _ { * } ^ { 2 } , . . . , \\theta _ { * } ^ { N _ { t } }", "type": "inline_equation" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 93, 361, 361, 380 ] }, { "type": "title", "bbox": [ 107, 411, 242, 424 ], "lines": [ { "bbox": [ 105, 410, 243, 425 ], "spans": [ { "bbox": [ 105, 410, 243, 425 ], "score": 1.0, "content": "4 THEORETICAL RESULT", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 436, 504, 459 ], "lines": [ { "bbox": [ 105, 436, 505, 449 ], "spans": [ { "bbox": [ 105, 436, 505, 449 ], "score": 1.0, "content": "In this section, we analyze the TL-DPINN method and give an error estimate result to approximate", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 447, 464, 460 ], "spans": [ { "bbox": [ 106, 447, 464, 460 ], "score": 1.0, "content": "the evolutionary differential equation 2. We have two reasonable assumptions as follows.", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 105, 436, 505, 460 ] }, { "type": "text", "bbox": [ 106, 461, 506, 514 ], "lines": [ { "bbox": [ 105, 461, 506, 475 ], "spans": [ { "bbox": [ 105, 461, 362, 475 ], "score": 1.0, "content": "Assumption 4.1. The equation equation 2 is dissipative, i.e.", "type": "text" }, { "bbox": [ 362, 461, 444, 475 ], "score": 0.91, "content": "\\textstyle \\int _ { \\Omega } u \\cdot { \\mathcal { N } } ( u ) d x \\leq 0", "type": "inline_equation" }, { "bbox": [ 444, 461, 474, 475 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 475, 462, 502, 474 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 503, 461, 506, 475 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 474, 506, 488 ], "spans": [ { "bbox": [ 105, 474, 237, 488 ], "score": 1.0, "content": "Moreover, if N is nonlinear, then", "type": "text" }, { "bbox": [ 237, 474, 393, 488 ], "score": 0.91, "content": "\\begin{array} { r } { \\int _ { \\Omega } ( u _ { 1 } - u _ { 2 } ) \\cdot ( \\mathcal { N } ( u _ { 1 } ) - \\mathcal { N } ( u _ { 2 } ) ) \\ddot { d x } \\le 0 } \\end{array}", "type": "inline_equation" }, { "bbox": [ 393, 474, 419, 488 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 420, 475, 452, 487 ], "score": 0.92, "content": "u _ { 1 } ( t , x )", "type": "inline_equation" }, { "bbox": [ 452, 474, 470, 488 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 470, 475, 502, 487 ], "score": 0.93, "content": "u _ { 2 } ( t , x )", "type": "inline_equation" }, { "bbox": [ 503, 474, 506, 488 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 490, 505, 504 ], "spans": [ { "bbox": [ 105, 490, 237, 504 ], "score": 1.0, "content": "Assumption 4.2. The solution", "type": "text" }, { "bbox": [ 237, 491, 265, 503 ], "score": 0.92, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 266, 490, 462, 504 ], "score": 1.0, "content": "to equation 2 and the neural network solution", "type": "text" }, { "bbox": [ 463, 491, 492, 503 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 493, 490, 505, 504 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 502, 424, 515 ], "spans": [ { "bbox": [ 106, 502, 424, 515 ], "score": 1.0, "content": "equation 5 are all smooth and bounded, as well as their high order derivatives.", "type": "text" } ], "index": 27 } ], "index": 25.5, "bbox_fs": [ 105, 461, 506, 515 ] }, { "type": "text", "bbox": [ 106, 522, 506, 574 ], "lines": [ { "bbox": [ 105, 521, 505, 535 ], "spans": [ { "bbox": [ 105, 521, 468, 535 ], "score": 1.0, "content": "The first assumption is to guarantee that the solution is not increasing over time. Consider the", "type": "text" }, { "bbox": [ 469, 522, 481, 533 ], "score": 0.86, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 481, 521, 505, 535 ], "score": 1.0, "content": "norm", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 530, 505, 553 ], "spans": [ { "bbox": [ 107, 533, 216, 548 ], "score": 0.92, "content": "\\begin{array} { r } { \\left\\| u ( t , \\cdot ) \\right\\| ^ { 2 } = \\bar { \\int _ { \\Omega } u } ( t , x ) ^ { 2 } \\bar { d } x } \\end{array}", "type": "inline_equation" }, { "bbox": [ 217, 530, 331, 553 ], "score": 1.0, "content": ", we multiply equation 2 by", "type": "text" }, { "bbox": [ 331, 537, 338, 545 ], "score": 0.7, "content": "u", "type": "inline_equation" }, { "bbox": [ 339, 530, 406, 553 ], "score": 1.0, "content": "and integrate in", "type": "text" }, { "bbox": [ 406, 538, 413, 545 ], "score": 0.73, "content": "x", "type": "inline_equation" }, { "bbox": [ 414, 530, 441, 553 ], "score": 1.0, "content": "to get", "type": "text" }, { "bbox": [ 441, 533, 505, 548 ], "score": 0.9, "content": "\\begin{array} { r l } { \\frac { 1 } { 2 } \\frac { d } { d t } \\left\\| u \\right\\| ^ { 2 } ( t ) = } & { { } } \\end{array}", "type": "inline_equation" } ], "index": 29 }, { "bbox": [ 106, 546, 508, 563 ], "spans": [ { "bbox": [ 106, 549, 258, 561 ], "score": 0.56, "content": "\\begin{array} { r } { \\int _ { \\Omega } u \\cdot \\mathcal { N } u d x \\leq 0 , \\mathrm { s o } \\left\\| u ( t , \\cdot ) \\right\\| \\leq \\left\\| u _ { 0 } \\right\\| } \\end{array}", "type": "inline_equation" }, { "bbox": [ 258, 546, 283, 563 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 283, 548, 306, 558 ], "score": 0.87, "content": "t > 0", "type": "inline_equation" }, { "bbox": [ 307, 546, 450, 563 ], "score": 1.0, "content": ". For the simplest Heat equation with", "type": "text" }, { "bbox": [ 450, 548, 502, 560 ], "score": 0.9, "content": "\\mathcal { N } ( u ) = u _ { x x }", "type": "inline_equation" }, { "bbox": [ 503, 546, 508, 563 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 30 }, { "bbox": [ 104, 559, 457, 575 ], "spans": [ { "bbox": [ 104, 559, 198, 575 ], "score": 1.0, "content": "it is easy to verify that", "type": "text" }, { "bbox": [ 198, 560, 344, 574 ], "score": 0.94, "content": "\\begin{array} { r } { \\int _ { \\Omega } u \\cdot \\mathcal { N } ( u ) d x = - \\int _ { \\Omega } | u _ { x } | ^ { 2 } d x \\leq 0 } \\end{array}", "type": "inline_equation" }, { "bbox": [ 344, 559, 457, 575 ], "score": 1.0, "content": ", satisfying Assumption 4.1.", "type": "text" } ], "index": 31 } ], "index": 29.5, "bbox_fs": [ 104, 521, 508, 575 ] }, { "type": "text", "bbox": [ 107, 577, 505, 600 ], "lines": [ { "bbox": [ 105, 577, 505, 590 ], "spans": [ { "bbox": [ 105, 577, 505, 590 ], "score": 1.0, "content": "The second assumption can be verified by the standard regularity estimate result of PDEs Evans", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 588, 263, 600 ], "spans": [ { "bbox": [ 106, 588, 263, 600 ], "score": 1.0, "content": "(2022), and we omit it here for brevity.", "type": "text" } ], "index": 33 } ], "index": 32.5, "bbox_fs": [ 105, 577, 505, 600 ] }, { "type": "text", "bbox": [ 105, 605, 504, 628 ], "lines": [ { "bbox": [ 105, 603, 505, 619 ], "spans": [ { "bbox": [ 105, 603, 182, 619 ], "score": 1.0, "content": "Denote the symbol", "type": "text" }, { "bbox": [ 183, 606, 214, 615 ], "score": 0.91, "content": "\\tau = \\Delta t", "type": "inline_equation" }, { "bbox": [ 215, 603, 232, 619 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 232, 606, 267, 617 ], "score": 0.91, "content": "t _ { n } = n \\tau", "type": "inline_equation" }, { "bbox": [ 267, 603, 505, 619 ], "score": 1.0, "content": ", we show that the error can be strictly controlled by the time", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 616, 395, 629 ], "spans": [ { "bbox": [ 105, 616, 124, 629 ], "score": 1.0, "content": "step", "type": "text" }, { "bbox": [ 125, 619, 132, 627 ], "score": 0.73, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 132, 616, 225, 629 ], "score": 1.0, "content": ", the training loss value", "type": "text" }, { "bbox": [ 226, 617, 239, 626 ], "score": 0.88, "content": "{ \\mathcal { L } } ^ { n }", "type": "inline_equation" }, { "bbox": [ 239, 616, 378, 629 ], "score": 1.0, "content": "and the collocation points number", "type": "text" }, { "bbox": [ 379, 617, 392, 627 ], "score": 0.88, "content": "N _ { r }", "type": "inline_equation" }, { "bbox": [ 392, 616, 395, 629 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 35 } ], "index": 34.5, "bbox_fs": [ 105, 603, 505, 629 ] }, { "type": "text", "bbox": [ 107, 631, 505, 665 ], "lines": [ { "bbox": [ 105, 630, 505, 644 ], "spans": [ { "bbox": [ 105, 630, 505, 644 ], "score": 1.0, "content": "Theorem 4.1. With the assumptions equation 4.1 and equation 4.2 hold, then the error between", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 641, 507, 656 ], "spans": [ { "bbox": [ 105, 641, 159, 656 ], "score": 1.0, "content": "the solution", "type": "text" }, { "bbox": [ 159, 642, 193, 654 ], "score": 0.92, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 193, 641, 394, 656 ], "score": 1.0, "content": "to equation 2 and the neural network solution", "type": "text" }, { "bbox": [ 394, 642, 424, 654 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 424, 641, 507, 656 ], "score": 1.0, "content": "to equation 5, i.e.,", "type": "text" } ], "index": 37 }, { "bbox": [ 107, 652, 366, 666 ], "spans": [ { "bbox": [ 107, 653, 218, 665 ], "score": 0.91, "content": "e ^ { n } ( x ) = u ( t _ { n } , x ) - u _ { \\theta ^ { n } } ( x ) ,", "type": "inline_equation" }, { "bbox": [ 218, 652, 316, 666 ], "score": 1.0, "content": ", can be estimated in the", "type": "text" }, { "bbox": [ 316, 653, 328, 663 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 329, 652, 366, 666 ], "score": 1.0, "content": "norm by", "type": "text" } ], "index": 38 } ], "index": 37, "bbox_fs": [ 105, 630, 507, 666 ] }, { "type": "interline_equation", "bbox": [ 180, 670, 429, 693 ], "lines": [ { "bbox": [ 180, 670, 429, 693 ], "spans": [ { "bbox": [ 180, 670, 429, 693 ], "score": 0.92, "content": "\\| e ^ { n } \\| \\leq C \\sqrt { 1 + t _ { n } } ( \\tau ^ { 2 } + \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } ) , \\quad n = 1 , . . . , N _ { t } ,", "type": "interline_equation", "image_path": "d02a32cfd56d92270f897654e9d9746cac878d5e910a8e52b7eb6068b5b0fe33.jpg" } ] } ], "index": 39, "virtual_lines": [ { "bbox": [ 180, 670, 429, 693 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 107, 699, 362, 712 ], "lines": [ { "bbox": [ 105, 697, 362, 713 ], "spans": [ { "bbox": [ 105, 697, 133, 713 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 700, 142, 709 ], "score": 0.81, "content": "C", "type": "inline_equation" }, { "bbox": [ 142, 697, 277, 713 ], "score": 1.0, "content": "is a bounded constant depend on", "type": "text" }, { "bbox": [ 277, 701, 310, 712 ], "score": 0.93, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 311, 697, 330, 713 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 330, 701, 358, 712 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 359, 697, 362, 713 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 40 } ], "index": 40, "bbox_fs": [ 105, 697, 362, 713 ] }, { "type": "text", "bbox": [ 106, 720, 337, 732 ], "lines": [ { "bbox": [ 105, 719, 338, 734 ], "spans": [ { "bbox": [ 105, 719, 338, 734 ], "score": 1.0, "content": "The proof of Theorem 4.1 can be found in Appendix A.3.", "type": "text" } ], "index": 41 } ], "index": 41, "bbox_fs": [ 105, 719, 338, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 112, 101, 502, 210 ], "blocks": [ { "type": "table_caption", "bbox": [ 112, 80, 493, 93 ], "group_id": 0, "lines": [ { "bbox": [ 116, 79, 494, 93 ], "spans": [ { "bbox": [ 116, 79, 268, 93 ], "score": 1.0, "content": "Table 1: A comparison of the relative", "type": "text" }, { "bbox": [ 269, 80, 281, 91 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 281, 79, 494, 93 ], "score": 1.0, "content": "error and training time (seconds) for different PDEs.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 112, 101, 502, 210 ], "group_id": 0, "lines": [ { "bbox": [ 112, 101, 502, 210 ], "spans": [ { "bbox": [ 112, 101, 502, 210 ], "score": 0.983, "html": "
MethodL2RDEqtimeL² ACEqtimeL2KS EqimeL2 NS Eq time
Original PINN4.17e-0213978.23e-0114121.00e+0011.32e+001
Adaptive sampling1.65e-0215618.64e-0314609.98e-0169018.45e-0125385
Self-attention1.14e-0214501.05e-0117708.22e-0154159.28e-0121296
Time marching3.98e-0332152.01e-0237158.02e-0155278.85e-0126200
Causal PINN3.99e-0573581.66e-0392644.16e-02220294.73e-02 5 days
TL-DPINN1 (ours)1.82e-0514635.92e-0423287.17e-0350503.44e-0212440
TL-DPINN2 (ours)9.34e-057489.82e-0411003.55e-0251713.66e-0256875
", "type": "table", "image_path": "7b0a36c753a6b95d8b776ff03b0246d2cd870a78c026939f3c9969ac26a0a76e.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 112, 101, 502, 137.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 112, 137.33333333333334, 502, 173.66666666666669 ], "spans": [], "index": 2 }, { "bbox": [ 112, 173.66666666666669, 502, 210.00000000000003 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "title", "bbox": [ 108, 242, 263, 255 ], "lines": [ { "bbox": [ 105, 240, 265, 257 ], "spans": [ { "bbox": [ 105, 240, 265, 257 ], "score": 1.0, "content": "5 COMPUTATIONAL RESULTS", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 275, 505, 365 ], "lines": [ { "bbox": [ 105, 275, 506, 289 ], "spans": [ { "bbox": [ 105, 275, 506, 289 ], "score": 1.0, "content": "This section compares the accuracy and training efficiency of the TL-DPINN approach to existing", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 286, 506, 299 ], "spans": [ { "bbox": [ 105, 286, 506, 299 ], "score": 1.0, "content": "PINN methods using various key evolutionary PDEs, including the Reaction-Diffusion (RD) equation,", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 297, 506, 311 ], "spans": [ { "bbox": [ 105, 297, 506, 311 ], "score": 1.0, "content": "Allen-Cahn (AC) equation, Kuramoto–Sivashinsky (KS) equation, Navier-Stokes (NS) equation. All", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 308, 505, 322 ], "spans": [ { "bbox": [ 105, 308, 505, 322 ], "score": 1.0, "content": "the code is implemented in JAX Bradbury et al. (2018), a framework that is gaining popularity in", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 320, 505, 333 ], "spans": [ { "bbox": [ 106, 320, 439, 333 ], "score": 1.0, "content": "scientific computing and deep learning. In all examples, the activation function is", "type": "text" }, { "bbox": [ 440, 320, 471, 332 ], "score": 0.66, "content": "\\operatorname { t a n h } ( { \\cdot } )", "type": "inline_equation" }, { "bbox": [ 472, 320, 505, 333 ], "score": 1.0, "content": "and the", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 330, 506, 345 ], "spans": [ { "bbox": [ 104, 330, 506, 345 ], "score": 1.0, "content": "optimizer is Adam Kingma & Ba (2014). Appendix A.4.1 discusses the Fourier feature embedding", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 342, 505, 354 ], "spans": [ { "bbox": [ 106, 342, 505, 354 ], "score": 1.0, "content": "and modified fully-connected neural networks used in Wang et al. (2022a). Appendix A.4.2 details", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 353, 402, 366 ], "spans": [ { "bbox": [ 105, 353, 402, 366 ], "score": 1.0, "content": "the error metric, neural network hyper-parameters, and training approach.", "type": "text" } ], "index": 12 } ], "index": 8.5 }, { "type": "text", "bbox": [ 106, 369, 505, 447 ], "lines": [ { "bbox": [ 106, 370, 505, 382 ], "spans": [ { "bbox": [ 106, 370, 505, 382 ], "score": 1.0, "content": "The Crank-Nicolson time differencing is denoted as TL-DPINN1, while the Gauss-Legendre time", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 381, 505, 393 ], "spans": [ { "bbox": [ 106, 381, 505, 393 ], "score": 1.0, "content": "differencing is denoted as TL-DPINN2. Our study involves a comparison of these methods with", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 392, 506, 404 ], "spans": [ { "bbox": [ 105, 392, 506, 404 ], "score": 1.0, "content": "several robust baselines: 1) original PINN Raissi et al. (2019); 2) adaptive sampling L. Wight &", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 402, 506, 416 ], "spans": [ { "bbox": [ 105, 402, 506, 416 ], "score": 1.0, "content": "Zhao (2021); 3) self-attention McClenny & Braga-Neto (2023); 4) time marching Mattey & Ghosh", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 413, 504, 426 ], "spans": [ { "bbox": [ 105, 413, 491, 426 ], "score": 1.0, "content": "(2022) and 5) causal PINN Wang et al. (2022a) Table 1 summarizes a comparison of the relative", "type": "text" }, { "bbox": [ 492, 413, 504, 424 ], "score": 0.86, "content": "L ^ { 2 }", "type": "inline_equation" } ], "index": 17 }, { "bbox": [ 105, 424, 506, 437 ], "spans": [ { "bbox": [ 105, 424, 506, 437 ], "score": 1.0, "content": "error and running time (seconds) for different equations by different methods. We note that all neural", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 435, 402, 448 ], "spans": [ { "bbox": [ 106, 435, 402, 448 ], "score": 1.0, "content": "networks are trained on an NVIDIA GeForce RTX 3080 Ti graphics card.", "type": "text" } ], "index": 19 } ], "index": 16 }, { "type": "title", "bbox": [ 108, 475, 273, 486 ], "lines": [ { "bbox": [ 105, 474, 275, 488 ], "spans": [ { "bbox": [ 105, 474, 275, 488 ], "score": 1.0, "content": "5.1 REACTION-DIFFUSION EQUATION", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 107, 501, 505, 535 ], "lines": [ { "bbox": [ 105, 501, 505, 515 ], "spans": [ { "bbox": [ 105, 501, 505, 515 ], "score": 1.0, "content": "This study begins with the Reaction-Diffusion (RD) equation, which is significant to nonlinear", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 513, 505, 525 ], "spans": [ { "bbox": [ 106, 513, 505, 525 ], "score": 1.0, "content": "physics, chemistry, and developmental biology. We consider the one-dimensional Reaction-Diffusion", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 524, 244, 537 ], "spans": [ { "bbox": [ 105, 524, 244, 537 ], "score": 1.0, "content": "equation with the following form:", "type": "text" } ], "index": 23 } ], "index": 22 }, { "type": "interline_equation", "bbox": [ 193, 553, 391, 593 ], "lines": [ { "bbox": [ 193, 553, 391, 593 ], "spans": [ { "bbox": [ 193, 553, 391, 593 ], "score": 0.93, "content": "\\begin{array} { r } { \\left\\{ \\begin{array} { l c c } { u _ { t } = d _ { 1 } u _ { x x } + d _ { 2 } u ^ { 2 } , \\quad t \\in [ 0 , 1 ] , x \\in [ - 1 , 1 ] , } \\\\ { u ( 0 , x ) = \\sin ( 2 \\pi x ) \\big ( 1 + \\cos ( 2 \\pi x ) \\big ) , } \\\\ { u ( t , - 1 ) = u ( t , 1 ) = 0 , } \\end{array} \\right. } \\end{array}", "type": "interline_equation", "image_path": "d85581d0b2c57ec6084ac00958a9a719f6dec96391f8db5bc7edf4eb731da93e.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 193, 553, 391, 566.3333333333334 ], "spans": [], "index": 24 }, { "bbox": [ 193, 566.3333333333334, 391, 579.6666666666667 ], "spans": [], "index": 25 }, { "bbox": [ 193, 579.6666666666667, 391, 593.0000000000001 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 106, 610, 506, 732 ], "lines": [ { "bbox": [ 105, 610, 506, 623 ], "spans": [ { "bbox": [ 105, 610, 133, 623 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 611, 200, 622 ], "score": 0.91, "content": "d _ { 1 } = d _ { 2 } = 0 . 0 1", "type": "inline_equation" }, { "bbox": [ 200, 610, 506, 623 ], "score": 1.0, "content": ". The solution changes slowly over time, and Table 1 demonstrates that all", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 621, 506, 635 ], "spans": [ { "bbox": [ 105, 621, 253, 635 ], "score": 1.0, "content": "methods succeed with small relative", "type": "text" }, { "bbox": [ 253, 622, 266, 632 ], "score": 0.86, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 266, 621, 506, 635 ], "score": 1.0, "content": "norm error in this instance. Our methods enhance accuracy", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 632, 506, 645 ], "spans": [ { "bbox": [ 105, 632, 506, 645 ], "score": 1.0, "content": "by 2 3 orders of magnitude compared to other PINN frameworks Raissi et al. (2019); L. Wight &", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 642, 507, 657 ], "spans": [ { "bbox": [ 105, 642, 507, 657 ], "score": 1.0, "content": "Zhao (2021); McClenny & Braga-Neto (2023); Mattey & Ghosh (2022) even with less training time.", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 654, 505, 667 ], "spans": [ { "bbox": [ 106, 654, 505, 667 ], "score": 1.0, "content": "We see that our method TL-DPINN1 is more accurate than causal PINN Wang et al. (2022a) with", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 665, 506, 679 ], "spans": [ { "bbox": [ 105, 665, 506, 679 ], "score": 1.0, "content": "much less computational time. We acknowledge that our methods TL-DPINN2 may be slightly less", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 677, 506, 689 ], "spans": [ { "bbox": [ 105, 677, 365, 689 ], "score": 1.0, "content": "accurate than causal PINN, but the training time is only nearly", "type": "text" }, { "bbox": [ 366, 677, 387, 689 ], "score": 0.7, "content": "1 / 1 0", "type": "inline_equation" }, { "bbox": [ 388, 677, 506, 689 ], "score": 1.0, "content": "of their method. In fact, the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 268, 701 ], "score": 1.0, "content": "casual PINN can only achieve a relative", "type": "text" }, { "bbox": [ 268, 687, 281, 698 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 281, 687, 315, 701 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 315, 688, 361, 698 ], "score": 0.89, "content": "1 . 1 3 e \\mathrm { ~ - ~ } 0 1", "type": "inline_equation" }, { "bbox": [ 361, 687, 505, 701 ], "score": 1.0, "content": "if we stop early at the training time", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 506, 712 ], "score": 1.0, "content": "of our methods ( 748 seconds). Figure 2 shows the predicted solution against the reference solution,", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 709, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 285, 722 ], "score": 1.0, "content": "and our proposed method achieves a relative", "type": "text" }, { "bbox": [ 286, 709, 298, 720 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 299, 710, 332, 722 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 333, 710, 378, 720 ], "score": 0.86, "content": "1 . 8 2 e \\mathrm { ~ - ~ } 0 5", "type": "inline_equation" }, { "bbox": [ 378, 710, 506, 722 ], "score": 1.0, "content": ". More computational results of", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 721, 306, 732 ], "spans": [ { "bbox": [ 105, 721, 306, 732 ], "score": 1.0, "content": "the RD equation are provided in Appendix A.4.3.", "type": "text" } ], "index": 37 } ], "index": 32 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 309, 760 ], "lines": [ { "bbox": [ 302, 751, 310, 762 ], "spans": [ { "bbox": [ 302, 751, 310, 762 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 112, 101, 502, 210 ], "blocks": [ { "type": "table_caption", "bbox": [ 112, 80, 493, 93 ], "group_id": 0, "lines": [ { "bbox": [ 116, 79, 494, 93 ], "spans": [ { "bbox": [ 116, 79, 268, 93 ], "score": 1.0, "content": "Table 1: A comparison of the relative", "type": "text" }, { "bbox": [ 269, 80, 281, 91 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 281, 79, 494, 93 ], "score": 1.0, "content": "error and training time (seconds) for different PDEs.", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 112, 101, 502, 210 ], "group_id": 0, "lines": [ { "bbox": [ 112, 101, 502, 210 ], "spans": [ { "bbox": [ 112, 101, 502, 210 ], "score": 0.983, "html": "
MethodL2RDEqtimeL² ACEqtimeL2KS EqimeL2 NS Eq time
Original PINN4.17e-0213978.23e-0114121.00e+0011.32e+001
Adaptive sampling1.65e-0215618.64e-0314609.98e-0169018.45e-0125385
Self-attention1.14e-0214501.05e-0117708.22e-0154159.28e-0121296
Time marching3.98e-0332152.01e-0237158.02e-0155278.85e-0126200
Causal PINN3.99e-0573581.66e-0392644.16e-02220294.73e-02 5 days
TL-DPINN1 (ours)1.82e-0514635.92e-0423287.17e-0350503.44e-0212440
TL-DPINN2 (ours)9.34e-057489.82e-0411003.55e-0251713.66e-0256875
", "type": "table", "image_path": "7b0a36c753a6b95d8b776ff03b0246d2cd870a78c026939f3c9969ac26a0a76e.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 112, 101, 502, 137.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 112, 137.33333333333334, 502, 173.66666666666669 ], "spans": [], "index": 2 }, { "bbox": [ 112, 173.66666666666669, 502, 210.00000000000003 ], "spans": [], "index": 3 } ] } ], "index": 1.0 }, { "type": "title", "bbox": [ 108, 242, 263, 255 ], "lines": [ { "bbox": [ 105, 240, 265, 257 ], "spans": [ { "bbox": [ 105, 240, 265, 257 ], "score": 1.0, "content": "5 COMPUTATIONAL RESULTS", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 275, 505, 365 ], "lines": [ { "bbox": [ 105, 275, 506, 289 ], "spans": [ { "bbox": [ 105, 275, 506, 289 ], "score": 1.0, "content": "This section compares the accuracy and training efficiency of the TL-DPINN approach to existing", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 286, 506, 299 ], "spans": [ { "bbox": [ 105, 286, 506, 299 ], "score": 1.0, "content": "PINN methods using various key evolutionary PDEs, including the Reaction-Diffusion (RD) equation,", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 297, 506, 311 ], "spans": [ { "bbox": [ 105, 297, 506, 311 ], "score": 1.0, "content": "Allen-Cahn (AC) equation, Kuramoto–Sivashinsky (KS) equation, Navier-Stokes (NS) equation. All", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 308, 505, 322 ], "spans": [ { "bbox": [ 105, 308, 505, 322 ], "score": 1.0, "content": "the code is implemented in JAX Bradbury et al. (2018), a framework that is gaining popularity in", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 320, 505, 333 ], "spans": [ { "bbox": [ 106, 320, 439, 333 ], "score": 1.0, "content": "scientific computing and deep learning. In all examples, the activation function is", "type": "text" }, { "bbox": [ 440, 320, 471, 332 ], "score": 0.66, "content": "\\operatorname { t a n h } ( { \\cdot } )", "type": "inline_equation" }, { "bbox": [ 472, 320, 505, 333 ], "score": 1.0, "content": "and the", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 330, 506, 345 ], "spans": [ { "bbox": [ 104, 330, 506, 345 ], "score": 1.0, "content": "optimizer is Adam Kingma & Ba (2014). Appendix A.4.1 discusses the Fourier feature embedding", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 342, 505, 354 ], "spans": [ { "bbox": [ 106, 342, 505, 354 ], "score": 1.0, "content": "and modified fully-connected neural networks used in Wang et al. (2022a). Appendix A.4.2 details", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 353, 402, 366 ], "spans": [ { "bbox": [ 105, 353, 402, 366 ], "score": 1.0, "content": "the error metric, neural network hyper-parameters, and training approach.", "type": "text" } ], "index": 12 } ], "index": 8.5, "bbox_fs": [ 104, 275, 506, 366 ] }, { "type": "text", "bbox": [ 106, 369, 505, 447 ], "lines": [ { "bbox": [ 106, 370, 505, 382 ], "spans": [ { "bbox": [ 106, 370, 505, 382 ], "score": 1.0, "content": "The Crank-Nicolson time differencing is denoted as TL-DPINN1, while the Gauss-Legendre time", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 381, 505, 393 ], "spans": [ { "bbox": [ 106, 381, 505, 393 ], "score": 1.0, "content": "differencing is denoted as TL-DPINN2. Our study involves a comparison of these methods with", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 392, 506, 404 ], "spans": [ { "bbox": [ 105, 392, 506, 404 ], "score": 1.0, "content": "several robust baselines: 1) original PINN Raissi et al. (2019); 2) adaptive sampling L. Wight &", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 402, 506, 416 ], "spans": [ { "bbox": [ 105, 402, 506, 416 ], "score": 1.0, "content": "Zhao (2021); 3) self-attention McClenny & Braga-Neto (2023); 4) time marching Mattey & Ghosh", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 413, 504, 426 ], "spans": [ { "bbox": [ 105, 413, 491, 426 ], "score": 1.0, "content": "(2022) and 5) causal PINN Wang et al. (2022a) Table 1 summarizes a comparison of the relative", "type": "text" }, { "bbox": [ 492, 413, 504, 424 ], "score": 0.86, "content": "L ^ { 2 }", "type": "inline_equation" } ], "index": 17 }, { "bbox": [ 105, 424, 506, 437 ], "spans": [ { "bbox": [ 105, 424, 506, 437 ], "score": 1.0, "content": "error and running time (seconds) for different equations by different methods. We note that all neural", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 435, 402, 448 ], "spans": [ { "bbox": [ 106, 435, 402, 448 ], "score": 1.0, "content": "networks are trained on an NVIDIA GeForce RTX 3080 Ti graphics card.", "type": "text" } ], "index": 19 } ], "index": 16, "bbox_fs": [ 105, 370, 506, 448 ] }, { "type": "title", "bbox": [ 108, 475, 273, 486 ], "lines": [ { "bbox": [ 105, 474, 275, 488 ], "spans": [ { "bbox": [ 105, 474, 275, 488 ], "score": 1.0, "content": "5.1 REACTION-DIFFUSION EQUATION", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 107, 501, 505, 535 ], "lines": [ { "bbox": [ 105, 501, 505, 515 ], "spans": [ { "bbox": [ 105, 501, 505, 515 ], "score": 1.0, "content": "This study begins with the Reaction-Diffusion (RD) equation, which is significant to nonlinear", "type": "text" } ], "index": 21 }, { "bbox": [ 106, 513, 505, 525 ], "spans": [ { "bbox": [ 106, 513, 505, 525 ], "score": 1.0, "content": "physics, chemistry, and developmental biology. We consider the one-dimensional Reaction-Diffusion", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 524, 244, 537 ], "spans": [ { "bbox": [ 105, 524, 244, 537 ], "score": 1.0, "content": "equation with the following form:", "type": "text" } ], "index": 23 } ], "index": 22, "bbox_fs": [ 105, 501, 505, 537 ] }, { "type": "interline_equation", "bbox": [ 193, 553, 391, 593 ], "lines": [ { "bbox": [ 193, 553, 391, 593 ], "spans": [ { "bbox": [ 193, 553, 391, 593 ], "score": 0.93, "content": "\\begin{array} { r } { \\left\\{ \\begin{array} { l c c } { u _ { t } = d _ { 1 } u _ { x x } + d _ { 2 } u ^ { 2 } , \\quad t \\in [ 0 , 1 ] , x \\in [ - 1 , 1 ] , } \\\\ { u ( 0 , x ) = \\sin ( 2 \\pi x ) \\big ( 1 + \\cos ( 2 \\pi x ) \\big ) , } \\\\ { u ( t , - 1 ) = u ( t , 1 ) = 0 , } \\end{array} \\right. } \\end{array}", "type": "interline_equation", "image_path": "d85581d0b2c57ec6084ac00958a9a719f6dec96391f8db5bc7edf4eb731da93e.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 193, 553, 391, 566.3333333333334 ], "spans": [], "index": 24 }, { "bbox": [ 193, 566.3333333333334, 391, 579.6666666666667 ], "spans": [], "index": 25 }, { "bbox": [ 193, 579.6666666666667, 391, 593.0000000000001 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 106, 610, 506, 732 ], "lines": [ { "bbox": [ 105, 610, 506, 623 ], "spans": [ { "bbox": [ 105, 610, 133, 623 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 611, 200, 622 ], "score": 0.91, "content": "d _ { 1 } = d _ { 2 } = 0 . 0 1", "type": "inline_equation" }, { "bbox": [ 200, 610, 506, 623 ], "score": 1.0, "content": ". The solution changes slowly over time, and Table 1 demonstrates that all", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 621, 506, 635 ], "spans": [ { "bbox": [ 105, 621, 253, 635 ], "score": 1.0, "content": "methods succeed with small relative", "type": "text" }, { "bbox": [ 253, 622, 266, 632 ], "score": 0.86, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 266, 621, 506, 635 ], "score": 1.0, "content": "norm error in this instance. Our methods enhance accuracy", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 632, 506, 645 ], "spans": [ { "bbox": [ 105, 632, 506, 645 ], "score": 1.0, "content": "by 2 3 orders of magnitude compared to other PINN frameworks Raissi et al. (2019); L. Wight &", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 642, 507, 657 ], "spans": [ { "bbox": [ 105, 642, 507, 657 ], "score": 1.0, "content": "Zhao (2021); McClenny & Braga-Neto (2023); Mattey & Ghosh (2022) even with less training time.", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 654, 505, 667 ], "spans": [ { "bbox": [ 106, 654, 505, 667 ], "score": 1.0, "content": "We see that our method TL-DPINN1 is more accurate than causal PINN Wang et al. (2022a) with", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 665, 506, 679 ], "spans": [ { "bbox": [ 105, 665, 506, 679 ], "score": 1.0, "content": "much less computational time. We acknowledge that our methods TL-DPINN2 may be slightly less", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 677, 506, 689 ], "spans": [ { "bbox": [ 105, 677, 365, 689 ], "score": 1.0, "content": "accurate than causal PINN, but the training time is only nearly", "type": "text" }, { "bbox": [ 366, 677, 387, 689 ], "score": 0.7, "content": "1 / 1 0", "type": "inline_equation" }, { "bbox": [ 388, 677, 506, 689 ], "score": 1.0, "content": "of their method. In fact, the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 268, 701 ], "score": 1.0, "content": "casual PINN can only achieve a relative", "type": "text" }, { "bbox": [ 268, 687, 281, 698 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 281, 687, 315, 701 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 315, 688, 361, 698 ], "score": 0.89, "content": "1 . 1 3 e \\mathrm { ~ - ~ } 0 1", "type": "inline_equation" }, { "bbox": [ 361, 687, 505, 701 ], "score": 1.0, "content": "if we stop early at the training time", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 699, 506, 712 ], "spans": [ { "bbox": [ 105, 699, 506, 712 ], "score": 1.0, "content": "of our methods ( 748 seconds). Figure 2 shows the predicted solution against the reference solution,", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 709, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 285, 722 ], "score": 1.0, "content": "and our proposed method achieves a relative", "type": "text" }, { "bbox": [ 286, 709, 298, 720 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 299, 710, 332, 722 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 333, 710, 378, 720 ], "score": 0.86, "content": "1 . 8 2 e \\mathrm { ~ - ~ } 0 5", "type": "inline_equation" }, { "bbox": [ 378, 710, 506, 722 ], "score": 1.0, "content": ". More computational results of", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 721, 306, 732 ], "spans": [ { "bbox": [ 105, 721, 306, 732 ], "score": 1.0, "content": "the RD equation are provided in Appendix A.4.3.", "type": "text" } ], "index": 37 } ], "index": 32, "bbox_fs": [ 105, 610, 507, 732 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 109, 80, 505, 189 ], "blocks": [ { "type": "image_body", "bbox": [ 109, 80, 505, 189 ], "group_id": 0, "lines": [ { "bbox": [ 109, 80, 505, 189 ], "spans": [ { "bbox": [ 109, 80, 505, 189 ], "score": 0.956, "type": "image", "image_path": "d6a0edef5ecd6c2623b6cb33d92fde8ac93f506033b0633c1b44a7bf11413f3f.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 109, 80, 505, 116.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 109, 116.33333333333334, 505, 152.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 109, 152.66666666666669, 505, 189.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 198, 504, 221 ], "group_id": 0, "lines": [ { "bbox": [ 106, 198, 505, 210 ], "spans": [ { "bbox": [ 106, 198, 505, 210 ], "score": 1.0, "content": "Figure 2: Comparison between the reference and predicted solutions for the Reaction-Diffusion", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 209, 272, 221 ], "spans": [ { "bbox": [ 105, 209, 177, 221 ], "score": 1.0, "content": "equation, and the", "type": "text" }, { "bbox": [ 177, 209, 190, 219 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 190, 209, 222, 221 ], "score": 1.0, "content": "error is", "type": "text" }, { "bbox": [ 222, 209, 268, 220 ], "score": 0.88, "content": "1 . 8 2 e \\mathrm { ~ - ~ } 0 5", "type": "inline_equation" }, { "bbox": [ 268, 209, 272, 221 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "image", "bbox": [ 134, 239, 477, 376 ], "blocks": [ { "type": "image_body", "bbox": [ 134, 239, 477, 376 ], "group_id": 1, "lines": [ { "bbox": [ 134, 239, 477, 376 ], "spans": [ { "bbox": [ 134, 239, 477, 376 ], "score": 0.97, "type": "image", "image_path": "910bdfb85943d3ed45636596e30261e855bbc46383d9ac9fd1fa2c512c225c32.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 134, 239, 477, 284.6666666666667 ], "spans": [], "index": 5 }, { "bbox": [ 134, 284.6666666666667, 477, 330.33333333333337 ], "spans": [], "index": 6 }, { "bbox": [ 134, 330.33333333333337, 477, 376.00000000000006 ], "spans": [], "index": 7 } ] }, { "type": "image_caption", "bbox": [ 194, 385, 415, 398 ], "group_id": 1, "lines": [ { "bbox": [ 195, 385, 416, 398 ], "spans": [ { "bbox": [ 195, 385, 416, 398 ], "score": 1.0, "content": "Figure 3: Training results for the Allen-Cahn equation.", "type": "text" } ], "index": 8 } ], "index": 8 } ], "index": 7.0 }, { "type": "title", "bbox": [ 107, 416, 238, 427 ], "lines": [ { "bbox": [ 105, 414, 239, 429 ], "spans": [ { "bbox": [ 105, 414, 239, 429 ], "score": 1.0, "content": "5.2 ALLEN-CAHN EQUATION", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 106, 436, 503, 459 ], "lines": [ { "bbox": [ 105, 434, 505, 450 ], "spans": [ { "bbox": [ 105, 434, 505, 450 ], "score": 1.0, "content": "We consider the one-dimensional Allen-Cahn (AC) equation, a benchmark problem for PINN training", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 447, 394, 460 ], "spans": [ { "bbox": [ 106, 447, 394, 460 ], "score": 1.0, "content": "L. Wight & Zhao (2021); Mattey & Ghosh (2022); Wang et al. (2022a):", "type": "text" } ], "index": 11 } ], "index": 10.5 }, { "type": "interline_equation", "bbox": [ 178, 461, 407, 499 ], "lines": [ { "bbox": [ 178, 461, 407, 499 ], "spans": [ { "bbox": [ 178, 461, 407, 499 ], "score": 0.93, "content": "\\left\\{ \\begin{array} { l } { u _ { t } = \\gamma _ { 1 } u _ { x x } + \\gamma _ { 2 } u ( 1 - u ^ { 2 } ) , \\quad t \\in [ 0 , 1 ] , x \\in [ - 1 , 1 ] , } \\\\ { u ( x , 0 ) = u _ { 0 } ( x ) , } \\\\ { u ( t , - 1 ) = u ( t , 1 ) , \\quad u _ { x } ( t , - 1 ) = u _ { x } ( t , 1 ) . } \\end{array} \\right.", "type": "interline_equation", "image_path": "9673ee6dd8f1e5d7c5c70b984f7a578cc3e990f7dcdc4c82592d14a9818c70b9.jpg" } ] } ], "index": 12.5, "virtual_lines": [ { "bbox": [ 178, 461, 407, 480.0 ], "spans": [], "index": 12 }, { "bbox": [ 178, 480.0, 407, 499.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 501, 506, 580 ], "lines": [ { "bbox": [ 105, 501, 506, 515 ], "spans": [ { "bbox": [ 105, 501, 134, 515 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 502, 189, 514 ], "score": 0.85, "content": "\\gamma _ { 1 } = 0 . 0 0 0 1", "type": "inline_equation" }, { "bbox": [ 190, 501, 194, 515 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 194, 503, 227, 514 ], "score": 0.88, "content": "\\gamma _ { 2 } = 5", "type": "inline_equation" }, { "bbox": [ 228, 501, 248, 515 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 248, 502, 334, 514 ], "score": 0.92, "content": "u _ { 0 } ( x ) = x ^ { 2 } \\cos ( \\pi x )", "type": "inline_equation" }, { "bbox": [ 334, 501, 506, 515 ], "score": 1.0, "content": ". For the original PINN, the Allen-Cahn", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 513, 506, 526 ], "spans": [ { "bbox": [ 106, 513, 506, 526 ], "score": 1.0, "content": "equation is hard to solve, but our approach performs well in accuracy and training efficiency. Figure 1", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 524, 506, 536 ], "spans": [ { "bbox": [ 105, 524, 469, 536 ], "score": 1.0, "content": "compares the predicted solution to the reference solution. 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Existing PINN frameworks are challenging to solve the KS equation as the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 634, 497, 647 ], "spans": [ { "bbox": [ 106, 634, 497, 647 ], "score": 1.0, "content": "solution exhibits fast transit and chaotic behaviors Raissi (2018). 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Figure 4 visualizes the predicted solution against the reference solution, and our", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 256, 712 ], "score": 1.0, "content": "proposed method achieves a relative", "type": "text" }, { "bbox": [ 257, 698, 269, 709 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 269, 698, 304, 712 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 304, 699, 350, 709 ], "score": 0.77, "content": "7 . 1 7 e \\mathrm { ~ - ~ } 0 3", "type": "inline_equation" }, { "bbox": [ 350, 698, 379, 712 ], "score": 1.0, "content": ". From", "type": "text" }, { "bbox": [ 379, 699, 410, 709 ], "score": 0.87, "content": "t = 0 . 4", "type": "inline_equation" }, { "bbox": [ 411, 698, 505, 712 ], "score": 1.0, "content": ", the reference solution", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "begins to quickly transition, and our method is able to capture this feature. More computational", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 720, 344, 732 ], "spans": [ { "bbox": [ 105, 720, 344, 732 ], "score": 1.0, "content": "results of the KS equation are provided in Appendix A.4.5.", "type": "text" } ], "index": 31 } ], "index": 29 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "7", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 109, 80, 505, 189 ], "blocks": [ { "type": "image_body", "bbox": [ 109, 80, 505, 189 ], "group_id": 0, "lines": [ { "bbox": [ 109, 80, 505, 189 ], "spans": [ { "bbox": [ 109, 80, 505, 189 ], "score": 0.956, "type": "image", "image_path": "d6a0edef5ecd6c2623b6cb33d92fde8ac93f506033b0633c1b44a7bf11413f3f.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 109, 80, 505, 116.33333333333334 ], "spans": [], "index": 0 }, { "bbox": [ 109, 116.33333333333334, 505, 152.66666666666669 ], "spans": [], "index": 1 }, { "bbox": [ 109, 152.66666666666669, 505, 189.00000000000003 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 198, 504, 221 ], "group_id": 0, "lines": [ { "bbox": [ 106, 198, 505, 210 ], "spans": [ { "bbox": [ 106, 198, 505, 210 ], "score": 1.0, "content": "Figure 2: Comparison between the reference and predicted solutions for the Reaction-Diffusion", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 209, 272, 221 ], "spans": [ { "bbox": [ 105, 209, 177, 221 ], "score": 1.0, "content": "equation, and the", "type": "text" }, { "bbox": [ 177, 209, 190, 219 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 190, 209, 222, 221 ], "score": 1.0, "content": "error is", "type": "text" }, { "bbox": [ 222, 209, 268, 220 ], "score": 0.88, "content": "1 . 8 2 e \\mathrm { ~ - ~ } 0 5", "type": "inline_equation" }, { "bbox": [ 268, 209, 272, 221 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "image", "bbox": [ 134, 239, 477, 376 ], "blocks": [ { "type": "image_body", "bbox": [ 134, 239, 477, 376 ], "group_id": 1, "lines": [ { "bbox": [ 134, 239, 477, 376 ], "spans": [ { "bbox": [ 134, 239, 477, 376 ], "score": 0.97, "type": "image", "image_path": "910bdfb85943d3ed45636596e30261e855bbc46383d9ac9fd1fa2c512c225c32.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 134, 239, 477, 284.6666666666667 ], "spans": [], "index": 5 }, { "bbox": [ 134, 284.6666666666667, 477, 330.33333333333337 ], "spans": [], "index": 6 }, { "bbox": [ 134, 330.33333333333337, 477, 376.00000000000006 ], "spans": [], "index": 7 } ] }, { "type": "image_caption", "bbox": [ 194, 385, 415, 398 ], "group_id": 1, "lines": [ { "bbox": [ 195, 385, 416, 398 ], "spans": [ { "bbox": [ 195, 385, 416, 398 ], "score": 1.0, "content": "Figure 3: Training results for the Allen-Cahn equation.", "type": "text" } ], "index": 8 } ], "index": 8 } ], "index": 7.0 }, { "type": "title", "bbox": [ 107, 416, 238, 427 ], "lines": [ { "bbox": [ 105, 414, 239, 429 ], "spans": [ { "bbox": [ 105, 414, 239, 429 ], "score": 1.0, "content": "5.2 ALLEN-CAHN EQUATION", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 106, 436, 503, 459 ], "lines": [ { "bbox": [ 105, 434, 505, 450 ], "spans": [ { "bbox": [ 105, 434, 505, 450 ], "score": 1.0, "content": "We consider the one-dimensional Allen-Cahn (AC) equation, a benchmark problem for PINN training", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 447, 394, 460 ], "spans": [ { "bbox": [ 106, 447, 394, 460 ], "score": 1.0, "content": "L. Wight & Zhao (2021); Mattey & Ghosh (2022); Wang et al. (2022a):", "type": "text" } ], "index": 11 } ], "index": 10.5, "bbox_fs": [ 105, 434, 505, 460 ] }, { "type": "interline_equation", "bbox": [ 178, 461, 407, 499 ], "lines": [ { "bbox": [ 178, 461, 407, 499 ], "spans": [ { "bbox": [ 178, 461, 407, 499 ], "score": 0.93, "content": "\\left\\{ \\begin{array} { l } { u _ { t } = \\gamma _ { 1 } u _ { x x } + \\gamma _ { 2 } u ( 1 - u ^ { 2 } ) , \\quad t \\in [ 0 , 1 ] , x \\in [ - 1 , 1 ] , } \\\\ { u ( x , 0 ) = u _ { 0 } ( x ) , } \\\\ { u ( t , - 1 ) = u ( t , 1 ) , \\quad u _ { x } ( t , - 1 ) = u _ { x } ( t , 1 ) . } \\end{array} \\right.", "type": "interline_equation", "image_path": "9673ee6dd8f1e5d7c5c70b984f7a578cc3e990f7dcdc4c82592d14a9818c70b9.jpg" } ] } ], "index": 12.5, "virtual_lines": [ { "bbox": [ 178, 461, 407, 480.0 ], "spans": [], "index": 12 }, { "bbox": [ 178, 480.0, 407, 499.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 501, 506, 580 ], "lines": [ { "bbox": [ 105, 501, 506, 515 ], "spans": [ { "bbox": [ 105, 501, 134, 515 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 502, 189, 514 ], "score": 0.85, "content": "\\gamma _ { 1 } = 0 . 0 0 0 1", "type": "inline_equation" }, { "bbox": [ 190, 501, 194, 515 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 194, 503, 227, 514 ], "score": 0.88, "content": "\\gamma _ { 2 } = 5", "type": "inline_equation" }, { "bbox": [ 228, 501, 248, 515 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 248, 502, 334, 514 ], "score": 0.92, "content": "u _ { 0 } ( x ) = x ^ { 2 } \\cos ( \\pi x )", "type": "inline_equation" }, { "bbox": [ 334, 501, 506, 515 ], "score": 1.0, "content": ". 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The", "type": "text" }, { "bbox": [ 228, 546, 241, 556 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 241, 545, 506, 558 ], "score": 1.0, "content": "error increases as the AC equation develops more complicated.", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 557, 505, 569 ], "spans": [ { "bbox": [ 106, 557, 505, 569 ], "score": 1.0, "content": "Each timestamp’s training epoch is small across the time domain, reducing training time. More", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 568, 405, 581 ], "spans": [ { "bbox": [ 106, 568, 405, 581 ], "score": 1.0, "content": "computational results of the AC equation are provided in Appendix A.4.4.", "type": "text" } ], "index": 20 } ], "index": 17, "bbox_fs": [ 105, 501, 506, 581 ] }, { "type": "title", "bbox": [ 108, 592, 293, 604 ], "lines": [ { "bbox": [ 106, 592, 294, 605 ], "spans": [ { "bbox": [ 106, 592, 294, 605 ], "score": 1.0, "content": "5.3 KURAMOTO–SIVASHINSKY EQUATION", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 613, 505, 646 ], "lines": [ { "bbox": [ 106, 613, 506, 625 ], "spans": [ { "bbox": [ 106, 613, 506, 625 ], "score": 1.0, "content": "The Kuramoto-Sivashinsky (KS) equation is used to model the diffusive–thermal instabilities in a", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 624, 505, 636 ], "spans": [ { "bbox": [ 106, 624, 505, 636 ], "score": 1.0, "content": "laminar flame front. 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Figure 4 visualizes the predicted solution against the reference solution, and our", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 256, 712 ], "score": 1.0, "content": "proposed method achieves a relative", "type": "text" }, { "bbox": [ 257, 698, 269, 709 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 269, 698, 304, 712 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 304, 699, 350, 709 ], "score": 0.77, "content": "7 . 1 7 e \\mathrm { ~ - ~ } 0 3", "type": "inline_equation" }, { "bbox": [ 350, 698, 379, 712 ], "score": 1.0, "content": ". From", "type": "text" }, { "bbox": [ 379, 699, 410, 709 ], "score": 0.87, "content": "t = 0 . 4", "type": "inline_equation" }, { "bbox": [ 411, 698, 505, 712 ], "score": 1.0, "content": ", the reference solution", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "begins to quickly transition, and our method is able to capture this feature. 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(2022a)", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 203, 435, 381, 475 ], "lines": [ { "bbox": [ 203, 435, 381, 475 ], "spans": [ { "bbox": [ 203, 435, 381, 475 ], "score": 0.92, "content": "\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { w _ { t } + u \\cdot \\nabla w = \\frac { 1 } { \\mathrm { R e } } \\Delta w , \\quad \\mathrm { i n } \\ [ 0 , \\mathrm { T } ] \\times \\Omega , } \\\\ { \\nabla \\cdot \\pmb { u } = 0 , \\quad \\mathrm { i n } \\ [ 0 , \\mathrm { T } ] \\times \\Omega , } \\\\ { w ( 0 , x , y ) = w _ { 0 } ( x , y ) , \\quad \\mathrm { i n } \\ \\Omega . } \\end{array} \\right. } \\end{array}", "type": "interline_equation", "image_path": "e094ad79ef358afd8c3fca11f5d26426eb67fc39344ce8d8a42ce408b0e8266a.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 203, 435, 381, 448.3333333333333 ], "spans": [], "index": 12 }, { "bbox": [ 203, 448.3333333333333, 381, 461.66666666666663 ], "spans": [], "index": 13 }, { "bbox": [ 203, 461.66666666666663, 381, 474.99999999999994 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 106, 480, 505, 547 ], "lines": [ { "bbox": [ 105, 480, 504, 495 ], "spans": [ { "bbox": [ 105, 480, 274, 495 ], "score": 1.0, "content": "with periodic boundary conditions. Here,", "type": "text" }, { "bbox": [ 275, 482, 318, 493 ], "score": 0.93, "content": "\\mathbf { u } = ( u , v )", "type": "inline_equation" }, { "bbox": [ 319, 480, 455, 495 ], "score": 1.0, "content": "represents the flow velocity field,", "type": "text" }, { "bbox": [ 456, 482, 504, 492 ], "score": 0.9, "content": "w = \\nabla \\times u", "type": "inline_equation" } ], "index": 15 }, { "bbox": [ 105, 492, 506, 505 ], "spans": [ { "bbox": [ 105, 492, 217, 505 ], "score": 1.0, "content": "represents the vorticity, and", "type": "text" }, { "bbox": [ 217, 492, 230, 502 ], "score": 0.3, "content": "\\mathrm { R e }", "type": "inline_equation" }, { "bbox": [ 231, 492, 376, 505 ], "score": 1.0, "content": "is the Reynolds number. 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Our", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 514, 506, 527 ], "spans": [ { "bbox": [ 104, 514, 506, 527 ], "score": 1.0, "content": "proposed method can obtain a result similar to that in Wang et al. (2022a), while the training time is", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 524, 505, 538 ], "spans": [ { "bbox": [ 105, 524, 505, 538 ], "score": 1.0, "content": "only 1/58 of their method. More computational results of the NS equation are provided in Appendix", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 536, 134, 548 ], "spans": [ { "bbox": [ 105, 536, 134, 548 ], "score": 1.0, "content": "A.4.6.", "type": "text" } ], "index": 20 } ], "index": 17.5 }, { "type": "title", "bbox": [ 108, 562, 208, 573 ], "lines": [ { "bbox": [ 105, 561, 209, 575 ], "spans": [ { "bbox": [ 105, 561, 209, 575 ], "score": 1.0, "content": "5.5 ABLATION STUDY", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 108, 583, 504, 605 ], "lines": [ { "bbox": [ 105, 581, 505, 596 ], "spans": [ { "bbox": [ 105, 581, 505, 596 ], "score": 1.0, "content": "We conduct ablation studies on the relatively simpler RD Eq. and AC Eq. to ablate the main designs", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 594, 176, 606 ], "spans": [ { "bbox": [ 106, 594, 176, 606 ], "score": 1.0, "content": "in our algorithm.", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "text", "bbox": [ 107, 618, 505, 663 ], "lines": [ { "bbox": [ 106, 619, 505, 631 ], "spans": [ { "bbox": [ 106, 619, 505, 631 ], "score": 1.0, "content": "Time differencing scheme study. Numerous time differencing schemes have been developed in the", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 630, 506, 643 ], "spans": [ { "bbox": [ 105, 630, 506, 643 ], "score": 1.0, "content": "last decades. We list some commonly used schemes in Appendix A.2. We do experiments on different", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 641, 506, 653 ], "spans": [ { "bbox": [ 105, 641, 506, 653 ], "score": 1.0, "content": "time differencing schemes to validate that implicit time differencing schemes (2nd Crank-Nicolson or", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 652, 506, 664 ], "spans": [ { "bbox": [ 105, 652, 506, 664 ], "score": 1.0, "content": "4th Gauss-Legendre) are more stable and lead to better performance. The results are given in Table 2.", "type": "text" } ], "index": 27 } ], "index": 25.5 }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 106, 676, 505, 690 ], "spans": [ { "bbox": [ 106, 676, 505, 690 ], "score": 1.0, "content": "Transfer learning study. To see weather the transfer learning part is effective, we do ablation", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 505, 701 ], "score": 1.0, "content": "studies without using transfer learning. Besides, since our strategy of transfer learning is to fine tune", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 699, 506, 710 ], "spans": [ { "bbox": [ 105, 699, 506, 710 ], "score": 1.0, "content": "all the network parameters, we also do experiments to fine tune the last 1/2/3 layers of the network.", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 707, 506, 723 ], "spans": [ { "bbox": [ 105, 707, 506, 723 ], "score": 1.0, "content": "The results are given in Table 3. 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In addition,", "type": "text" }, { "bbox": [ 377, 492, 385, 502 ], "score": 0.82, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 385, 492, 419, 505 ], "score": 1.0, "content": "is set to", "type": "text" }, { "bbox": [ 419, 492, 450, 504 ], "score": 0.93, "content": "[ 0 , 2 \\pi ] ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 451, 492, 506, 505 ], "score": 1.0, "content": "and Re is set", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 503, 506, 516 ], "spans": [ { "bbox": [ 105, 503, 305, 516 ], "score": 1.0, "content": "to 100. Figure 5 presents the predicted solution of", "type": "text" }, { "bbox": [ 305, 503, 344, 515 ], "score": 0.92, "content": "w ( t , x , y )", "type": "inline_equation" }, { "bbox": [ 345, 503, 506, 516 ], "score": 1.0, "content": "compared to the reference solution. Our", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 514, 506, 527 ], "spans": [ { "bbox": [ 104, 514, 506, 527 ], "score": 1.0, "content": "proposed method can obtain a result similar to that in Wang et al. (2022a), while the training time is", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 524, 505, 538 ], "spans": [ { "bbox": [ 105, 524, 505, 538 ], "score": 1.0, "content": "only 1/58 of their method. More computational results of the NS equation are provided in Appendix", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 536, 134, 548 ], "spans": [ { "bbox": [ 105, 536, 134, 548 ], "score": 1.0, "content": "A.4.6.", "type": "text" } ], "index": 20 } ], "index": 17.5, "bbox_fs": [ 104, 480, 506, 548 ] }, { "type": "title", "bbox": [ 108, 562, 208, 573 ], "lines": [ { "bbox": [ 105, 561, 209, 575 ], "spans": [ { "bbox": [ 105, 561, 209, 575 ], "score": 1.0, "content": "5.5 ABLATION STUDY", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 108, 583, 504, 605 ], "lines": [ { "bbox": [ 105, 581, 505, 596 ], "spans": [ { "bbox": [ 105, 581, 505, 596 ], "score": 1.0, "content": "We conduct ablation studies on the relatively simpler RD Eq. and AC Eq. to ablate the main designs", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 594, 176, 606 ], "spans": [ { "bbox": [ 106, 594, 176, 606 ], "score": 1.0, "content": "in our algorithm.", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 105, 581, 505, 606 ] }, { "type": "text", "bbox": [ 107, 618, 505, 663 ], "lines": [ { "bbox": [ 106, 619, 505, 631 ], "spans": [ { "bbox": [ 106, 619, 505, 631 ], "score": 1.0, "content": "Time differencing scheme study. Numerous time differencing schemes have been developed in the", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 630, 506, 643 ], "spans": [ { "bbox": [ 105, 630, 506, 643 ], "score": 1.0, "content": "last decades. We list some commonly used schemes in Appendix A.2. We do experiments on different", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 641, 506, 653 ], "spans": [ { "bbox": [ 105, 641, 506, 653 ], "score": 1.0, "content": "time differencing schemes to validate that implicit time differencing schemes (2nd Crank-Nicolson or", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 652, 506, 664 ], "spans": [ { "bbox": [ 105, 652, 506, 664 ], "score": 1.0, "content": "4th Gauss-Legendre) are more stable and lead to better performance. The results are given in Table 2.", "type": "text" } ], "index": 27 } ], "index": 25.5, "bbox_fs": [ 105, 619, 506, 664 ] }, { "type": "text", "bbox": [ 107, 676, 505, 732 ], "lines": [ { "bbox": [ 106, 676, 505, 690 ], "spans": [ { "bbox": [ 106, 676, 505, 690 ], "score": 1.0, "content": "Transfer learning study. To see weather the transfer learning part is effective, we do ablation", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 505, 701 ], "score": 1.0, "content": "studies without using transfer learning. Besides, since our strategy of transfer learning is to fine tune", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 699, 506, 710 ], "spans": [ { "bbox": [ 105, 699, 506, 710 ], "score": 1.0, "content": "all the network parameters, we also do experiments to fine tune the last 1/2/3 layers of the network.", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 707, 506, 723 ], "spans": [ { "bbox": [ 105, 707, 506, 723 ], "score": 1.0, "content": "The results are given in Table 3. We can see that transfer learning is effective both in the efficiency", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 721, 222, 732 ], "spans": [ { "bbox": [ 106, 721, 222, 732 ], "score": 1.0, "content": "and accuracy of our method.", "type": "text" } ], "index": 32 } ], "index": 30, "bbox_fs": [ 105, 676, 506, 732 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 110, 101, 300, 187 ], "blocks": [ { "type": "table_caption", "bbox": [ 122, 80, 286, 92 ], "group_id": 0, "lines": [ { "bbox": [ 120, 77, 287, 95 ], "spans": [ { "bbox": [ 120, 77, 287, 95 ], "score": 1.0, "content": "Table 2: Time differencing scheme study", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 110, 101, 300, 187 ], "group_id": 0, "lines": [ { "bbox": [ 110, 101, 300, 187 ], "spans": [ { "bbox": [ 110, 101, 300, 187 ], "score": 0.979, "html": "
MethodL2RDEqimeL2 AC Eqtime
Forward Euler1.32e-032089.57e-03304
Backward Euler2.74e-032061.64e-02444
2nd RK1.97e-037611.17e-031054
4th RK2.11e-0311871.31e-031779
TL-DPINN11.82e-0514635.92e-042328
TL-DPINN29.34e-057489.82e-041100
", "type": "table", "image_path": "59abcc302b028ddbae40a10ac119fcbb80c25fdff03ea1526ad429cd1b831aed.jpg" } ] } ], "index": 4.5, "virtual_lines": [ { "bbox": [ 110, 101, 300, 115.33333333333333 ], "spans": [], "index": 2 }, { "bbox": [ 110, 115.33333333333333, 300, 129.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 110, 129.66666666666666, 300, 144.0 ], "spans": [], "index": 4 }, { "bbox": [ 110, 144.0, 300, 158.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 110, 158.33333333333334, 300, 172.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 110, 172.66666666666669, 300, 187.00000000000003 ], "spans": [], "index": 7 } ] } ], "index": 2.25 }, { "type": "table", "bbox": [ 312, 101, 493, 187 ], "blocks": [ { "type": "table_caption", "bbox": [ 342, 81, 471, 92 ], "group_id": 2, "lines": [ { "bbox": [ 340, 78, 472, 94 ], "spans": [ { "bbox": [ 340, 78, 472, 94 ], "score": 1.0, "content": "Table 3: Transfer learning study", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "table_body", "bbox": [ 312, 101, 493, 187 ], "group_id": 2, "lines": [ { "bbox": [ 312, 101, 493, 187 ], "spans": [ { "bbox": [ 312, 101, 493, 187 ], "score": 0.978, "html": "
MethodL2RDEqtimeL AC Eqime
Without TL4.01e-0458801.35e-029170
last layer3.31e-046381.01e-023624
last 2 layers3.22e-042211.01e-024029
last 3 layers4.08e-042321.01e-024685
TL-DPINN11.82e-0514635.92e-042328
TL-DPINN29.34e-057489.82e-041100
", "type": "table", "image_path": "37702a38d5de5e70ea046c834665cc2d2944027117c191c6d0def9a8e4c561c3.jpg" } ] } ], "index": 10.5, "virtual_lines": [ { "bbox": [ 312, 101, 493, 115.33333333333333 ], "spans": [], "index": 8 }, { "bbox": [ 312, 115.33333333333333, 493, 129.66666666666666 ], "spans": [], "index": 9 }, { "bbox": [ 312, 129.66666666666666, 493, 144.0 ], "spans": [], "index": 10 }, { "bbox": [ 312, 144.0, 493, 158.33333333333334 ], "spans": [], "index": 11 }, { "bbox": [ 312, 158.33333333333334, 493, 172.66666666666669 ], "spans": [], "index": 12 }, { "bbox": [ 312, 172.66666666666669, 493, 187.00000000000003 ], "spans": [], "index": 13 } ] } ], "index": 5.75 }, { "type": "text", "bbox": [ 107, 210, 506, 244 ], "lines": [ { "bbox": [ 106, 210, 505, 223 ], "spans": [ { "bbox": [ 106, 210, 505, 223 ], "score": 1.0, "content": "Repeated test. To further demonstrate the well-performance of our TL-DPINN method through", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 222, 505, 234 ], "spans": [ { "bbox": [ 106, 222, 505, 234 ], "score": 1.0, "content": "accuracy and efficiency, we do 5 random runs for RD and AC Eq. by casual PINN and our method", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 233, 303, 245 ], "spans": [ { "bbox": [ 106, 233, 303, 245 ], "score": 1.0, "content": "for comparison. The results are given in Table 4.", "type": "text" } ], "index": 16 } ], "index": 15 }, { "type": "table", "bbox": [ 121, 278, 490, 348 ], "blocks": [ { "type": "table_caption", "bbox": [ 259, 258, 351, 270 ], "group_id": 1, "lines": [ { "bbox": [ 258, 256, 352, 271 ], "spans": [ { "bbox": [ 258, 256, 352, 271 ], "score": 1.0, "content": "Table 4: Repeated test.", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "table_body", "bbox": [ 121, 278, 490, 348 ], "group_id": 1, "lines": [ { "bbox": [ 121, 278, 490, 348 ], "spans": [ { "bbox": [ 121, 278, 490, 348 ], "score": 0.982, "html": "
MethodL2 rrRD Eq.L² rrAC Eq.
timetime
Causal PINN3.73e-05 ± 4.66e-067207± 2191.51e-03 ± 2.12e-049060± 341
TL-DPINN11.76e-05 ± 1.06e-061463 ± 536.08e-04 ± 3.06e-052328±89
TL-DPINN29.89e-05 ± 8.94e-06811 ± 1229.29e-04 ± 8.06e-051291 ± 178
", "type": "table", "image_path": "a449cb94e098ba65aab6718353ed992c57201d4b3b5046b8687c360466d72230.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 121, 278, 490, 301.3333333333333 ], "spans": [], "index": 18 }, { "bbox": [ 121, 301.3333333333333, 490, 324.66666666666663 ], "spans": [], "index": 19 }, { "bbox": [ 121, 324.66666666666663, 490, 347.99999999999994 ], "spans": [], "index": 20 } ] } ], "index": 18.0 }, { "type": "title", "bbox": [ 107, 374, 227, 385 ], "lines": [ { "bbox": [ 105, 373, 229, 387 ], "spans": [ { "bbox": [ 105, 373, 229, 387 ], "score": 1.0, "content": "5.6 TRAINING EFFICIENCY", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 106, 396, 505, 451 ], "lines": [ { "bbox": [ 106, 396, 505, 408 ], "spans": [ { "bbox": [ 106, 396, 505, 408 ], "score": 1.0, "content": "Table 5 illustrates how the computation efficiency is affected by different time discretization methods", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 407, 505, 419 ], "spans": [ { "bbox": [ 106, 407, 505, 419 ], "score": 1.0, "content": "on different equations. In addition, the casual PINN method is also compared. All neural networks", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 417, 505, 431 ], "spans": [ { "bbox": [ 105, 417, 505, 431 ], "score": 1.0, "content": "are trained on an NVIDIA GeForce RTX 3080 Ti graphics card. We note that the total training epochs", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 427, 506, 443 ], "spans": [ { "bbox": [ 104, 427, 506, 443 ], "score": 1.0, "content": "of our methods are not fixed due to the stopping criterion (see Algorithm 1). The training efficiency", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 440, 334, 452 ], "spans": [ { "bbox": [ 106, 440, 334, 452 ], "score": 1.0, "content": "in Table 5 is consistent with the training time in Table 1.", "type": "text" } ], "index": 26 } ], "index": 24 }, { "type": "table", "bbox": [ 113, 486, 498, 555 ], "blocks": [ { "type": "table_caption", "bbox": [ 168, 465, 441, 477 ], "group_id": 3, "lines": [ { "bbox": [ 168, 464, 442, 479 ], "spans": [ { "bbox": [ 168, 464, 442, 479 ], "score": 1.0, "content": "Table 5: A comparison of training efficiency for different equations.", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "table_body", "bbox": [ 113, 486, 498, 555 ], "group_id": 3, "lines": [ { "bbox": [ 113, 486, 498, 555 ], "spans": [ { "bbox": [ 113, 486, 498, 555 ], "score": 0.981, "html": "
Methodrn
Reaction-DiffusionNavier-Stokes
Casual PINN61.7052.3326.242.77
TL-DPINN1439.37384.47259.208.32
TL-DPINN2276.40239.52127.556.37
", "type": "table", "image_path": "51668a27a9c7b5fb86ddb3016897af68a98e4231803408d950a145e55a286486.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 113, 486, 498, 509.0 ], "spans": [], "index": 28 }, { "bbox": [ 113, 509.0, 498, 532.0 ], "spans": [], "index": 29 }, { "bbox": [ 113, 532.0, 498, 555.0 ], "spans": [], "index": 30 } ] } ], "index": 28.0 }, { "type": "title", "bbox": [ 107, 583, 195, 596 ], "lines": [ { "bbox": [ 105, 581, 198, 599 ], "spans": [ { "bbox": [ 105, 581, 198, 599 ], "score": 1.0, "content": "6 CONCLUSION", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 106, 610, 506, 732 ], "lines": [ { "bbox": [ 105, 610, 506, 624 ], "spans": [ { "bbox": [ 105, 610, 506, 624 ], "score": 1.0, "content": "In this paper, we propose a method for solving evolutionary partial differential equations via transfer-", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 622, 506, 635 ], "spans": [ { "bbox": [ 105, 622, 506, 635 ], "score": 1.0, "content": "learning enhanced discrete physics-informed neural networks (TL-DPINN). The discrete PINNs were", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 632, 506, 646 ], "spans": [ { "bbox": [ 105, 632, 506, 646 ], "score": 1.0, "content": "thought to be time-consuming and seldom applied in the PINNs literature. We contribute to the PINN", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 643, 506, 657 ], "spans": [ { "bbox": [ 105, 643, 506, 657 ], "score": 1.0, "content": "community by rediscovering the good performance of the discrete PINNs applied to solve evolutionary", "type": "text" } ], "index": 35 }, { "bbox": [ 104, 654, 506, 668 ], "spans": [ { "bbox": [ 104, 654, 506, 668 ], "score": 1.0, "content": "PDEs, both theoretically and numerically. Our method first employs a classical numerical implicit", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 666, 506, 679 ], "spans": [ { "bbox": [ 105, 666, 506, 679 ], "score": 1.0, "content": "time differencing scheme to produce a series of stable propagation equations in space, and then applies", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "PINN approximation to sequentially solve. Transfer learning is used to reduce computational costs", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 687, 506, 701 ], "spans": [ { "bbox": [ 105, 687, 506, 701 ], "score": 1.0, "content": "while maintaining precision. We demonstrate the convergence property, accuracy, and computational", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "effectiveness of our TL-DPINN method both theoretically and numerically. Our proposed method", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 506, 723 ], "score": 1.0, "content": "achieves state-of-the-art results among different PINN frameworks while significantly reducing the", "type": "text" } ], "index": 41 }, { "bbox": [ 106, 721, 187, 733 ], "spans": [ { "bbox": [ 106, 721, 187, 733 ], "score": 1.0, "content": "computational cost.", "type": "text" } ], "index": 42 } ], "index": 37 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 25, 308, 38 ], "spans": [ { "bbox": [ 107, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 759 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "9", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 110, 101, 300, 187 ], "blocks": [ { "type": "table_caption", "bbox": [ 122, 80, 286, 92 ], "group_id": 0, "lines": [ { "bbox": [ 120, 77, 287, 95 ], "spans": [ { "bbox": [ 120, 77, 287, 95 ], "score": 1.0, "content": "Table 2: Time differencing scheme study", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 110, 101, 300, 187 ], "group_id": 0, "lines": [ { "bbox": [ 110, 101, 300, 187 ], "spans": [ { "bbox": [ 110, 101, 300, 187 ], "score": 0.979, "html": "
MethodL2RDEqimeL2 AC Eqtime
Forward Euler1.32e-032089.57e-03304
Backward Euler2.74e-032061.64e-02444
2nd RK1.97e-037611.17e-031054
4th RK2.11e-0311871.31e-031779
TL-DPINN11.82e-0514635.92e-042328
TL-DPINN29.34e-057489.82e-041100
", "type": "table", "image_path": "59abcc302b028ddbae40a10ac119fcbb80c25fdff03ea1526ad429cd1b831aed.jpg" } ] } ], "index": 4.5, "virtual_lines": [ { "bbox": [ 110, 101, 300, 115.33333333333333 ], "spans": [], "index": 2 }, { "bbox": [ 110, 115.33333333333333, 300, 129.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 110, 129.66666666666666, 300, 144.0 ], "spans": [], "index": 4 }, { "bbox": [ 110, 144.0, 300, 158.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 110, 158.33333333333334, 300, 172.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 110, 172.66666666666669, 300, 187.00000000000003 ], "spans": [], "index": 7 } ] } ], "index": 2.25 }, { "type": "table", "bbox": [ 312, 101, 493, 187 ], "blocks": [ { "type": "table_caption", "bbox": [ 342, 81, 471, 92 ], "group_id": 2, "lines": [ { "bbox": [ 340, 78, 472, 94 ], "spans": [ { "bbox": [ 340, 78, 472, 94 ], "score": 1.0, "content": "Table 3: Transfer learning study", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "table_body", "bbox": [ 312, 101, 493, 187 ], "group_id": 2, "lines": [ { "bbox": [ 312, 101, 493, 187 ], "spans": [ { "bbox": [ 312, 101, 493, 187 ], "score": 0.978, "html": "
MethodL2RDEqtimeL AC Eqime
Without TL4.01e-0458801.35e-029170
last layer3.31e-046381.01e-023624
last 2 layers3.22e-042211.01e-024029
last 3 layers4.08e-042321.01e-024685
TL-DPINN11.82e-0514635.92e-042328
TL-DPINN29.34e-057489.82e-041100
", "type": "table", "image_path": "37702a38d5de5e70ea046c834665cc2d2944027117c191c6d0def9a8e4c561c3.jpg" } ] } ], "index": 10.5, "virtual_lines": [ { "bbox": [ 312, 101, 493, 115.33333333333333 ], "spans": [], "index": 8 }, { "bbox": [ 312, 115.33333333333333, 493, 129.66666666666666 ], "spans": [], "index": 9 }, { "bbox": [ 312, 129.66666666666666, 493, 144.0 ], "spans": [], "index": 10 }, { "bbox": [ 312, 144.0, 493, 158.33333333333334 ], "spans": [], "index": 11 }, { "bbox": [ 312, 158.33333333333334, 493, 172.66666666666669 ], "spans": [], "index": 12 }, { "bbox": [ 312, 172.66666666666669, 493, 187.00000000000003 ], "spans": [], "index": 13 } ] } ], "index": 5.75 }, { "type": "text", "bbox": [ 107, 210, 506, 244 ], "lines": [ { "bbox": [ 106, 210, 505, 223 ], "spans": [ { "bbox": [ 106, 210, 505, 223 ], "score": 1.0, "content": "Repeated test. To further demonstrate the well-performance of our TL-DPINN method through", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 222, 505, 234 ], "spans": [ { "bbox": [ 106, 222, 505, 234 ], "score": 1.0, "content": "accuracy and efficiency, we do 5 random runs for RD and AC Eq. by casual PINN and our method", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 233, 303, 245 ], "spans": [ { "bbox": [ 106, 233, 303, 245 ], "score": 1.0, "content": "for comparison. The results are given in Table 4.", "type": "text" } ], "index": 16 } ], "index": 15, "bbox_fs": [ 106, 210, 505, 245 ] }, { "type": "table", "bbox": [ 121, 278, 490, 348 ], "blocks": [ { "type": "table_caption", "bbox": [ 259, 258, 351, 270 ], "group_id": 1, "lines": [ { "bbox": [ 258, 256, 352, 271 ], "spans": [ { "bbox": [ 258, 256, 352, 271 ], "score": 1.0, "content": "Table 4: Repeated test.", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "table_body", "bbox": [ 121, 278, 490, 348 ], "group_id": 1, "lines": [ { "bbox": [ 121, 278, 490, 348 ], "spans": [ { "bbox": [ 121, 278, 490, 348 ], "score": 0.982, "html": "
MethodL2 rrRD Eq.L² rrAC Eq.
timetime
Causal PINN3.73e-05 ± 4.66e-067207± 2191.51e-03 ± 2.12e-049060± 341
TL-DPINN11.76e-05 ± 1.06e-061463 ± 536.08e-04 ± 3.06e-052328±89
TL-DPINN29.89e-05 ± 8.94e-06811 ± 1229.29e-04 ± 8.06e-051291 ± 178
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Methodrn
Reaction-DiffusionNavier-Stokes
Casual PINN61.7052.3326.242.77
TL-DPINN1439.37384.47259.208.32
TL-DPINN2276.40239.52127.556.37
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The discrete PINNs were", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 632, 506, 646 ], "spans": [ { "bbox": [ 105, 632, 506, 646 ], "score": 1.0, "content": "thought to be time-consuming and seldom applied in the PINNs literature. We contribute to the PINN", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 643, 506, 657 ], "spans": [ { "bbox": [ 105, 643, 506, 657 ], "score": 1.0, "content": "community by rediscovering the good performance of the discrete PINNs applied to solve evolutionary", "type": "text" } ], "index": 35 }, { "bbox": [ 104, 654, 506, 668 ], "spans": [ { "bbox": [ 104, 654, 506, 668 ], "score": 1.0, "content": "PDEs, both theoretically and numerically. Our method first employs a classical numerical implicit", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 666, 506, 679 ], "spans": [ { "bbox": [ 105, 666, 506, 679 ], "score": 1.0, "content": "time differencing scheme to produce a series of stable propagation equations in space, and then applies", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 676, 505, 690 ], "spans": [ { "bbox": [ 105, 676, 505, 690 ], "score": 1.0, "content": "PINN approximation to sequentially solve. Transfer learning is used to reduce computational costs", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 687, 506, 701 ], "spans": [ { "bbox": [ 105, 687, 506, 701 ], "score": 1.0, "content": "while maintaining precision. We demonstrate the convergence property, accuracy, and computational", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 698, 506, 712 ], "spans": [ { "bbox": [ 105, 698, 506, 712 ], "score": 1.0, "content": "effectiveness of our TL-DPINN method both theoretically and numerically. 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URL", "type": "text" } ], "index": 4 }, { "bbox": [ 114, 151, 291, 168 ], "spans": [ { "bbox": [ 114, 151, 291, 168 ], "score": 1.0, "content": "http://github.com/google/jax.", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 171, 385, 185 ], "spans": [ { "bbox": [ 105, 171, 385, 185 ], "score": 1.0, "content": "John Butcher. Runge-Kutta methods. Scholarpedia, 2(9):3147, 2007.", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 190, 508, 205 ], "spans": [ { "bbox": [ 104, 190, 508, 205 ], "score": 1.0, "content": "Xinhai Chen, Chunye Gong, Qian Wan, Liang Deng, Yunbo Wan, Yang Liu, Bo Chen, and Jie Liu.", "type": "text" } ], "index": 7 }, { "bbox": [ 115, 202, 506, 215 ], "spans": [ { "bbox": [ 115, 202, 506, 215 ], "score": 1.0, "content": "Transfer learning for deep neural network-based partial differential equations solving. Advances in", "type": "text" } ], "index": 8 }, { "bbox": [ 114, 212, 248, 226 ], "spans": [ { "bbox": [ 114, 212, 248, 226 ], "score": 1.0, "content": "Aerodynamics, 3(1):1–14, 2021.", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 230, 506, 247 ], "spans": [ { "bbox": [ 104, 230, 506, 247 ], "score": 1.0, "content": "Constantine M Dafermos and Milan Pokorny. Handbook of differential equations: evolutionary", "type": "text" } ], "index": 10 }, { "bbox": [ 114, 243, 223, 256 ], "spans": [ { "bbox": [ 114, 243, 223, 256 ], "score": 1.0, "content": "equations. Elsevier, 2008.", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 260, 507, 276 ], "spans": [ { "bbox": [ 105, 260, 507, 276 ], "score": 1.0, "content": "Shaan Desai, Marios Mattheakis, Hayden Joy, Pavlos Protopapas, and Stephen J Roberts. One-shot", "type": "text" } ], "index": 12 }, { "bbox": [ 114, 270, 508, 288 ], "spans": [ { "bbox": [ 114, 270, 508, 288 ], "score": 1.0, "content": "transfer learning of physics-informed neural networks. In ICML 2022 2nd AI for Science Workshop,", "type": "text" } ], "index": 13 }, { "bbox": [ 115, 280, 143, 299 ], "spans": [ { "bbox": [ 115, 280, 143, 299 ], "score": 1.0, "content": "2022.", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 304, 506, 315 ], "spans": [ { "bbox": [ 106, 304, 506, 315 ], "score": 1.0, "content": "Lawrence C Evans. Partial differential equations, volume 19. American Mathematical Society, 2022.", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 321, 506, 335 ], "spans": [ { "bbox": [ 105, 321, 506, 335 ], "score": 1.0, "content": "Somdatta Goswami, Cosmin Anitescu, Souvik Chakraborty, and Timon Rabczuk. Transfer learning", "type": "text" } ], "index": 16 }, { "bbox": [ 115, 333, 506, 346 ], "spans": [ { "bbox": [ 115, 333, 506, 346 ], "score": 1.0, "content": "enhanced physics informed neural network for phase-field modeling of fracture. Theoretical and", "type": "text" } ], "index": 17 }, { "bbox": [ 114, 344, 312, 357 ], "spans": [ { "bbox": [ 114, 344, 312, 357 ], "score": 1.0, "content": "Applied Fracture Mechanics, 106:102447, 2020.", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 361, 506, 377 ], "spans": [ { "bbox": [ 104, 361, 506, 377 ], "score": 1.0, "content": "Somdatta Goswami, Katiana Kontolati, Michael D Shields, and George Em Karniadakis. Deep", "type": "text" } ], "index": 19 }, { "bbox": [ 115, 374, 506, 387 ], "spans": [ { "bbox": [ 115, 374, 506, 387 ], "score": 1.0, "content": "transfer operator learning for partial differential equations under conditional shift. Nature Machine", "type": "text" } ], "index": 20 }, { "bbox": [ 114, 384, 234, 399 ], "spans": [ { "bbox": [ 114, 384, 234, 399 ], "score": 1.0, "content": "Intelligence, pp. 1–10, 2022.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 404, 506, 417 ], "spans": [ { "bbox": [ 105, 404, 506, 417 ], "score": 1.0, "content": "Jiequn Han, Arnulf Jentzen, and Weinan E. Solving high-dimensional partial differential equations", "type": "text" } ], "index": 22 }, { "bbox": [ 114, 414, 508, 429 ], "spans": [ { "bbox": [ 114, 414, 508, 429 ], "score": 1.0, "content": "using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 433, 507, 447 ], "spans": [ { "bbox": [ 105, 433, 507, 447 ], "score": 1.0, "content": "Ameya D Jagtap and George E Karniadakis. Extended Physics-informed Neural Networks (XPINNs):", "type": "text" } ], "index": 24 }, { "bbox": [ 115, 445, 506, 459 ], "spans": [ { "bbox": [ 115, 445, 506, 459 ], "score": 1.0, "content": "A generalized space-time domain decomposition based deep learning framework for nonlinear", "type": "text" } ], "index": 25 }, { "bbox": [ 114, 455, 470, 469 ], "spans": [ { "bbox": [ 114, 455, 470, 469 ], "score": 1.0, "content": "partial differential equations. In AAAI Spring Symposium: MLPS, pp. 2002–2041, 2021.", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 474, 506, 488 ], "spans": [ { "bbox": [ 106, 474, 506, 488 ], "score": 1.0, "content": "Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conservative physics-informed", "type": "text" } ], "index": 27 }, { "bbox": [ 115, 485, 505, 498 ], "spans": [ { "bbox": [ 115, 485, 505, 498 ], "score": 1.0, "content": "neural networks on discrete domains for conservation laws: Applications to forward and inverse", "type": "text" } ], "index": 28 }, { "bbox": [ 114, 496, 478, 511 ], "spans": [ { "bbox": [ 114, 496, 478, 511 ], "score": 1.0, "content": "problems. Computer Methods in Applied Mechanics and Engineering, 365:113028, 2020.", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 514, 508, 530 ], "spans": [ { "bbox": [ 105, 514, 508, 530 ], "score": 1.0, "content": "Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis. hp-VPINNs: Variational physics-", "type": "text" } ], "index": 30 }, { "bbox": [ 115, 527, 506, 540 ], "spans": [ { "bbox": [ 115, 527, 506, 540 ], "score": 1.0, "content": "informed neural networks with domain decomposition. Computer Methods in Applied Mechanics", "type": "text" } ], "index": 31 }, { "bbox": [ 115, 537, 266, 551 ], "spans": [ { "bbox": [ 115, 537, 266, 551 ], "score": 1.0, "content": "and Engineering, 374:113547, 2021.", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 556, 506, 572 ], "spans": [ { "bbox": [ 106, 556, 506, 572 ], "score": 1.0, "content": "Yuehaw Khoo, Jianfeng Lu, and Lexing Ying. Solving parametric PDE problems with artificial neural", "type": "text" } ], "index": 33 }, { "bbox": [ 115, 568, 423, 581 ], "spans": [ { "bbox": [ 115, 568, 423, 581 ], "score": 1.0, "content": "networks. European Journal of Applied Mathematics, 32(3):421–435, 2021.", "type": "text" } ], "index": 34 }, { "bbox": [ 104, 584, 506, 601 ], "spans": [ { "bbox": [ 104, 584, 506, 601 ], "score": 1.0, "content": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint", "type": "text" } ], "index": 35 }, { "bbox": [ 114, 597, 215, 610 ], "spans": [ { "bbox": [ 114, 597, 215, 610 ], "score": 1.0, "content": "arXiv:1412.6980, 2014.", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 616, 506, 630 ], "spans": [ { "bbox": [ 106, 616, 506, 630 ], "score": 1.0, "content": "Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Char-", "type": "text" } ], "index": 37 }, { "bbox": [ 115, 628, 506, 641 ], "spans": [ { "bbox": [ 115, 628, 506, 641 ], "score": 1.0, "content": "acterizing possible failure modes in physics-informed neural networks. Advances in Neural", "type": "text" } ], "index": 38 }, { "bbox": [ 114, 638, 347, 653 ], "spans": [ { "bbox": [ 114, 638, 347, 653 ], "score": 1.0, "content": "Information Processing Systems, 34:26548–26560, 2021.", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 657, 506, 671 ], "spans": [ { "bbox": [ 105, 657, 506, 671 ], "score": 1.0, "content": "Colby L. Wight and Jia Zhao. Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive", "type": "text" } ], "index": 40 }, { "bbox": [ 115, 668, 506, 682 ], "spans": [ { "bbox": [ 115, 668, 506, 682 ], "score": 1.0, "content": "physics informed neural networks. Communications in Computational Physics, 29(3):930–954,", "type": "text" } ], "index": 41 }, { "bbox": [ 115, 677, 145, 694 ], "spans": [ { "bbox": [ 115, 677, 145, 694 ], "score": 1.0, "content": "2021.", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 698, 508, 713 ], "spans": [ { "bbox": [ 105, 698, 508, 713 ], "score": 1.0, "content": "Songming Liu, Hao Zhongkai, Chengyang Ying, Hang Su, Jun Zhu, and Ze Cheng. A unified hard-", "type": "text" } ], "index": 43 }, { "bbox": [ 115, 710, 506, 723 ], "spans": [ { "bbox": [ 115, 710, 506, 723 ], "score": 1.0, "content": "constraint framework for solving geometrically complex PDEs. 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One-shot", "type": "text" } ], "index": 12, "is_list_start_line": true }, { "bbox": [ 114, 270, 508, 288 ], "spans": [ { "bbox": [ 114, 270, 508, 288 ], "score": 1.0, "content": "transfer learning of physics-informed neural networks. In ICML 2022 2nd AI for Science Workshop,", "type": "text" } ], "index": 13 }, { "bbox": [ 115, 280, 143, 299 ], "spans": [ { "bbox": [ 115, 280, 143, 299 ], "score": 1.0, "content": "2022.", "type": "text" } ], "index": 14, "is_list_end_line": true }, { "bbox": [ 106, 304, 506, 315 ], "spans": [ { "bbox": [ 106, 304, 506, 315 ], "score": 1.0, "content": "Lawrence C Evans. Partial differential equations, volume 19. American Mathematical Society, 2022.", "type": "text" } ], "index": 15, "is_list_start_line": true }, { "bbox": [ 105, 321, 506, 335 ], "spans": [ { "bbox": [ 105, 321, 506, 335 ], "score": 1.0, "content": "Somdatta Goswami, Cosmin Anitescu, Souvik Chakraborty, and Timon Rabczuk. Transfer learning", "type": "text" } ], "index": 16, "is_list_start_line": true }, { "bbox": [ 115, 333, 506, 346 ], "spans": [ { "bbox": [ 115, 333, 506, 346 ], "score": 1.0, "content": "enhanced physics informed neural network for phase-field modeling of fracture. Theoretical and", "type": "text" } ], "index": 17 }, { "bbox": [ 114, 344, 312, 357 ], "spans": [ { "bbox": [ 114, 344, 312, 357 ], "score": 1.0, "content": "Applied Fracture Mechanics, 106:102447, 2020.", "type": "text" } ], "index": 18, "is_list_end_line": true }, { "bbox": [ 104, 361, 506, 377 ], "spans": [ { "bbox": [ 104, 361, 506, 377 ], "score": 1.0, "content": "Somdatta Goswami, Katiana Kontolati, Michael D Shields, and George Em Karniadakis. Deep", "type": "text" } ], "index": 19, "is_list_start_line": true }, { "bbox": [ 115, 374, 506, 387 ], "spans": [ { "bbox": [ 115, 374, 506, 387 ], "score": 1.0, "content": "transfer operator learning for partial differential equations under conditional shift. Nature Machine", "type": "text" } ], "index": 20 }, { "bbox": [ 114, 384, 234, 399 ], "spans": [ { "bbox": [ 114, 384, 234, 399 ], "score": 1.0, "content": "Intelligence, pp. 1–10, 2022.", "type": "text" } ], "index": 21, "is_list_end_line": true }, { "bbox": [ 105, 404, 506, 417 ], "spans": [ { "bbox": [ 105, 404, 506, 417 ], "score": 1.0, "content": "Jiequn Han, Arnulf Jentzen, and Weinan E. Solving high-dimensional partial differential equations", "type": "text" } ], "index": 22, "is_list_start_line": true }, { "bbox": [ 114, 414, 508, 429 ], "spans": [ { "bbox": [ 114, 414, 508, 429 ], "score": 1.0, "content": "using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 433, 507, 447 ], "spans": [ { "bbox": [ 105, 433, 507, 447 ], "score": 1.0, "content": "Ameya D Jagtap and George E Karniadakis. Extended Physics-informed Neural Networks (XPINNs):", "type": "text" } ], "index": 24, "is_list_start_line": true }, { "bbox": [ 115, 445, 506, 459 ], "spans": [ { "bbox": [ 115, 445, 506, 459 ], "score": 1.0, "content": "A generalized space-time domain decomposition based deep learning framework for nonlinear", "type": "text" } ], "index": 25 }, { "bbox": [ 114, 455, 470, 469 ], "spans": [ { "bbox": [ 114, 455, 470, 469 ], "score": 1.0, "content": "partial differential equations. In AAAI Spring Symposium: MLPS, pp. 2002–2041, 2021.", "type": "text" } ], "index": 26, "is_list_end_line": true }, { "bbox": [ 106, 474, 506, 488 ], "spans": [ { "bbox": [ 106, 474, 506, 488 ], "score": 1.0, "content": "Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conservative physics-informed", "type": "text" } ], "index": 27, "is_list_start_line": true }, { "bbox": [ 115, 485, 505, 498 ], "spans": [ { "bbox": [ 115, 485, 505, 498 ], "score": 1.0, "content": "neural networks on discrete domains for conservation laws: Applications to forward and inverse", "type": "text" } ], "index": 28 }, { "bbox": [ 114, 496, 478, 511 ], "spans": [ { "bbox": [ 114, 496, 478, 511 ], "score": 1.0, "content": "problems. Computer Methods in Applied Mechanics and Engineering, 365:113028, 2020.", "type": "text" } ], "index": 29, "is_list_end_line": true }, { "bbox": [ 105, 514, 508, 530 ], "spans": [ { "bbox": [ 105, 514, 508, 530 ], "score": 1.0, "content": "Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis. hp-VPINNs: Variational physics-", "type": "text" } ], "index": 30, "is_list_start_line": true }, { "bbox": [ 115, 527, 506, 540 ], "spans": [ { "bbox": [ 115, 527, 506, 540 ], "score": 1.0, "content": "informed neural networks with domain decomposition. Computer Methods in Applied Mechanics", "type": "text" } ], "index": 31 }, { "bbox": [ 115, 537, 266, 551 ], "spans": [ { "bbox": [ 115, 537, 266, 551 ], "score": 1.0, "content": "and Engineering, 374:113547, 2021.", "type": "text" } ], "index": 32, "is_list_end_line": true }, { "bbox": [ 106, 556, 506, 572 ], "spans": [ { "bbox": [ 106, 556, 506, 572 ], "score": 1.0, "content": "Yuehaw Khoo, Jianfeng Lu, and Lexing Ying. Solving parametric PDE problems with artificial neural", "type": "text" } ], "index": 33, "is_list_start_line": true }, { "bbox": [ 115, 568, 423, 581 ], "spans": [ { "bbox": [ 115, 568, 423, 581 ], "score": 1.0, "content": "networks. European Journal of Applied Mathematics, 32(3):421–435, 2021.", "type": "text" } ], "index": 34, "is_list_end_line": true }, { "bbox": [ 104, 584, 506, 601 ], "spans": [ { "bbox": [ 104, 584, 506, 601 ], "score": 1.0, "content": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint", "type": "text" } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 114, 597, 215, 610 ], "spans": [ { "bbox": [ 114, 597, 215, 610 ], "score": 1.0, "content": "arXiv:1412.6980, 2014.", "type": "text" } ], "index": 36, "is_list_end_line": true }, { "bbox": [ 106, 616, 506, 630 ], "spans": [ { "bbox": [ 106, 616, 506, 630 ], "score": 1.0, "content": "Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Char-", "type": "text" } ], "index": 37, "is_list_start_line": true }, { "bbox": [ 115, 628, 506, 641 ], "spans": [ { "bbox": [ 115, 628, 506, 641 ], "score": 1.0, "content": "acterizing possible failure modes in physics-informed neural networks. Advances in Neural", "type": "text" } ], "index": 38 }, { "bbox": [ 114, 638, 347, 653 ], "spans": [ { "bbox": [ 114, 638, 347, 653 ], "score": 1.0, "content": "Information Processing Systems, 34:26548–26560, 2021.", "type": "text" } ], "index": 39, "is_list_end_line": true }, { "bbox": [ 105, 657, 506, 671 ], "spans": [ { "bbox": [ 105, 657, 506, 671 ], "score": 1.0, "content": "Colby L. Wight and Jia Zhao. Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive", "type": "text" } ], "index": 40, "is_list_start_line": true }, { "bbox": [ 115, 668, 506, 682 ], "spans": [ { "bbox": [ 115, 668, 506, 682 ], "score": 1.0, "content": "physics informed neural networks. Communications in Computational Physics, 29(3):930–954,", "type": "text" } ], "index": 41 }, { "bbox": [ 115, 677, 145, 694 ], "spans": [ { "bbox": [ 115, 677, 145, 694 ], "score": 1.0, "content": "2021.", "type": "text" } ], "index": 42, "is_list_end_line": true }, { "bbox": [ 105, 698, 508, 713 ], "spans": [ { "bbox": [ 105, 698, 508, 713 ], "score": 1.0, "content": "Songming Liu, Hao Zhongkai, Chengyang Ying, Hang Su, Jun Zhu, and Ze Cheng. A unified hard-", "type": "text" } ], "index": 43, "is_list_start_line": true }, { "bbox": [ 115, 710, 506, 723 ], "spans": [ { "bbox": [ 115, 710, 506, 723 ], "score": 1.0, "content": "constraint framework for solving geometrically complex PDEs. Advances in Neural Information", "type": "text" } ], "index": 44 }, { "bbox": [ 114, 718, 303, 735 ], "spans": [ { "bbox": [ 114, 718, 303, 735 ], "score": 1.0, "content": "Processing Systems, 35:20287–20299, 2022a.", "type": "text" } ], "index": 45, "is_list_end_line": true }, { "bbox": [ 105, 82, 506, 96 ], "spans": [ { "bbox": [ 105, 82, 506, 96 ], "score": 1.0, "content": "Xu Liu, Xiaoya Zhang, Wei Peng, Weien Zhou, and Wen Yao. A novel meta-learning initialization", "type": "text", "cross_page": true } ], "index": 0, "is_list_start_line": true }, { "bbox": [ 115, 94, 506, 108 ], "spans": [ { "bbox": [ 115, 94, 506, 108 ], "score": 1.0, "content": "method for physics-informed neural networks. Neural Computing and Applications, 34(17):", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 114, 103, 206, 118 ], "spans": [ { "bbox": [ 114, 103, 206, 118 ], "score": 1.0, "content": "14511–14534, 2022b.", "type": "text", "cross_page": true } ], "index": 2, "is_list_end_line": true }, { "bbox": [ 106, 124, 505, 136 ], "spans": [ { "bbox": [ 106, 124, 505, 136 ], "score": 1.0, "content": "Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong. PDE-Net: Learning PDEs from data. In", "type": "text", "cross_page": true } ], "index": 3, "is_list_start_line": true }, { "bbox": [ 114, 134, 432, 148 ], "spans": [ { "bbox": [ 114, 134, 432, 148 ], "score": 1.0, "content": "International Conference on Machine Learning, pp. 3208–3216. PMLR, 2018.", "type": "text", "cross_page": true } ], "index": 4, "is_list_end_line": true }, { "bbox": [ 104, 152, 505, 167 ], "spans": [ { "bbox": [ 104, 152, 505, 167 ], "score": 1.0, "content": "Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. DeepXDE: A deep learning library", "type": "text", "cross_page": true } ], "index": 5, "is_list_start_line": true }, { "bbox": [ 115, 164, 403, 176 ], "spans": [ { "bbox": [ 115, 164, 403, 176 ], "score": 1.0, "content": "for solving differential equations. SIAM review, 63(1):208–228, 2021a.", "type": "text", "cross_page": true } ], "index": 6, "is_list_end_line": true }, { "bbox": [ 104, 182, 507, 196 ], "spans": [ { "bbox": [ 104, 182, 507, 196 ], "score": 1.0, "content": "Lu Lu, Raphael Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G Johnson.", "type": "text", "cross_page": true } ], "index": 7, "is_list_start_line": true }, { "bbox": [ 116, 193, 506, 207 ], "spans": [ { "bbox": [ 116, 193, 506, 207 ], "score": 1.0, "content": "Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 115, 204, 322, 218 ], "spans": [ { "bbox": [ 115, 204, 322, 218 ], "score": 1.0, "content": "Scientific Computing, 43(6):B1105–B1132, 2021b.", "type": "text", "cross_page": true } ], "index": 9, "is_list_end_line": true }, { "bbox": [ 106, 223, 504, 235 ], "spans": [ { "bbox": [ 106, 223, 504, 235 ], "score": 1.0, "content": "Revanth Mattey and Susanta Ghosh. A novel sequential method to train physics-informed neural", "type": "text", "cross_page": true } ], "index": 10, "is_list_start_line": true }, { "bbox": [ 114, 234, 505, 248 ], "spans": [ { "bbox": [ 114, 234, 505, 248 ], "score": 1.0, "content": "networks for Allen Cahn and Cahn Hilliard equations. Computer Methods in Applied Mechanics", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 116, 245, 265, 257 ], "spans": [ { "bbox": [ 116, 245, 265, 257 ], "score": 1.0, "content": "and Engineering, 390:114474, 2022.", "type": "text", "cross_page": true } ], "index": 12, "is_list_end_line": true }, { "bbox": [ 105, 263, 507, 277 ], "spans": [ { "bbox": [ 105, 263, 507, 277 ], "score": 1.0, "content": "Levi D McClenny and Ulisses M Braga-Neto. Self-adaptive physics-informed neural networks.", "type": "text", "cross_page": true } ], "index": 13, "is_list_start_line": true }, { "bbox": [ 115, 275, 335, 287 ], "spans": [ { "bbox": [ 115, 275, 335, 287 ], "score": 1.0, "content": "Journal of Computational Physics, 474:111722, 2023.", "type": "text", "cross_page": true } ], "index": 14, "is_list_end_line": true }, { "bbox": [ 105, 293, 505, 307 ], "spans": [ { "bbox": [ 105, 293, 505, 307 ], "score": 1.0, "content": "Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics-informed", "type": "text", "cross_page": true } ], "index": 15, "is_list_start_line": true }, { "bbox": [ 114, 304, 506, 318 ], "spans": [ { "bbox": [ 114, 304, 506, 318 ], "score": 1.0, "content": "neural networks for approximating PDEs. IMA Journal of Numerical Analysis, 43(1):1–43, 01", "type": "text", "cross_page": true } ], "index": 16 }, { "bbox": [ 115, 315, 505, 329 ], "spans": [ { "bbox": [ 115, 315, 505, 329 ], "score": 1.0, "content": "2023. ISSN 0272-4979. doi: 10.1093/imanum/drab093. URL https://doi.org/10.1093/", "type": "text", "cross_page": true } ], "index": 17 }, { "bbox": [ 117, 326, 205, 338 ], "spans": [ { "bbox": [ 117, 326, 205, 338 ], "score": 1.0, "content": "imanum/drab093.", "type": "text", "cross_page": true } ], "index": 18, "is_list_end_line": true }, { "bbox": [ 104, 342, 507, 360 ], "spans": [ { "bbox": [ 104, 342, 507, 360 ], "score": 1.0, "content": "Christian Moya and Guang Lin. DAE-PINN: A physics-informed neural network model for simulating", "type": "text", "cross_page": true } ], "index": 19, "is_list_start_line": true }, { "bbox": [ 115, 356, 506, 369 ], "spans": [ { "bbox": [ 115, 356, 506, 369 ], "score": 1.0, "content": "differential algebraic equations with application to power networks. Neural Computing and", "type": "text", "cross_page": true } ], "index": 20 }, { "bbox": [ 114, 366, 271, 380 ], "spans": [ { "bbox": [ 114, 366, 271, 380 ], "score": 1.0, "content": "Applications, 35(5):3789–3804, 2023.", "type": "text", "cross_page": true } ], "index": 21, "is_list_end_line": true }, { "bbox": [ 105, 385, 507, 399 ], "spans": [ { "bbox": [ 105, 385, 507, 399 ], "score": 1.0, "content": "Michael Penwarden, Ameya D Jagtap, Shandian Zhe, George Em Karniadakis, and Robert M Kirby.", "type": "text", "cross_page": true } ], "index": 22, "is_list_start_line": true }, { "bbox": [ 115, 396, 506, 410 ], "spans": [ { "bbox": [ 115, 396, 506, 410 ], "score": 1.0, "content": "A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks", "type": "text", "cross_page": true } ], "index": 23 }, { "bbox": [ 115, 407, 458, 421 ], "spans": [ { "bbox": [ 115, 407, 458, 421 ], "score": 1.0, "content": "(PINNs) and their temporal decompositions. arXiv preprint arXiv:2302.14227, 2023.", "type": "text", "cross_page": true } ], "index": 24, "is_list_end_line": true }, { "bbox": [ 105, 425, 507, 439 ], "spans": [ { "bbox": [ 105, 425, 507, 439 ], "score": 1.0, "content": "Maziar Raissi. Deep hidden physics models: Deep learning of nonlinear partial differential equations.", "type": "text", "cross_page": true } ], "index": 25, "is_list_start_line": true }, { "bbox": [ 115, 437, 385, 451 ], "spans": [ { "bbox": [ 115, 437, 385, 451 ], "score": 1.0, "content": "The Journal of Machine Learning Research, 19(1):932–955, 2018.", "type": "text", "cross_page": true } ], "index": 26, "is_list_end_line": true }, { "bbox": [ 104, 453, 507, 469 ], "spans": [ { "bbox": [ 104, 453, 507, 469 ], "score": 1.0, "content": "Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part i):", "type": "text", "cross_page": true } ], "index": 27, "is_list_start_line": true }, { "bbox": [ 115, 466, 507, 480 ], "spans": [ { "bbox": [ 115, 466, 507, 480 ], "score": 1.0, "content": "Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561,", "type": "text", "cross_page": true } ], "index": 28 }, { "bbox": [ 115, 476, 144, 491 ], "spans": [ { "bbox": [ 115, 476, 144, 491 ], "score": 1.0, "content": "2017.", "type": "text", "cross_page": true } ], "index": 29, "is_list_end_line": true }, { "bbox": [ 106, 496, 505, 508 ], "spans": [ { "bbox": [ 106, 496, 505, 508 ], "score": 1.0, "content": "Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A", "type": "text", "cross_page": true } ], "index": 30, "is_list_start_line": true }, { "bbox": [ 115, 506, 506, 520 ], "spans": [ { "bbox": [ 115, 506, 506, 520 ], "score": 1.0, "content": "deep learning framework for solving forward and inverse problems involving nonlinear partial", "type": "text", "cross_page": true } ], "index": 31 }, { "bbox": [ 116, 518, 430, 532 ], "spans": [ { "bbox": [ 116, 518, 430, 532 ], "score": 1.0, "content": "differential equations. Journal of Computational physics, 378:686–707, 2019.", "type": "text", "cross_page": true } ], "index": 32, "is_list_end_line": true }, { "bbox": [ 104, 535, 507, 551 ], "spans": [ { "bbox": [ 104, 535, 507, 551 ], "score": 1.0, "content": "Justin Sirignano and Konstantinos Spiliopoulos. DGM: A deep learning algorithm for solving partial", "type": "text", "cross_page": true } ], "index": 33, "is_list_start_line": true }, { "bbox": [ 115, 547, 438, 561 ], "spans": [ { "bbox": [ 115, 547, 438, 561 ], "score": 1.0, "content": "differential equations. Journal of computational physics, 375:1339–1364, 2018.", "type": "text", "cross_page": true } ], "index": 34, "is_list_end_line": true }, { "bbox": [ 105, 565, 505, 579 ], "spans": [ { "bbox": [ 105, 565, 505, 579 ], "score": 1.0, "content": "Dong H Song and Daniel M Tartakovsky. Transfer learning on multifidelity data. Journal of Machine", "type": "text", "cross_page": true } ], "index": 35, "is_list_start_line": true }, { "bbox": [ 114, 577, 324, 591 ], "spans": [ { "bbox": [ 114, 577, 324, 591 ], "score": 1.0, "content": "Learning for Modeling and Computing, 3(1), 2022.", "type": "text", "cross_page": true } ], "index": 36, "is_list_end_line": true }, { "bbox": [ 105, 595, 506, 609 ], "spans": [ { "bbox": [ 105, 595, 506, 609 ], "score": 1.0, "content": "Jochen Stiasny, Samuel Chevalier, and Spyros Chatzivasileiadis. Learning without data: Physics-", "type": "text", "cross_page": true } ], "index": 37, "is_list_start_line": true }, { "bbox": [ 115, 606, 506, 620 ], "spans": [ { "bbox": [ 115, 606, 506, 620 ], "score": 1.0, "content": "informed neural networks for fast time-domain simulation. In 2021 IEEE International Conference", "type": "text", "cross_page": true } ], "index": 38 }, { "bbox": [ 116, 619, 506, 630 ], "spans": [ { "bbox": [ 116, 619, 506, 630 ], "score": 1.0, "content": "on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm),", "type": "text", "cross_page": true } ], "index": 39 }, { "bbox": [ 114, 628, 225, 642 ], "spans": [ { "bbox": [ 114, 628, 225, 642 ], "score": 1.0, "content": "pp. 438–443. IEEE, 2021.", "type": "text", "cross_page": true } ], "index": 40, "is_list_end_line": true }, { "bbox": [ 104, 646, 505, 660 ], "spans": [ { "bbox": [ 104, 646, 505, 660 ], "score": 1.0, "content": "N Sukumar and Ankit Srivastava. Exact imposition of boundary conditions with distance functions in", "type": "text", "cross_page": true } ], "index": 41, "is_list_start_line": true }, { "bbox": [ 114, 656, 507, 674 ], "spans": [ { "bbox": [ 114, 656, 507, 674 ], "score": 1.0, "content": "physics-informed deep neural networks. Computer Methods in Applied Mechanics and Engineering,", "type": "text", "cross_page": true } ], "index": 42 }, { "bbox": [ 115, 668, 195, 682 ], "spans": [ { "bbox": [ 115, 668, 195, 682 ], "score": 1.0, "content": "389:114333, 2022.", "type": "text", "cross_page": true } ], "index": 43, "is_list_end_line": true }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 505, 701 ], "score": 1.0, "content": "Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh", "type": "text", "cross_page": true } ], "index": 44, "is_list_start_line": true }, { "bbox": [ 116, 700, 505, 712 ], "spans": [ { "bbox": [ 116, 700, 505, 712 ], "score": 1.0, "content": "Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn", "type": "text", "cross_page": true } ], "index": 45 }, { "bbox": [ 114, 708, 505, 724 ], "spans": [ { "bbox": [ 114, 708, 505, 724 ], "score": 1.0, "content": "high frequency functions in low dimensional domains. Advances in Neural Information Processing", "type": "text", "cross_page": true } ], "index": 46 }, { "bbox": [ 115, 720, 240, 734 ], "spans": [ { "bbox": [ 115, 720, 240, 734 ], "score": 1.0, "content": "Systems, 33:7537–7547, 2020.", "type": "text", "cross_page": true } ], "index": 47, "is_list_end_line": true } ], "index": 23, "bbox_fs": [ 104, 101, 508, 735 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 100, 73, 507, 744 ], "lines": [ { "bbox": [ 105, 82, 506, 96 ], "spans": [ { "bbox": [ 105, 82, 506, 96 ], "score": 1.0, "content": "Xu Liu, Xiaoya Zhang, Wei Peng, Weien Zhou, and Wen Yao. A novel meta-learning initialization", "type": "text" } ], "index": 0 }, { "bbox": [ 115, 94, 506, 108 ], "spans": [ { "bbox": [ 115, 94, 506, 108 ], "score": 1.0, "content": "method for physics-informed neural networks. Neural Computing and Applications, 34(17):", "type": "text" } ], "index": 1 }, { "bbox": [ 114, 103, 206, 118 ], "spans": [ { "bbox": [ 114, 103, 206, 118 ], "score": 1.0, "content": "14511–14534, 2022b.", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 124, 505, 136 ], "spans": [ { "bbox": [ 106, 124, 505, 136 ], "score": 1.0, "content": "Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong. PDE-Net: Learning PDEs from data. In", "type": "text" } ], "index": 3 }, { "bbox": [ 114, 134, 432, 148 ], "spans": [ { "bbox": [ 114, 134, 432, 148 ], "score": 1.0, "content": "International Conference on Machine Learning, pp. 3208–3216. PMLR, 2018.", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 152, 505, 167 ], "spans": [ { "bbox": [ 104, 152, 505, 167 ], "score": 1.0, "content": "Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. DeepXDE: A deep learning library", "type": "text" } ], "index": 5 }, { "bbox": [ 115, 164, 403, 176 ], "spans": [ { "bbox": [ 115, 164, 403, 176 ], "score": 1.0, "content": "for solving differential equations. SIAM review, 63(1):208–228, 2021a.", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 182, 507, 196 ], "spans": [ { "bbox": [ 104, 182, 507, 196 ], "score": 1.0, "content": "Lu Lu, Raphael Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G Johnson.", "type": "text" } ], "index": 7 }, { "bbox": [ 116, 193, 506, 207 ], "spans": [ { "bbox": [ 116, 193, 506, 207 ], "score": 1.0, "content": "Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on", "type": "text" } ], "index": 8 }, { "bbox": [ 115, 204, 322, 218 ], "spans": [ { "bbox": [ 115, 204, 322, 218 ], "score": 1.0, "content": "Scientific Computing, 43(6):B1105–B1132, 2021b.", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 223, 504, 235 ], "spans": [ { "bbox": [ 106, 223, 504, 235 ], "score": 1.0, "content": "Revanth Mattey and Susanta Ghosh. A novel sequential method to train physics-informed neural", "type": "text" } ], "index": 10 }, { "bbox": [ 114, 234, 505, 248 ], "spans": [ { "bbox": [ 114, 234, 505, 248 ], "score": 1.0, "content": "networks for Allen Cahn and Cahn Hilliard equations. Computer Methods in Applied Mechanics", "type": "text" } ], "index": 11 }, { "bbox": [ 116, 245, 265, 257 ], "spans": [ { "bbox": [ 116, 245, 265, 257 ], "score": 1.0, "content": "and Engineering, 390:114474, 2022.", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 263, 507, 277 ], "spans": [ { "bbox": [ 105, 263, 507, 277 ], "score": 1.0, "content": "Levi D McClenny and Ulisses M Braga-Neto. Self-adaptive physics-informed neural networks.", "type": "text" } ], "index": 13 }, { "bbox": [ 115, 275, 335, 287 ], "spans": [ { "bbox": [ 115, 275, 335, 287 ], "score": 1.0, "content": "Journal of Computational Physics, 474:111722, 2023.", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 293, 505, 307 ], "spans": [ { "bbox": [ 105, 293, 505, 307 ], "score": 1.0, "content": "Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics-informed", "type": "text" } ], "index": 15 }, { "bbox": [ 114, 304, 506, 318 ], "spans": [ { "bbox": [ 114, 304, 506, 318 ], "score": 1.0, "content": "neural networks for approximating PDEs. IMA Journal of Numerical Analysis, 43(1):1–43, 01", "type": "text" } ], "index": 16 }, { "bbox": [ 115, 315, 505, 329 ], "spans": [ { "bbox": [ 115, 315, 505, 329 ], "score": 1.0, "content": "2023. ISSN 0272-4979. doi: 10.1093/imanum/drab093. URL https://doi.org/10.1093/", "type": "text" } ], "index": 17 }, { "bbox": [ 117, 326, 205, 338 ], "spans": [ { "bbox": [ 117, 326, 205, 338 ], "score": 1.0, "content": "imanum/drab093.", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 342, 507, 360 ], "spans": [ { "bbox": [ 104, 342, 507, 360 ], "score": 1.0, "content": "Christian Moya and Guang Lin. DAE-PINN: A physics-informed neural network model for simulating", "type": "text" } ], "index": 19 }, { "bbox": [ 115, 356, 506, 369 ], "spans": [ { "bbox": [ 115, 356, 506, 369 ], "score": 1.0, "content": "differential algebraic equations with application to power networks. Neural Computing and", "type": "text" } ], "index": 20 }, { "bbox": [ 114, 366, 271, 380 ], "spans": [ { "bbox": [ 114, 366, 271, 380 ], "score": 1.0, "content": "Applications, 35(5):3789–3804, 2023.", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 385, 507, 399 ], "spans": [ { "bbox": [ 105, 385, 507, 399 ], "score": 1.0, "content": "Michael Penwarden, Ameya D Jagtap, Shandian Zhe, George Em Karniadakis, and Robert M Kirby.", "type": "text" } ], "index": 22 }, { "bbox": [ 115, 396, 506, 410 ], "spans": [ { "bbox": [ 115, 396, 506, 410 ], "score": 1.0, "content": "A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks", "type": "text" } ], "index": 23 }, { "bbox": [ 115, 407, 458, 421 ], "spans": [ { "bbox": [ 115, 407, 458, 421 ], "score": 1.0, "content": "(PINNs) and their temporal decompositions. arXiv preprint arXiv:2302.14227, 2023.", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 425, 507, 439 ], "spans": [ { "bbox": [ 105, 425, 507, 439 ], "score": 1.0, "content": "Maziar Raissi. Deep hidden physics models: Deep learning of nonlinear partial differential equations.", "type": "text" } ], "index": 25 }, { "bbox": [ 115, 437, 385, 451 ], "spans": [ { "bbox": [ 115, 437, 385, 451 ], "score": 1.0, "content": "The Journal of Machine Learning Research, 19(1):932–955, 2018.", "type": "text" } ], "index": 26 }, { "bbox": [ 104, 453, 507, 469 ], "spans": [ { "bbox": [ 104, 453, 507, 469 ], "score": 1.0, "content": "Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part i):", "type": "text" } ], "index": 27 }, { "bbox": [ 115, 466, 507, 480 ], "spans": [ { "bbox": [ 115, 466, 507, 480 ], "score": 1.0, "content": "Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561,", "type": "text" } ], "index": 28 }, { "bbox": [ 115, 476, 144, 491 ], "spans": [ { "bbox": [ 115, 476, 144, 491 ], "score": 1.0, "content": "2017.", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 496, 505, 508 ], "spans": [ { "bbox": [ 106, 496, 505, 508 ], "score": 1.0, "content": "Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A", "type": "text" } ], "index": 30 }, { "bbox": [ 115, 506, 506, 520 ], "spans": [ { "bbox": [ 115, 506, 506, 520 ], "score": 1.0, "content": "deep learning framework for solving forward and inverse problems involving nonlinear partial", "type": "text" } ], "index": 31 }, { "bbox": [ 116, 518, 430, 532 ], "spans": [ { "bbox": [ 116, 518, 430, 532 ], "score": 1.0, "content": "differential equations. Journal of Computational physics, 378:686–707, 2019.", "type": "text" } ], "index": 32 }, { "bbox": [ 104, 535, 507, 551 ], "spans": [ { "bbox": [ 104, 535, 507, 551 ], "score": 1.0, "content": "Justin Sirignano and Konstantinos Spiliopoulos. DGM: A deep learning algorithm for solving partial", "type": "text" } ], "index": 33 }, { "bbox": [ 115, 547, 438, 561 ], "spans": [ { "bbox": [ 115, 547, 438, 561 ], "score": 1.0, "content": "differential equations. Journal of computational physics, 375:1339–1364, 2018.", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 565, 505, 579 ], "spans": [ { "bbox": [ 105, 565, 505, 579 ], "score": 1.0, "content": "Dong H Song and Daniel M Tartakovsky. Transfer learning on multifidelity data. Journal of Machine", "type": "text" } ], "index": 35 }, { "bbox": [ 114, 577, 324, 591 ], "spans": [ { "bbox": [ 114, 577, 324, 591 ], "score": 1.0, "content": "Learning for Modeling and Computing, 3(1), 2022.", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 595, 506, 609 ], "spans": [ { "bbox": [ 105, 595, 506, 609 ], "score": 1.0, "content": "Jochen Stiasny, Samuel Chevalier, and Spyros Chatzivasileiadis. Learning without data: Physics-", "type": "text" } ], "index": 37 }, { "bbox": [ 115, 606, 506, 620 ], "spans": [ { "bbox": [ 115, 606, 506, 620 ], "score": 1.0, "content": "informed neural networks for fast time-domain simulation. In 2021 IEEE International Conference", "type": "text" } ], "index": 38 }, { "bbox": [ 116, 619, 506, 630 ], "spans": [ { "bbox": [ 116, 619, 506, 630 ], "score": 1.0, "content": "on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm),", "type": "text" } ], "index": 39 }, { "bbox": [ 114, 628, 225, 642 ], "spans": [ { "bbox": [ 114, 628, 225, 642 ], "score": 1.0, "content": "pp. 438–443. IEEE, 2021.", "type": "text" } ], "index": 40 }, { "bbox": [ 104, 646, 505, 660 ], "spans": [ { "bbox": [ 104, 646, 505, 660 ], "score": 1.0, "content": "N Sukumar and Ankit Srivastava. Exact imposition of boundary conditions with distance functions in", "type": "text" } ], "index": 41 }, { "bbox": [ 114, 656, 507, 674 ], "spans": [ { "bbox": [ 114, 656, 507, 674 ], "score": 1.0, "content": "physics-informed deep neural networks. Computer Methods in Applied Mechanics and Engineering,", "type": "text" } ], "index": 42 }, { "bbox": [ 115, 668, 195, 682 ], "spans": [ { "bbox": [ 115, 668, 195, 682 ], "score": 1.0, "content": "389:114333, 2022.", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 687, 505, 701 ], "spans": [ { "bbox": [ 105, 687, 505, 701 ], "score": 1.0, "content": "Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh", "type": "text" } ], "index": 44 }, { "bbox": [ 116, 700, 505, 712 ], "spans": [ { "bbox": [ 116, 700, 505, 712 ], "score": 1.0, "content": "Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn", "type": "text" } ], "index": 45 }, { "bbox": [ 114, 708, 505, 724 ], "spans": [ { "bbox": [ 114, 708, 505, 724 ], "score": 1.0, "content": "high frequency functions in low dimensional domains. Advances in Neural Information Processing", "type": "text" } ], "index": 46 }, { "bbox": [ 115, 720, 240, 734 ], "spans": [ { "bbox": [ 115, 720, 240, 734 ], "score": 1.0, "content": "Systems, 33:7537–7547, 2020.", "type": "text" } ], "index": 47 } ], "index": 23.5 } ], "page_idx": 10, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 751, 309, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 765 ], "spans": [ { "bbox": [ 299, 750, 312, 765 ], "score": 1.0, "content": "", "type": "text", "height": 15, "width": 13 } ] } ] }, { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 26, 308, 38 ], "spans": [ { "bbox": [ 107, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 100, 73, 507, 744 ], "lines": [], "index": 23.5, "bbox_fs": [ 104, 82, 507, 734 ], "lines_deleted": true } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 115 ], "lines": [ { "bbox": [ 105, 81, 505, 96 ], "spans": [ { "bbox": [ 105, 81, 505, 96 ], "score": 1.0, "content": "Sifan Wang, Yujun Teng, and Paris Perdikaris. Understanding and mitigating gradient flow pathologies", "type": "text" } ], "index": 0 }, { "bbox": [ 115, 94, 507, 106 ], "spans": [ { "bbox": [ 115, 94, 507, 106 ], "score": 1.0, "content": "in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081,", "type": "text" } ], "index": 1 }, { "bbox": [ 115, 103, 147, 117 ], "spans": [ { "bbox": [ 115, 103, 147, 117 ], "score": 1.0, "content": "2021a.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "text", "bbox": [ 107, 127, 506, 162 ], "lines": [ { "bbox": [ 106, 128, 506, 141 ], "spans": [ { "bbox": [ 106, 128, 506, 141 ], "score": 1.0, "content": "Sifan Wang, Hanwen Wang, and Paris Perdikaris. On the eigenvector bias of Fourier feature networks:", "type": "text" } ], "index": 3 }, { "bbox": [ 115, 139, 506, 151 ], "spans": [ { "bbox": [ 115, 139, 506, 151 ], "score": 1.0, "content": "From regression to solving multi-scale PDEs with physics-informed neural networks. Computer", "type": "text" } ], "index": 4 }, { "bbox": [ 115, 150, 396, 163 ], "spans": [ { "bbox": [ 115, 150, 396, 163 ], "score": 1.0, "content": "Methods in Applied Mechanics and Engineering, 384:113938, 2021b.", "type": "text" } ], "index": 5 } ], "index": 4 }, { "type": "text", "bbox": [ 105, 173, 505, 196 ], "lines": [ { "bbox": [ 105, 172, 506, 187 ], "spans": [ { "bbox": [ 105, 172, 506, 187 ], "score": 1.0, "content": "Sifan Wang, Shyam Sankaran, and Paris Perdikaris. Respecting causality is all you need for training", "type": "text" } ], "index": 6 }, { "bbox": [ 115, 185, 425, 197 ], "spans": [ { "bbox": [ 115, 185, 425, 197 ], "score": 1.0, "content": "physics-informed neural networks. arXiv preprint arXiv:2203.07404, 2022a.", "type": "text" } ], "index": 7 } ], "index": 6.5 }, { "type": "text", "bbox": [ 106, 208, 504, 231 ], "lines": [ { "bbox": [ 106, 208, 505, 221 ], "spans": [ { "bbox": [ 106, 208, 505, 221 ], "score": 1.0, "content": "Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why PINNs fail to train: A neural tangent", "type": "text" } ], "index": 8 }, { "bbox": [ 115, 219, 418, 232 ], "spans": [ { "bbox": [ 115, 219, 418, 232 ], "score": 1.0, "content": "kernel perspective. Journal of Computational Physics, 449:110768, 2022b.", "type": "text" } ], "index": 9 } ], "index": 8.5 }, { "type": "text", "bbox": [ 107, 243, 506, 277 ], "lines": [ { "bbox": [ 106, 242, 506, 256 ], "spans": [ { "bbox": [ 106, 242, 506, 256 ], "score": 1.0, "content": "Chen Xu, Ba Trung Cao, Yong Yuan, and Günther Meschke. Transfer learning based physics-", "type": "text" } ], "index": 10 }, { "bbox": [ 116, 254, 505, 266 ], "spans": [ { "bbox": [ 116, 254, 505, 266 ], "score": 1.0, "content": "informed neural networks for solving inverse problems in engineering structures under different", "type": "text" } ], "index": 11 }, { "bbox": [ 116, 265, 506, 279 ], "spans": [ { "bbox": [ 116, 265, 506, 279 ], "score": 1.0, "content": "loading scenarios. Computer Methods in Applied Mechanics and Engineering, 405:115852, 2023.", "type": "text" } ], "index": 12 } ], "index": 11 }, { "type": "text", "bbox": [ 107, 288, 504, 311 ], "lines": [ { "bbox": [ 106, 287, 505, 301 ], "spans": [ { "bbox": [ 106, 287, 505, 301 ], "score": 1.0, "content": "Wuzhe Xu, Yulong Lu, and Li Wang. Transfer learning enhanced DeepONet for long-time prediction", "type": "text" } ], "index": 13 }, { "bbox": [ 116, 299, 373, 312 ], "spans": [ { "bbox": [ 116, 299, 373, 312 ], "score": 1.0, "content": "of evolution equations. arXiv preprint arXiv:2212.04663, 2022.", "type": "text" } ], "index": 14 } ], "index": 13.5 }, { "type": "text", "bbox": [ 108, 323, 504, 345 ], "lines": [ { "bbox": [ 106, 323, 505, 335 ], "spans": [ { "bbox": [ 106, 323, 505, 335 ], "score": 1.0, "content": "Bing Yu and Weinan E. The deep Ritz method: A deep learning-based numerical algorithm for", "type": "text" } ], "index": 15 }, { "bbox": [ 115, 334, 498, 346 ], "spans": [ { "bbox": [ 115, 334, 498, 346 ], "score": 1.0, "content": "solving variational problems. Communications in Mathematics and Statistics, 6(1):1–12, 2018.", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "text", "bbox": [ 108, 371, 183, 384 ], "lines": [ { "bbox": [ 105, 370, 184, 386 ], "spans": [ { "bbox": [ 105, 370, 184, 386 ], "score": 1.0, "content": "A APPENDIX", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "text", "bbox": [ 108, 397, 229, 409 ], "lines": [ { "bbox": [ 106, 397, 229, 410 ], "spans": [ { "bbox": [ 106, 397, 229, 410 ], "score": 1.0, "content": "A.1 TABLE OF NOTATIONS", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "text", "bbox": [ 107, 419, 265, 431 ], "lines": [ { "bbox": [ 106, 419, 265, 432 ], "spans": [ { "bbox": [ 106, 419, 265, 432 ], "score": 1.0, "content": "A table of notations is given in Table 1.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 446, 267, 459 ], "lines": [ { "bbox": [ 106, 447, 268, 459 ], "spans": [ { "bbox": [ 106, 447, 268, 459 ], "score": 1.0, "content": "A.2 TIME DIFFERENCING SCHEMES", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 108, 468, 226, 480 ], "lines": [ { "bbox": [ 106, 469, 226, 480 ], "spans": [ { "bbox": [ 106, 469, 226, 480 ], "score": 1.0, "content": "A.2.1 EXPLICIT SCHEMES", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "text", "bbox": [ 107, 488, 244, 500 ], "lines": [ { "bbox": [ 106, 489, 244, 501 ], "spans": [ { "bbox": [ 106, 489, 244, 501 ], "score": 1.0, "content": "First-order forward Euler scheme:", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "interline_equation", "bbox": [ 238, 505, 373, 532 ], "lines": [ { "bbox": [ 238, 505, 373, 532 ], "spans": [ { "bbox": [ 238, 505, 373, 532 ], "score": 0.94, "content": "\\frac { \\boldsymbol { u } ^ { n + 1 } ( \\boldsymbol { x } ) - \\boldsymbol { u } ^ { n } ( \\boldsymbol { x } ) } { \\Delta t } = \\mathcal { N } \\left[ \\boldsymbol { u } ^ { n } ( \\boldsymbol { x } ) \\right] .", "type": "interline_equation", "image_path": "31a8e4fa8b749565c35ca5ba07a856690f8eae3858643b7fb0d656672825cb5c.jpg" } ] } ], "index": 23.5, "virtual_lines": [ { "bbox": [ 238, 505, 373, 518.5 ], "spans": [], "index": 23 }, { "bbox": [ 238, 518.5, 373, 532.0 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 107, 543, 324, 556 ], "lines": [ { "bbox": [ 105, 542, 325, 558 ], "spans": [ { "bbox": [ 105, 542, 325, 558 ], "score": 1.0, "content": "Second-order explicit Runge-Kutta (2nd RK) scheme:", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 202, 562, 409, 591 ], "lines": [ { "bbox": [ 202, 562, 409, 591 ], "spans": [ { "bbox": [ 202, 562, 409, 591 ], "score": 0.92, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { \\Delta t } { 2 } \\mathcal { N } [ u ^ { n } ( x ) ] \\right] .", "type": "interline_equation", "image_path": "146de5483eae76a7901a80a2db4033ba2b55e12ed7ae463ee61c20359e8ebdcd.jpg" } ] } ], "index": 26.5, "virtual_lines": [ { "bbox": [ 202, 562, 409, 576.5 ], "spans": [], "index": 26 }, { "bbox": [ 202, 576.5, 409, 591.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 604, 315, 616 ], "lines": [ { "bbox": [ 106, 604, 315, 617 ], "spans": [ { "bbox": [ 106, 604, 315, 617 ], "score": 1.0, "content": "Fouth-order explicit Runge-Kutta (4th RK) scheme:", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "interline_equation", "bbox": [ 182, 622, 429, 736 ], "lines": [ { "bbox": [ 182, 622, 429, 736 ], "spans": [ { "bbox": [ 182, 622, 429, 736 ], "score": 0.95, "content": "\\begin{array} { l } { \\displaystyle \\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\frac { 1 } { 6 } \\left[ k _ { 1 } ( x ) + 2 k _ { 2 } ( x ) + 2 k _ { 3 } ( x ) + k _ { 4 } ( x ) \\right] , } \\\\ { \\displaystyle k _ { 1 } ( x ) = \\mathcal { N } [ u ^ { n } ( x ) ] , } \\\\ { \\displaystyle k _ { 2 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { \\Delta t } { 2 } \\mathcal { N } [ k _ { 1 } ( x ) ] \\right] , } \\\\ { \\displaystyle k _ { 3 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { \\Delta t } { 2 } \\mathcal { N } [ k _ { 2 } ( x ) ] \\right] , } \\\\ { \\displaystyle k _ { 4 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\Delta t \\mathcal { N } [ k _ { 3 } ( x ) ] \\right] . } \\end{array}", "type": "interline_equation", "image_path": "0e397af9967fbd25bcce50d51803aa32dd29176f1d0ba59b3c0d2f7b50309aaf.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 182, 622, 429, 660.0 ], "spans": [], "index": 29 }, { "bbox": [ 182, 660.0, 429, 698.0 ], "spans": [], "index": 30 }, { "bbox": [ 182, 698.0, 429, 736.0 ], "spans": [], "index": 31 } ] } ], "page_idx": 11, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 765 ], "spans": [ { "bbox": [ 299, 750, 312, 765 ], "score": 1.0, "content": "12", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 107, 26, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 115 ], "lines": [ { "bbox": [ 105, 81, 505, 96 ], "spans": [ { "bbox": [ 105, 81, 505, 96 ], "score": 1.0, "content": "Sifan Wang, Yujun Teng, and Paris Perdikaris. Understanding and mitigating gradient flow pathologies", "type": "text" } ], "index": 0 }, { "bbox": [ 115, 94, 507, 106 ], "spans": [ { "bbox": [ 115, 94, 507, 106 ], "score": 1.0, "content": "in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5):A3055–A3081,", "type": "text" } ], "index": 1 }, { "bbox": [ 115, 103, 147, 117 ], "spans": [ { "bbox": [ 115, 103, 147, 117 ], "score": 1.0, "content": "2021a.", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 105, 81, 507, 117 ] }, { "type": "text", "bbox": [ 107, 127, 506, 162 ], "lines": [ { "bbox": [ 106, 128, 506, 141 ], "spans": [ { "bbox": [ 106, 128, 506, 141 ], "score": 1.0, "content": "Sifan Wang, Hanwen Wang, and Paris Perdikaris. On the eigenvector bias of Fourier feature networks:", "type": "text" } ], "index": 3 }, { "bbox": [ 115, 139, 506, 151 ], "spans": [ { "bbox": [ 115, 139, 506, 151 ], "score": 1.0, "content": "From regression to solving multi-scale PDEs with physics-informed neural networks. Computer", "type": "text" } ], "index": 4 }, { "bbox": [ 115, 150, 396, 163 ], "spans": [ { "bbox": [ 115, 150, 396, 163 ], "score": 1.0, "content": "Methods in Applied Mechanics and Engineering, 384:113938, 2021b.", "type": "text" } ], "index": 5 } ], "index": 4, "bbox_fs": [ 106, 128, 506, 163 ] }, { "type": "text", "bbox": [ 105, 173, 505, 196 ], "lines": [ { "bbox": [ 105, 172, 506, 187 ], "spans": [ { "bbox": [ 105, 172, 506, 187 ], "score": 1.0, "content": "Sifan Wang, Shyam Sankaran, and Paris Perdikaris. Respecting causality is all you need for training", "type": "text" } ], "index": 6 }, { "bbox": [ 115, 185, 425, 197 ], "spans": [ { "bbox": [ 115, 185, 425, 197 ], "score": 1.0, "content": "physics-informed neural networks. arXiv preprint arXiv:2203.07404, 2022a.", "type": "text" } ], "index": 7 } ], "index": 6.5, "bbox_fs": [ 105, 172, 506, 197 ] }, { "type": "text", "bbox": [ 106, 208, 504, 231 ], "lines": [ { "bbox": [ 106, 208, 505, 221 ], "spans": [ { "bbox": [ 106, 208, 505, 221 ], "score": 1.0, "content": "Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why PINNs fail to train: A neural tangent", "type": "text" } ], "index": 8 }, { "bbox": [ 115, 219, 418, 232 ], "spans": [ { "bbox": [ 115, 219, 418, 232 ], "score": 1.0, "content": "kernel perspective. Journal of Computational Physics, 449:110768, 2022b.", "type": "text" } ], "index": 9 } ], "index": 8.5, "bbox_fs": [ 106, 208, 505, 232 ] }, { "type": "text", "bbox": [ 107, 243, 506, 277 ], "lines": [ { "bbox": [ 106, 242, 506, 256 ], "spans": [ { "bbox": [ 106, 242, 506, 256 ], "score": 1.0, "content": "Chen Xu, Ba Trung Cao, Yong Yuan, and Günther Meschke. Transfer learning based physics-", "type": "text" } ], "index": 10 }, { "bbox": [ 116, 254, 505, 266 ], "spans": [ { "bbox": [ 116, 254, 505, 266 ], "score": 1.0, "content": "informed neural networks for solving inverse problems in engineering structures under different", "type": "text" } ], "index": 11 }, { "bbox": [ 116, 265, 506, 279 ], "spans": [ { "bbox": [ 116, 265, 506, 279 ], "score": 1.0, "content": "loading scenarios. Computer Methods in Applied Mechanics and Engineering, 405:115852, 2023.", "type": "text" } ], "index": 12 } ], "index": 11, "bbox_fs": [ 106, 242, 506, 279 ] }, { "type": "text", "bbox": [ 107, 288, 504, 311 ], "lines": [ { "bbox": [ 106, 287, 505, 301 ], "spans": [ { "bbox": [ 106, 287, 505, 301 ], "score": 1.0, "content": "Wuzhe Xu, Yulong Lu, and Li Wang. Transfer learning enhanced DeepONet for long-time prediction", "type": "text" } ], "index": 13 }, { "bbox": [ 116, 299, 373, 312 ], "spans": [ { "bbox": [ 116, 299, 373, 312 ], "score": 1.0, "content": "of evolution equations. arXiv preprint arXiv:2212.04663, 2022.", "type": "text" } ], "index": 14 } ], "index": 13.5, "bbox_fs": [ 106, 287, 505, 312 ] }, { "type": "text", "bbox": [ 108, 323, 504, 345 ], "lines": [ { "bbox": [ 106, 323, 505, 335 ], "spans": [ { "bbox": [ 106, 323, 505, 335 ], "score": 1.0, "content": "Bing Yu and Weinan E. The deep Ritz method: A deep learning-based numerical algorithm for", "type": "text" } ], "index": 15 }, { "bbox": [ 115, 334, 498, 346 ], "spans": [ { "bbox": [ 115, 334, 498, 346 ], "score": 1.0, "content": "solving variational problems. Communications in Mathematics and Statistics, 6(1):1–12, 2018.", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 106, 323, 505, 346 ] }, { "type": "text", "bbox": [ 108, 371, 183, 384 ], "lines": [ { "bbox": [ 105, 370, 184, 386 ], "spans": [ { "bbox": [ 105, 370, 184, 386 ], "score": 1.0, "content": "A APPENDIX", "type": "text" } ], "index": 17 } ], "index": 17, "bbox_fs": [ 105, 370, 184, 386 ] }, { "type": "text", "bbox": [ 108, 397, 229, 409 ], "lines": [ { "bbox": [ 106, 397, 229, 410 ], "spans": [ { "bbox": [ 106, 397, 229, 410 ], "score": 1.0, "content": "A.1 TABLE OF NOTATIONS", "type": "text" } ], "index": 18 } ], "index": 18, "bbox_fs": [ 106, 397, 229, 410 ] }, { "type": "text", "bbox": [ 107, 419, 265, 431 ], "lines": [ { "bbox": [ 106, 419, 265, 432 ], "spans": [ { "bbox": [ 106, 419, 265, 432 ], "score": 1.0, "content": "A table of notations is given in Table 1.", "type": "text" } ], "index": 19 } ], "index": 19, "bbox_fs": [ 106, 419, 265, 432 ] }, { "type": "text", "bbox": [ 107, 446, 267, 459 ], "lines": [ { "bbox": [ 106, 447, 268, 459 ], "spans": [ { "bbox": [ 106, 447, 268, 459 ], "score": 1.0, "content": "A.2 TIME DIFFERENCING SCHEMES", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 106, 447, 268, 459 ] }, { "type": "text", "bbox": [ 108, 468, 226, 480 ], "lines": [ { "bbox": [ 106, 469, 226, 480 ], "spans": [ { "bbox": [ 106, 469, 226, 480 ], "score": 1.0, "content": "A.2.1 EXPLICIT SCHEMES", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 106, 469, 226, 480 ] }, { "type": "text", "bbox": [ 107, 488, 244, 500 ], "lines": [ { "bbox": [ 106, 489, 244, 501 ], "spans": [ { "bbox": [ 106, 489, 244, 501 ], "score": 1.0, "content": "First-order forward Euler scheme:", "type": "text" } ], "index": 22 } ], "index": 22, "bbox_fs": [ 106, 489, 244, 501 ] }, { "type": "interline_equation", "bbox": [ 238, 505, 373, 532 ], "lines": [ { "bbox": [ 238, 505, 373, 532 ], "spans": [ { "bbox": [ 238, 505, 373, 532 ], "score": 0.94, "content": "\\frac { \\boldsymbol { u } ^ { n + 1 } ( \\boldsymbol { x } ) - \\boldsymbol { u } ^ { n } ( \\boldsymbol { x } ) } { \\Delta t } = \\mathcal { N } \\left[ \\boldsymbol { u } ^ { n } ( \\boldsymbol { x } ) \\right] .", "type": "interline_equation", "image_path": "31a8e4fa8b749565c35ca5ba07a856690f8eae3858643b7fb0d656672825cb5c.jpg" } ] } ], "index": 23.5, "virtual_lines": [ { "bbox": [ 238, 505, 373, 518.5 ], "spans": [], "index": 23 }, { "bbox": [ 238, 518.5, 373, 532.0 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 107, 543, 324, 556 ], "lines": [ { "bbox": [ 105, 542, 325, 558 ], "spans": [ { "bbox": [ 105, 542, 325, 558 ], "score": 1.0, "content": "Second-order explicit Runge-Kutta (2nd RK) scheme:", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 105, 542, 325, 558 ] }, { "type": "interline_equation", "bbox": [ 202, 562, 409, 591 ], "lines": [ { "bbox": [ 202, 562, 409, 591 ], "spans": [ { "bbox": [ 202, 562, 409, 591 ], "score": 0.92, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { \\Delta t } { 2 } \\mathcal { N } [ u ^ { n } ( x ) ] \\right] .", "type": "interline_equation", "image_path": "146de5483eae76a7901a80a2db4033ba2b55e12ed7ae463ee61c20359e8ebdcd.jpg" } ] } ], "index": 26.5, "virtual_lines": [ { "bbox": [ 202, 562, 409, 576.5 ], "spans": [], "index": 26 }, { "bbox": [ 202, 576.5, 409, 591.0 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 604, 315, 616 ], "lines": [ { "bbox": [ 106, 604, 315, 617 ], "spans": [ { "bbox": [ 106, 604, 315, 617 ], "score": 1.0, "content": "Fouth-order explicit Runge-Kutta (4th RK) scheme:", "type": "text" } ], "index": 28 } ], "index": 28, "bbox_fs": [ 106, 604, 315, 617 ] }, { "type": "interline_equation", "bbox": [ 182, 622, 429, 736 ], "lines": [ { "bbox": [ 182, 622, 429, 736 ], "spans": [ { "bbox": [ 182, 622, 429, 736 ], "score": 0.95, "content": "\\begin{array} { l } { \\displaystyle \\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\frac { 1 } { 6 } \\left[ k _ { 1 } ( x ) + 2 k _ { 2 } ( x ) + 2 k _ { 3 } ( x ) + k _ { 4 } ( x ) \\right] , } \\\\ { \\displaystyle k _ { 1 } ( x ) = \\mathcal { N } [ u ^ { n } ( x ) ] , } \\\\ { \\displaystyle k _ { 2 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { \\Delta t } { 2 } \\mathcal { N } [ k _ { 1 } ( x ) ] \\right] , } \\\\ { \\displaystyle k _ { 3 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { \\Delta t } { 2 } \\mathcal { N } [ k _ { 2 } ( x ) ] \\right] , } \\\\ { \\displaystyle k _ { 4 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\Delta t \\mathcal { N } [ k _ { 3 } ( x ) ] \\right] . } \\end{array}", "type": "interline_equation", "image_path": "0e397af9967fbd25bcce50d51803aa32dd29176f1d0ba59b3c0d2f7b50309aaf.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 182, 622, 429, 660.0 ], "spans": [], "index": 29 }, { "bbox": [ 182, 660.0, 429, 698.0 ], "spans": [], "index": 30 }, { "bbox": [ 182, 698.0, 429, 736.0 ], "spans": [], "index": 31 } ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 106, 259, 500, 572 ], "blocks": [ { "type": "table_caption", "bbox": [ 251, 238, 360, 249 ], "group_id": 0, "lines": [ { "bbox": [ 250, 236, 360, 250 ], "spans": [ { "bbox": [ 250, 236, 360, 250 ], "score": 1.0, "content": "Table 6: Table of notations", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 106, 259, 500, 572 ], "group_id": 0, "lines": [ { "bbox": [ 106, 259, 500, 572 ], "spans": [ { "bbox": [ 106, 259, 500, 572 ], "score": 0.979, "html": "
NotationMeaning
PINNPhysics-informed neural network
PDEPartial differential equation
TL-DPINNTransfer learning enhanced discrete PINN
TL-DPINN1Crank-Nicolson time differencing in TL-DPINN
TL-DPINN1Gauss-Legendre time differencing in TL-DPINN
Lor LnPhysics-informed loss function
Differential operator, such as N(u) = Ucr
RThe residual term of the evolutionary PDE, for example R(u) = ut - Uxx
ΩSpatial domain
8The boundary of the spatial domain
TEnd time
NtTimestamps number
NbThe collocation points number on ∂Ω
Nu,NrThe collocation points number in Ω or Ω × [0,T]
u(t,x）The exact solution to the evolutionary PDE
un(x)The time differencing scheme solution to the evolutionary PDE
ugn(x)The discrete PINN solution to the evolutionary PDE
hjThe j component in the output of the last hidden layer of the neural network
x,xr,xb tortnSpatial coordinate
0 or 0n,Wn,wnTemporal coordinate Neural network parameters
△t or TTime step, the interval time between two adjacent timestamps
Mo,M1
Number of maximum iterations in different training stages
nThe learning rate in gradient descent methods
EThe threshold value
I- The L² norm of a function, defined by |lfll = (Jo If(x)|²dx)²
", "type": "table", "image_path": "0a11662fb2727b2da545c6b426aaca387696f5558e6933dc1e28c6bb7717b785.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 259, 500, 363.3333333333333 ], "spans": [], "index": 1 }, { "bbox": [ 106, 363.3333333333333, 500, 467.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 467.66666666666663, 500, 572.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 } ], "page_idx": 12, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "13", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 309, 39 ], "spans": [ { "bbox": [ 106, 25, 309, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "table", "bbox": [ 106, 259, 500, 572 ], "blocks": [ { "type": "table_caption", "bbox": [ 251, 238, 360, 249 ], "group_id": 0, "lines": [ { "bbox": [ 250, 236, 360, 250 ], "spans": [ { "bbox": [ 250, 236, 360, 250 ], "score": 1.0, "content": "Table 6: Table of notations", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "table_body", "bbox": [ 106, 259, 500, 572 ], "group_id": 0, "lines": [ { "bbox": [ 106, 259, 500, 572 ], "spans": [ { "bbox": [ 106, 259, 500, 572 ], "score": 0.979, "html": "
NotationMeaning
PINNPhysics-informed neural network
PDEPartial differential equation
TL-DPINNTransfer learning enhanced discrete PINN
TL-DPINN1Crank-Nicolson time differencing in TL-DPINN
TL-DPINN1Gauss-Legendre time differencing in TL-DPINN
Lor LnPhysics-informed loss function
Differential operator, such as N(u) = Ucr
RThe residual term of the evolutionary PDE, for example R(u) = ut - Uxx
ΩSpatial domain
8The boundary of the spatial domain
TEnd time
NtTimestamps number
NbThe collocation points number on ∂Ω
Nu,NrThe collocation points number in Ω or Ω × [0,T]
u(t,x）The exact solution to the evolutionary PDE
un(x)The time differencing scheme solution to the evolutionary PDE
ugn(x)The discrete PINN solution to the evolutionary PDE
hjThe j component in the output of the last hidden layer of the neural network
x,xr,xb tortnSpatial coordinate
0 or 0n,Wn,wnTemporal coordinate Neural network parameters
△t or TTime step, the interval time between two adjacent timestamps
Mo,M1
Number of maximum iterations in different training stages
nThe learning rate in gradient descent methods
EThe threshold value
I- The L² norm of a function, defined by |lfll = (Jo If(x)|²dx)²
", "type": "table", "image_path": "0a11662fb2727b2da545c6b426aaca387696f5558e6933dc1e28c6bb7717b785.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 106, 259, 500, 363.3333333333333 ], "spans": [], "index": 1 }, { "bbox": [ 106, 363.3333333333333, 500, 467.66666666666663 ], "spans": [], "index": 2 }, { "bbox": [ 106, 467.66666666666663, 500, 572.0 ], "spans": [], "index": 3 } ] } ], "index": 1.0 } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 108, 82, 225, 94 ], "lines": [ { "bbox": [ 106, 82, 225, 94 ], "spans": [ { "bbox": [ 106, 82, 225, 94 ], "score": 1.0, "content": "A.2.2 IMPLICIT SCHEMES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 101, 252, 113 ], "lines": [ { "bbox": [ 105, 100, 252, 114 ], "spans": [ { "bbox": [ 105, 100, 252, 114 ], "score": 1.0, "content": "First-order backward Euler scheme:", "type": "text" } ], "index": 1 } ], "index": 1 }, { "type": "interline_equation", "bbox": [ 232, 113, 380, 138 ], "lines": [ { "bbox": [ 232, 113, 380, 138 ], "spans": [ { "bbox": [ 232, 113, 380, 138 ], "score": 0.94, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\mathcal { N } \\left[ u ^ { n + 1 } ( x ) \\right] .", "type": "interline_equation", "image_path": "1c0c38a8c84f8d4730eff665cf269b3e4fe43c5db5325e5ae3c89a06c27837e1.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 232, 113, 380, 138 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 107, 146, 247, 157 ], "lines": [ { "bbox": [ 106, 144, 248, 159 ], "spans": [ { "bbox": [ 106, 144, 248, 159 ], "score": 1.0, "content": "Second-order Trapezoidal scheme:", "type": "text" } ], "index": 3 } ], "index": 3 }, { "type": "interline_equation", "bbox": [ 204, 159, 407, 185 ], "lines": [ { "bbox": [ 204, 159, 407, 185 ], "spans": [ { "bbox": [ 204, 159, 407, 185 ], "score": 0.91, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\frac { \\mathcal { N } [ u ^ { n + 1 } ( x ) ] + \\mathcal { N } [ u ^ { n + } ( x ) ] } { 2 } .", "type": "interline_equation", "image_path": "9c1ed34751d41b17f37995bad7a811c69a4de309e33e1fc344964c125cbd8d1a.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 204, 159, 407, 185 ], "spans": [], "index": 4 } ] }, { "type": "text", "bbox": [ 107, 192, 354, 204 ], "lines": [ { "bbox": [ 106, 191, 355, 205 ], "spans": [ { "bbox": [ 106, 191, 355, 205 ], "score": 1.0, "content": "Second-order Crank-Nicolson scheme (used in TL-DPINN1):", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "interline_equation", "bbox": [ 208, 205, 403, 234 ], "lines": [ { "bbox": [ 208, 205, 403, 234 ], "spans": [ { "bbox": [ 208, 205, 403, 234 ], "score": 0.92, "content": "\\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\mathcal { N } \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n + } ( x ) } { 2 } \\right] .", "type": "interline_equation", "image_path": "874db3c9535de0c778f28e38bc46b1b229a03d3b4f738b468705420d53198b6a.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 208, 205, 403, 219.5 ], "spans": [], "index": 6 }, { "bbox": [ 208, 219.5, 403, 234.0 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 240, 347, 252 ], "lines": [ { "bbox": [ 105, 239, 349, 255 ], "spans": [ { "bbox": [ 105, 239, 349, 255 ], "score": 1.0, "content": "Forth-order Gauss-Legendre scheme (used in TL-DPINN2):", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "interline_equation", "bbox": [ 183, 254, 428, 348 ], "lines": [ { "bbox": [ 183, 254, 428, 348 ], "spans": [ { "bbox": [ 183, 254, 428, 348 ], "score": 0.96, "content": "\\begin{array} { l } { \\displaystyle \\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\frac { k _ { 1 } ( x ) + k _ { 2 } ( x ) } { 2 } , } \\\\ { \\displaystyle k _ { 1 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { 1 } { 4 } \\Delta t k _ { 1 } ( x ) + \\left( \\frac { 1 } { 4 } + \\frac { \\sqrt { 3 } } { 6 } \\right) \\Delta t k _ { 2 } ( x ) \\right] , } \\\\ { \\displaystyle k _ { 2 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\left( \\frac { 1 } { 4 } - \\frac { \\sqrt { 3 } } { 6 } \\right) \\Delta t k _ { 1 } ( x ) + \\frac { 1 } { 4 } \\Delta t k _ { 2 } ( x ) . \\right] } \\end{array}", "type": "interline_equation", "image_path": "c0f72c50dbcc3572906c94d3ff0ab11506d9fdbaefa6c7daaefa3d370f77fb30.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 183, 254, 428, 285.3333333333333 ], "spans": [], "index": 9 }, { "bbox": [ 183, 285.3333333333333, 428, 316.66666666666663 ], "spans": [], "index": 10 }, { "bbox": [ 183, 316.66666666666663, 428, 347.99999999999994 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 107, 354, 336, 366 ], "lines": [ { "bbox": [ 105, 353, 338, 369 ], "spans": [ { "bbox": [ 105, 353, 300, 369 ], "score": 1.0, "content": "The general form of Runge–Kutta schemes with", "type": "text" }, { "bbox": [ 301, 357, 307, 366 ], "score": 0.83, "content": "q", "type": "inline_equation" }, { "bbox": [ 307, 353, 338, 369 ], "score": 1.0, "content": "stages:", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "interline_equation", "bbox": [ 195, 368, 415, 442 ], "lines": [ { "bbox": [ 195, 368, 415, 442 ], "spans": [ { "bbox": [ 195, 368, 415, 442 ], "score": 0.94, "content": "\\begin{array} { l } { \\displaystyle \\frac { u ^ { n + 1 } ( { \\boldsymbol { x } } ) - u ^ { n } ( { \\boldsymbol { x } } ) } { \\Delta t } = \\sum _ { i = 1 } ^ { q } b _ { i } k _ { i } ( { \\boldsymbol { x } } ) , } \\\\ { \\displaystyle k _ { i } ( { \\boldsymbol { x } } ) = \\mathcal { N } \\left[ u ^ { n } ( { \\boldsymbol { x } } ) + \\Delta t \\sum _ { j = 1 } ^ { q } a _ { i j } k _ { j } ( { \\boldsymbol { x } } ) \\right] , i = 1 , . . . , q . } \\end{array}", "type": "interline_equation", "image_path": "b15b841b24e323ec5cb20112ed8395c5ba99b6d142774557fb968121d3d9f006.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 195, 368, 415, 382.8 ], "spans": [], "index": 13 }, { "bbox": [ 195, 382.8, 415, 397.6 ], "spans": [], "index": 14 }, { "bbox": [ 195, 397.6, 415, 412.40000000000003 ], "spans": [], "index": 15 }, { "bbox": [ 195, 412.40000000000003, 415, 427.20000000000005 ], "spans": [], "index": 16 }, { "bbox": [ 195, 427.20000000000005, 415, 442.00000000000006 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 107, 443, 505, 488 ], "lines": [ { "bbox": [ 105, 442, 506, 457 ], "spans": [ { "bbox": [ 105, 442, 196, 457 ], "score": 1.0, "content": "where the coefficients", "type": "text" }, { "bbox": [ 197, 443, 232, 456 ], "score": 0.95, "content": "\\{ a _ { i j } , b _ { i } \\}", "type": "inline_equation" }, { "bbox": [ 232, 442, 506, 457 ], "score": 1.0, "content": "are determined. Since there are no significant differences for PINN", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 455, 506, 467 ], "spans": [ { "bbox": [ 105, 455, 271, 467 ], "score": 1.0, "content": "approximation of explicit schemes (i.e.", "type": "text" }, { "bbox": [ 272, 455, 306, 467 ], "score": 0.92, "content": "a _ { i j } = 0", "type": "inline_equation" }, { "bbox": [ 306, 455, 335, 467 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 335, 455, 361, 466 ], "score": 0.89, "content": "j \\geq i", "type": "inline_equation" }, { "bbox": [ 362, 455, 506, 467 ], "score": 1.0, "content": ") and implicit schemes (i.e. not all", "type": "text" } ], "index": 19 }, { "bbox": [ 107, 466, 505, 478 ], "spans": [ { "bbox": [ 107, 466, 140, 478 ], "score": 0.91, "content": "a _ { i j } = 0", "type": "inline_equation" }, { "bbox": [ 140, 466, 155, 478 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 156, 466, 180, 477 ], "score": 0.88, "content": "j \\geq i", "type": "inline_equation" }, { "bbox": [ 180, 466, 505, 478 ], "score": 1.0, "content": "), we prefer implicit schemes as they possess the A-stable property to make the", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 477, 291, 489 ], "spans": [ { "bbox": [ 105, 477, 291, 489 ], "score": 1.0, "content": "time-marching process stable Butcher (2007).", "type": "text" } ], "index": 21 } ], "index": 19.5 }, { "type": "title", "bbox": [ 108, 500, 243, 512 ], "lines": [ { "bbox": [ 106, 501, 242, 513 ], "spans": [ { "bbox": [ 106, 501, 242, 513 ], "score": 1.0, "content": "A.3 THEORETICAL ANALYSIS", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "title", "bbox": [ 107, 520, 246, 533 ], "lines": [ { "bbox": [ 106, 521, 247, 533 ], "spans": [ { "bbox": [ 106, 521, 247, 533 ], "score": 1.0, "content": "A.3.1 PROOF OF THEOREM 4.1", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 540, 379, 553 ], "lines": [ { "bbox": [ 105, 539, 380, 555 ], "spans": [ { "bbox": [ 105, 539, 207, 555 ], "score": 1.0, "content": "Proof. We split the error", "type": "text" }, { "bbox": [ 208, 540, 319, 553 ], "score": 0.91, "content": "e ^ { n } ( x ) = u ( t _ { n } , x ) - u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 319, 539, 380, 555 ], "score": 1.0, "content": "into two parts:", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 210, 554, 402, 579 ], "lines": [ { "bbox": [ 210, 554, 402, 579 ], "spans": [ { "bbox": [ 210, 554, 402, 579 ], "score": 0.89, "content": "e ^ { n } ( x ) \\ = \\ { \\frac { u ( t _ { n } , x ) - u ^ { n } ( x ) } { { \\underline { { \\circ } } } \\varepsilon ^ { n } ( x ) } } + { \\underline { { u ^ { n } ( x ) - u _ { \\theta ^ { n } } ( x ) } } }", "type": "interline_equation", "image_path": "4c31e733fa3d0c26b1f0b541d0c3d15c9bf00ae3c1aa6b71e778cc8eb51cf083.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 210, 554, 402, 579 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 582, 505, 638 ], "lines": [ { "bbox": [ 106, 582, 505, 595 ], "spans": [ { "bbox": [ 106, 582, 163, 595 ], "score": 1.0, "content": "The first term", "type": "text" }, { "bbox": [ 163, 583, 188, 594 ], "score": 0.92, "content": "\\xi ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 188, 582, 505, 595 ], "score": 1.0, "content": "estimates the error from the Crank-Nicolson time differencing schemes. From", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 591, 505, 606 ], "spans": [ { "bbox": [ 105, 591, 198, 606 ], "score": 1.0, "content": "Lemma A.1 we have", "type": "text" }, { "bbox": [ 198, 593, 254, 605 ], "score": 0.94, "content": "\\| \\xi ^ { n } \\| \\le C \\tau ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 255, 591, 336, 606 ], "score": 1.0, "content": ". The second term", "type": "text" }, { "bbox": [ 336, 594, 356, 605 ], "score": 0.91, "content": "\\eta ( x )", "type": "inline_equation" }, { "bbox": [ 356, 591, 505, 606 ], "score": 1.0, "content": "estimates the error from the PINN", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 604, 505, 618 ], "spans": [ { "bbox": [ 105, 604, 469, 618 ], "score": 1.0, "content": "approximation in space and the cumulative effect of time. From Lemma A.2 we have", "type": "text" }, { "bbox": [ 470, 604, 505, 617 ], "score": 0.9, "content": "\\| \\eta ^ { n } \\| \\leq", "type": "inline_equation" } ], "index": 28 }, { "bbox": [ 106, 615, 505, 637 ], "spans": [ { "bbox": [ 106, 615, 212, 637 ], "score": 0.93, "content": "C \\sqrt { t _ { n } } ( \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } )", "type": "inline_equation" }, { "bbox": [ 213, 617, 433, 633 ], "score": 1.0, "content": ". Then by the triangular inequality, we finish the proof.", "type": "text" }, { "bbox": [ 495, 621, 505, 631 ], "score": 0.998, "content": "□", "type": "text" } ], "index": 29 } ], "index": 27.5 }, { "type": "title", "bbox": [ 107, 648, 347, 660 ], "lines": [ { "bbox": [ 106, 648, 348, 661 ], "spans": [ { "bbox": [ 106, 648, 348, 661 ], "score": 1.0, "content": "A.3.2 SOME LEMMAS IN THE PROOF OF THEOREM 4.1", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 105, 667, 505, 691 ], "lines": [ { "bbox": [ 105, 667, 505, 681 ], "spans": [ { "bbox": [ 105, 667, 193, 681 ], "score": 1.0, "content": "Lemma A.1. Denote", "type": "text" }, { "bbox": [ 193, 668, 299, 680 ], "score": 0.93, "content": "\\xi ^ { n } ( x ) = u ( t _ { n } , x ) - u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 299, 667, 329, 681 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 329, 668, 362, 680 ], "score": 0.92, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 363, 667, 505, 681 ], "score": 1.0, "content": "is the exact solution to evolutionary", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 678, 505, 691 ], "spans": [ { "bbox": [ 105, 678, 149, 691 ], "score": 1.0, "content": "PDEs and", "type": "text" }, { "bbox": [ 149, 679, 174, 691 ], "score": 0.93, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 174, 678, 505, 691 ], "score": 1.0, "content": "is the Crank-Nicolson time differencing discrete solution, then we have the estimate", "type": "text" } ], "index": 32 } ], "index": 31.5 }, { "type": "interline_equation", "bbox": [ 277, 693, 333, 707 ], "lines": [ { "bbox": [ 277, 693, 333, 707 ], "spans": [ { "bbox": [ 277, 693, 333, 707 ], "score": 0.92, "content": "\\| \\xi ^ { n } \\| \\leq C \\tau ^ { 2 } ,", "type": "interline_equation", "image_path": "813a42b1ba716880fd6176e26d29b07e8316ff4f1c36204c1e7d2218f4546e14.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 277, 693, 333, 707 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 105, 709, 506, 731 ], "lines": [ { "bbox": [ 104, 709, 505, 722 ], "spans": [ { "bbox": [ 104, 709, 178, 722 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 179, 711, 187, 720 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 187, 709, 286, 722 ], "score": 1.0, "content": "independent of time step", "type": "text" }, { "bbox": [ 286, 713, 293, 720 ], "score": 0.74, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 293, 709, 401, 722 ], "score": 1.0, "content": ", collocation points number", "type": "text" }, { "bbox": [ 402, 711, 415, 721 ], "score": 0.87, "content": "N _ { r }", "type": "inline_equation" }, { "bbox": [ 415, 709, 505, 722 ], "score": 1.0, "content": "and trained loss value", "type": "text" } ], "index": 34 }, { "bbox": [ 107, 718, 126, 734 ], "spans": [ { "bbox": [ 107, 722, 119, 730 ], "score": 0.83, "content": "{ \\mathcal { L } } ^ { n }", "type": "inline_equation" }, { "bbox": [ 120, 718, 126, 734 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 35 } ], "index": 34.5 } ], "page_idx": 13, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "14", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 108, 82, 225, 94 ], "lines": [ { "bbox": [ 106, 82, 225, 94 ], "spans": [ { "bbox": [ 106, 82, 225, 94 ], "score": 1.0, "content": "A.2.2 IMPLICIT SCHEMES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 101, 252, 113 ], "lines": [ { "bbox": [ 105, 100, 252, 114 ], "spans": [ { "bbox": [ 105, 100, 252, 114 ], "score": 1.0, "content": "First-order backward Euler scheme:", "type": "text" } ], "index": 1 } ], "index": 1, "bbox_fs": [ 105, 100, 252, 114 ] }, { "type": "interline_equation", "bbox": [ 232, 113, 380, 138 ], "lines": [ { "bbox": [ 232, 113, 380, 138 ], "spans": [ { "bbox": [ 232, 113, 380, 138 ], "score": 0.94, "content": "\\frac { u ^ { n + 1 } ( x ) - 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u ^ { n } ( x ) } { \\Delta t } = \\mathcal { N } \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n + } ( x ) } { 2 } \\right] .", "type": "interline_equation", "image_path": "874db3c9535de0c778f28e38bc46b1b229a03d3b4f738b468705420d53198b6a.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 208, 205, 403, 219.5 ], "spans": [], "index": 6 }, { "bbox": [ 208, 219.5, 403, 234.0 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 240, 347, 252 ], "lines": [ { "bbox": [ 105, 239, 349, 255 ], "spans": [ { "bbox": [ 105, 239, 349, 255 ], "score": 1.0, "content": "Forth-order Gauss-Legendre scheme (used in TL-DPINN2):", "type": "text" } ], "index": 8 } ], "index": 8, "bbox_fs": [ 105, 239, 349, 255 ] }, { "type": "interline_equation", "bbox": [ 183, 254, 428, 348 ], "lines": [ { "bbox": [ 183, 254, 428, 348 ], "spans": [ { "bbox": [ 183, 254, 428, 348 ], "score": 0.96, "content": "\\begin{array} { l } { \\displaystyle \\frac { u ^ { n + 1 } ( x ) - u ^ { n } ( x ) } { \\Delta t } = \\frac { k _ { 1 } ( x ) + k _ { 2 } ( x ) } { 2 } , } \\\\ { \\displaystyle k _ { 1 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\frac { 1 } { 4 } \\Delta t k _ { 1 } ( x ) + \\left( \\frac { 1 } { 4 } + \\frac { \\sqrt { 3 } } { 6 } \\right) \\Delta t k _ { 2 } ( x ) \\right] , } \\\\ { \\displaystyle k _ { 2 } ( x ) = \\mathcal { N } \\left[ u ^ { n } ( x ) + \\left( \\frac { 1 } { 4 } - \\frac { \\sqrt { 3 } } { 6 } \\right) \\Delta t k _ { 1 } ( x ) + \\frac { 1 } { 4 } \\Delta t k _ { 2 } ( x ) . \\right] } \\end{array}", "type": "interline_equation", "image_path": "c0f72c50dbcc3572906c94d3ff0ab11506d9fdbaefa6c7daaefa3d370f77fb30.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 183, 254, 428, 285.3333333333333 ], "spans": [], "index": 9 }, { "bbox": [ 183, 285.3333333333333, 428, 316.66666666666663 ], "spans": [], "index": 10 }, { "bbox": [ 183, 316.66666666666663, 428, 347.99999999999994 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 107, 354, 336, 366 ], "lines": [ { "bbox": [ 105, 353, 338, 369 ], "spans": [ { "bbox": [ 105, 353, 300, 369 ], "score": 1.0, "content": "The general form of Runge–Kutta schemes with", "type": "text" }, { "bbox": [ 301, 357, 307, 366 ], "score": 0.83, "content": "q", "type": "inline_equation" }, { "bbox": [ 307, 353, 338, 369 ], "score": 1.0, "content": "stages:", "type": "text" } ], "index": 12 } ], "index": 12, "bbox_fs": [ 105, 353, 338, 369 ] }, { "type": "interline_equation", "bbox": [ 195, 368, 415, 442 ], "lines": [ { "bbox": [ 195, 368, 415, 442 ], "spans": [ { "bbox": [ 195, 368, 415, 442 ], "score": 0.94, "content": "\\begin{array} { l } { \\displaystyle \\frac { u ^ { n + 1 } ( { \\boldsymbol { x } } ) - u ^ { n } ( { \\boldsymbol { x } } ) } { \\Delta t } = \\sum _ { i = 1 } ^ { q } b _ { i } k _ { i } ( { \\boldsymbol { x } } ) , } \\\\ { \\displaystyle k _ { i } ( { \\boldsymbol { x } } ) = \\mathcal { N } \\left[ u ^ { n } ( { \\boldsymbol { x } } ) + \\Delta t \\sum _ { j = 1 } ^ { q } a _ { i j } k _ { j } ( { \\boldsymbol { x } } ) \\right] , i = 1 , . . . , q . } \\end{array}", "type": "interline_equation", "image_path": "b15b841b24e323ec5cb20112ed8395c5ba99b6d142774557fb968121d3d9f006.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 195, 368, 415, 382.8 ], "spans": [], "index": 13 }, { "bbox": [ 195, 382.8, 415, 397.6 ], "spans": [], "index": 14 }, { "bbox": [ 195, 397.6, 415, 412.40000000000003 ], "spans": [], "index": 15 }, { "bbox": [ 195, 412.40000000000003, 415, 427.20000000000005 ], "spans": [], "index": 16 }, { "bbox": [ 195, 427.20000000000005, 415, 442.00000000000006 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 107, 443, 505, 488 ], "lines": [ { "bbox": [ 105, 442, 506, 457 ], "spans": [ { "bbox": [ 105, 442, 196, 457 ], "score": 1.0, "content": "where the coefficients", "type": "text" }, { "bbox": [ 197, 443, 232, 456 ], "score": 0.95, "content": "\\{ a _ { i j } , b _ { i } \\}", "type": "inline_equation" }, { "bbox": [ 232, 442, 506, 457 ], "score": 1.0, "content": "are determined. Since there are no significant differences for PINN", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 455, 506, 467 ], "spans": [ { "bbox": [ 105, 455, 271, 467 ], "score": 1.0, "content": "approximation of explicit schemes (i.e.", "type": "text" }, { "bbox": [ 272, 455, 306, 467 ], "score": 0.92, "content": "a _ { i j } = 0", "type": "inline_equation" }, { "bbox": [ 306, 455, 335, 467 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 335, 455, 361, 466 ], "score": 0.89, "content": "j \\geq i", "type": "inline_equation" }, { "bbox": [ 362, 455, 506, 467 ], "score": 1.0, "content": ") and implicit schemes (i.e. not all", "type": "text" } ], "index": 19 }, { "bbox": [ 107, 466, 505, 478 ], "spans": [ { "bbox": [ 107, 466, 140, 478 ], "score": 0.91, "content": "a _ { i j } = 0", "type": "inline_equation" }, { "bbox": [ 140, 466, 155, 478 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 156, 466, 180, 477 ], "score": 0.88, "content": "j \\geq i", "type": "inline_equation" }, { "bbox": [ 180, 466, 505, 478 ], "score": 1.0, "content": "), we prefer implicit schemes as they possess the A-stable property to make the", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 477, 291, 489 ], "spans": [ { "bbox": [ 105, 477, 291, 489 ], "score": 1.0, "content": "time-marching process stable Butcher (2007).", "type": "text" } ], "index": 21 } ], "index": 19.5, "bbox_fs": [ 105, 442, 506, 489 ] }, { "type": "title", "bbox": [ 108, 500, 243, 512 ], "lines": [ { "bbox": [ 106, 501, 242, 513 ], "spans": [ { "bbox": [ 106, 501, 242, 513 ], "score": 1.0, "content": "A.3 THEORETICAL ANALYSIS", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "title", "bbox": [ 107, 520, 246, 533 ], "lines": [ { "bbox": [ 106, 521, 247, 533 ], "spans": [ { "bbox": [ 106, 521, 247, 533 ], "score": 1.0, "content": "A.3.1 PROOF OF THEOREM 4.1", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 540, 379, 553 ], "lines": [ { "bbox": [ 105, 539, 380, 555 ], "spans": [ { "bbox": [ 105, 539, 207, 555 ], "score": 1.0, "content": "Proof. We split the error", "type": "text" }, { "bbox": [ 208, 540, 319, 553 ], "score": 0.91, "content": "e ^ { n } ( x ) = u ( t _ { n } , x ) - u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 319, 539, 380, 555 ], "score": 1.0, "content": "into two parts:", "type": "text" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 105, 539, 380, 555 ] }, { "type": "interline_equation", "bbox": [ 210, 554, 402, 579 ], "lines": [ { "bbox": [ 210, 554, 402, 579 ], "spans": [ { "bbox": [ 210, 554, 402, 579 ], "score": 0.89, "content": "e ^ { n } ( x ) \\ = \\ { \\frac { u ( t _ { n } , x ) - u ^ { n } ( x ) } { { \\underline { { \\circ } } } \\varepsilon ^ { n } ( x ) } } + { \\underline { { u ^ { n } ( x ) - u _ { \\theta ^ { n } } ( x ) } } }", "type": "interline_equation", "image_path": "4c31e733fa3d0c26b1f0b541d0c3d15c9bf00ae3c1aa6b71e778cc8eb51cf083.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 210, 554, 402, 579 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 106, 582, 505, 638 ], "lines": [ { "bbox": [ 106, 582, 505, 595 ], "spans": [ { "bbox": [ 106, 582, 163, 595 ], "score": 1.0, "content": "The first term", "type": "text" }, { "bbox": [ 163, 583, 188, 594 ], "score": 0.92, "content": "\\xi ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 188, 582, 505, 595 ], "score": 1.0, "content": "estimates the error from the Crank-Nicolson time differencing schemes. From", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 591, 505, 606 ], "spans": [ { "bbox": [ 105, 591, 198, 606 ], "score": 1.0, "content": "Lemma A.1 we have", "type": "text" }, { "bbox": [ 198, 593, 254, 605 ], "score": 0.94, "content": "\\| \\xi ^ { n } \\| \\le C \\tau ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 255, 591, 336, 606 ], "score": 1.0, "content": ". The second term", "type": "text" }, { "bbox": [ 336, 594, 356, 605 ], "score": 0.91, "content": "\\eta ( x )", "type": "inline_equation" }, { "bbox": [ 356, 591, 505, 606 ], "score": 1.0, "content": "estimates the error from the PINN", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 604, 505, 618 ], "spans": [ { "bbox": [ 105, 604, 469, 618 ], "score": 1.0, "content": "approximation in space and the cumulative effect of time. From Lemma A.2 we have", "type": "text" }, { "bbox": [ 470, 604, 505, 617 ], "score": 0.9, "content": "\\| \\eta ^ { n } \\| \\leq", "type": "inline_equation" } ], "index": 28 }, { "bbox": [ 106, 615, 505, 637 ], "spans": [ { "bbox": [ 106, 615, 212, 637 ], "score": 0.93, "content": "C \\sqrt { t _ { n } } ( \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } )", "type": "inline_equation" }, { "bbox": [ 213, 617, 433, 633 ], "score": 1.0, "content": ". Then by the triangular inequality, we finish the proof.", "type": "text" }, { "bbox": [ 495, 621, 505, 631 ], "score": 0.998, "content": "□", "type": "text" } ], "index": 29 } ], "index": 27.5, "bbox_fs": [ 105, 582, 505, 637 ] }, { "type": "title", "bbox": [ 107, 648, 347, 660 ], "lines": [ { "bbox": [ 106, 648, 348, 661 ], "spans": [ { "bbox": [ 106, 648, 348, 661 ], "score": 1.0, "content": "A.3.2 SOME LEMMAS IN THE PROOF OF THEOREM 4.1", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 105, 667, 505, 691 ], "lines": [ { "bbox": [ 105, 667, 505, 681 ], "spans": [ { "bbox": [ 105, 667, 193, 681 ], "score": 1.0, "content": "Lemma A.1. Denote", "type": "text" }, { "bbox": [ 193, 668, 299, 680 ], "score": 0.93, "content": "\\xi ^ { n } ( x ) = u ( t _ { n } , x ) - u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 299, 667, 329, 681 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 329, 668, 362, 680 ], "score": 0.92, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 363, 667, 505, 681 ], "score": 1.0, "content": "is the exact solution to evolutionary", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 678, 505, 691 ], "spans": [ { "bbox": [ 105, 678, 149, 691 ], "score": 1.0, "content": "PDEs and", "type": "text" }, { "bbox": [ 149, 679, 174, 691 ], "score": 0.93, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 174, 678, 505, 691 ], "score": 1.0, "content": "is the Crank-Nicolson time differencing discrete solution, then we have the estimate", "type": "text" } ], "index": 32 } ], "index": 31.5, "bbox_fs": [ 105, 667, 505, 691 ] }, { "type": "interline_equation", "bbox": [ 277, 693, 333, 707 ], "lines": [ { "bbox": [ 277, 693, 333, 707 ], "spans": [ { "bbox": [ 277, 693, 333, 707 ], "score": 0.92, "content": "\\| \\xi ^ { n } \\| \\leq C \\tau ^ { 2 } ,", "type": "interline_equation", "image_path": "813a42b1ba716880fd6176e26d29b07e8316ff4f1c36204c1e7d2218f4546e14.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 277, 693, 333, 707 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 105, 709, 506, 731 ], "lines": [ { "bbox": [ 104, 709, 505, 722 ], "spans": [ { "bbox": [ 104, 709, 178, 722 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 179, 711, 187, 720 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 187, 709, 286, 722 ], "score": 1.0, "content": "independent of time step", "type": "text" }, { "bbox": [ 286, 713, 293, 720 ], "score": 0.74, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 293, 709, 401, 722 ], "score": 1.0, "content": ", collocation points number", "type": "text" }, { "bbox": [ 402, 711, 415, 721 ], "score": 0.87, "content": "N _ { r }", "type": "inline_equation" }, { "bbox": [ 415, 709, 505, 722 ], "score": 1.0, "content": "and trained loss value", "type": "text" } ], "index": 34 }, { "bbox": [ 107, 718, 126, 734 ], "spans": [ { "bbox": [ 107, 722, 119, 730 ], "score": 0.83, "content": "{ \\mathcal { L } } ^ { n }", "type": "inline_equation" }, { "bbox": [ 120, 718, 126, 734 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 35 } ], "index": 34.5, "bbox_fs": [ 104, 709, 505, 734 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 117 ], "lines": [ { "bbox": [ 105, 81, 505, 96 ], "spans": [ { "bbox": [ 105, 81, 210, 96 ], "score": 1.0, "content": "Proof. Firstly, we replace", "type": "text" }, { "bbox": [ 210, 83, 236, 95 ], "score": 0.92, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 236, 81, 505, 96 ], "score": 1.0, "content": "in the Crank-Nicolson time differencing scheme by the evolutionary", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 92, 506, 106 ], "spans": [ { "bbox": [ 105, 92, 171, 106 ], "score": 1.0, "content": "PDE’s solution", "type": "text" }, { "bbox": [ 171, 94, 204, 106 ], "score": 0.92, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 205, 92, 506, 106 ], "score": 1.0, "content": "and compare the difference. This can be achieved by the standard Taylor", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 444, 119 ], "spans": [ { "bbox": [ 106, 104, 343, 118 ], "score": 1.0, "content": "expansion techniques. We do Taylor expansion at the point", "type": "text" }, { "bbox": [ 344, 104, 417, 119 ], "score": 0.94, "content": "\\begin{array} { r } { t _ { n + \\frac { 1 } { 2 } } = ( n + \\frac { 1 } { 2 } ) \\tau } \\end{array}", "type": "inline_equation" }, { "bbox": [ 417, 104, 444, 118 ], "score": 1.0, "content": "to get", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "interline_equation", "bbox": [ 209, 124, 401, 149 ], "lines": [ { "bbox": [ 209, 124, 401, 149 ], "spans": [ { "bbox": [ 209, 124, 401, 149 ], "score": 0.95, "content": "\\frac { u ( t _ { n + 1 } , x ) - u ( t _ { n } , x ) } { \\tau } = u _ { t } ( t _ { n + \\frac { 1 } { 2 } } , x ) + \\mathcal { O } ( \\tau ^ { 2 } ) ,", "type": "interline_equation", "image_path": "78622b632684f69fbc5a0e98a38a4c94791dde4669fe4fb5f8154f9cb9f88d78.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 209, 124, 401, 149 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 106, 154, 124, 165 ], "lines": [ { "bbox": [ 105, 155, 123, 165 ], "spans": [ { "bbox": [ 105, 155, 123, 165 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "interline_equation", "bbox": [ 189, 162, 422, 190 ], "lines": [ { "bbox": [ 189, 162, 422, 190 ], "spans": [ { "bbox": [ 189, 162, 422, 190 ], "score": 0.93, "content": "\\mathcal { N } \\left[ \\frac { u ( t _ { n + 1 } , x ) + u ( t _ { n } , x ) } { 2 } \\right] = \\mathcal { N } \\left[ u ( t _ { n + \\frac { 1 } { 2 } } , x ) \\right] + \\mathcal { O } ( \\tau ^ { 2 } ) .", "type": "interline_equation", "image_path": "a47eadef04a56cc4dcfb26fd0d23f4dd37e4817da3148e9aa00e921a8c28d785.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 189, 162, 422, 190 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 108, 193, 428, 205 ], "lines": [ { "bbox": [ 106, 192, 428, 207 ], "spans": [ { "bbox": [ 106, 192, 161, 207 ], "score": 1.0, "content": "Noticing that", "type": "text" }, { "bbox": [ 162, 193, 189, 205 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 190, 192, 343, 207 ], "score": 1.0, "content": "is satisfied with the evolutionary PDE", "type": "text" }, { "bbox": [ 344, 193, 388, 205 ], "score": 0.92, "content": "u _ { t } = \\mathcal { N } [ u ]", "type": "inline_equation" }, { "bbox": [ 389, 192, 428, 207 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "interline_equation", "bbox": [ 176, 210, 435, 239 ], "lines": [ { "bbox": [ 176, 210, 435, 239 ], "spans": [ { "bbox": [ 176, 210, 435, 239 ], "score": 0.93, "content": "\\frac { u ( t _ { n + 1 } , x ) - u ( t _ { n } , x ) } { \\tau } = \\mathcal { N } \\left[ \\frac { u ( t _ { n + 1 } , x ) + u ( t _ { n } , x ) } { 2 } \\right] + \\mathcal { O } ( \\tau ^ { 2 } ) .", "type": "interline_equation", "image_path": "3853576a90fe2d715164a697760719836d9252c6ff0d661393082cea1db861fd.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 176, 210, 435, 239 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 243, 504, 267 ], "lines": [ { "bbox": [ 105, 240, 506, 258 ], "spans": [ { "bbox": [ 105, 240, 506, 258 ], "score": 1.0, "content": "Now subtracting equation 28 from the Crank-Nicolson scheme, we obtain the relation of the propaga-", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 254, 267, 268 ], "spans": [ { "bbox": [ 105, 254, 146, 268 ], "score": 1.0, "content": "tion error", "type": "text" }, { "bbox": [ 146, 254, 254, 267 ], "score": 0.92, "content": "\\xi ^ { n } ( x ) = \\bar { u } ( t _ { n } , x ) - u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 254, 254, 267, 268 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 9 } ], "index": 8.5 }, { "type": "interline_equation", "bbox": [ 124, 272, 471, 300 ], "lines": [ { "bbox": [ 124, 272, 471, 300 ], "spans": [ { "bbox": [ 124, 272, 471, 300 ], "score": 0.93, "content": "\\frac { \\xi ^ { n + 1 } ( x ) - \\xi ^ { n } ( x ) } { \\tau } = \\mathcal { N } \\left[ \\frac { u ( t _ { n + 1 } , x ) + u ( t _ { n } , x ) } { 2 } \\right] - \\mathcal { N } \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n } ( x ) } { 2 } \\right] + \\mathcal { O } ( \\tau ^ { 2 } ) ,", "type": "interline_equation", "image_path": "4b02e39049024801ee71fa7e2e695e697143cba8408c2f878b9ccbfe03cb7fe5.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 124, 272, 471, 300 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 108, 306, 504, 340 ], "lines": [ { "bbox": [ 106, 306, 505, 319 ], "spans": [ { "bbox": [ 106, 306, 214, 319 ], "score": 1.0, "content": "Secondly, we estimate the", "type": "text" }, { "bbox": [ 215, 306, 227, 317 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 228, 306, 322, 319 ], "score": 1.0, "content": "norm error estimate of", "type": "text" }, { "bbox": [ 323, 307, 347, 319 ], "score": 0.92, "content": "\\xi ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 348, 306, 505, 319 ], "score": 1.0, "content": ". This can be achieved by the standard", "type": "text" } ], "index": 11 }, { "bbox": [ 104, 316, 506, 332 ], "spans": [ { "bbox": [ 104, 316, 368, 332 ], "score": 1.0, "content": "Ho¨der inequality estimate techniques. We multiply equation 29 by", "type": "text" }, { "bbox": [ 368, 317, 450, 331 ], "score": 0.92, "content": "\\begin{array} { r } { \\frac 1 2 ( \\xi ^ { n + 1 } ( x ) + \\xi ^ { n } ( x ) ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 451, 316, 506, 332 ], "score": 1.0, "content": "and integrate", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 329, 351, 341 ], "spans": [ { "bbox": [ 106, 329, 120, 341 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 331, 127, 339 ], "score": 0.76, "content": "x", "type": "inline_equation" }, { "bbox": [ 128, 329, 189, 341 ], "score": 1.0, "content": "on the domain", "type": "text" }, { "bbox": [ 189, 329, 196, 339 ], "score": 0.79, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 197, 329, 351, 341 ], "score": 1.0, "content": ". With Assumption 4.1 holds, we have", "type": "text" } ], "index": 13 } ], "index": 12 }, { "type": "interline_equation", "bbox": [ 195, 345, 415, 403 ], "lines": [ { "bbox": [ 195, 345, 415, 403 ], "spans": [ { "bbox": [ 195, 345, 415, 403 ], "score": 0.94, "content": "\\begin{array} { r l r } { { \\frac { \\| \\xi ^ { n + 1 } \\| ^ { 2 } - \\| \\xi ^ { n } \\| ^ { 2 } } { 2 \\tau } \\leq \\int _ { \\Omega } \\mathcal { O } ( \\tau ^ { 2 } ) \\cdot \\frac { \\xi ^ { n + 1 } ( x ) + \\xi ^ { n } ( x ) } { 2 } } } \\\\ & { } & { \\leq C _ { 0 } \\tau ^ { 4 } + \\frac { 1 } { 2 } \\| \\xi ^ { n + 1 } \\| ^ { 2 } + \\frac { 1 } { 2 } \\| \\xi ^ { n } \\| ^ { 2 } , } \\end{array}", "type": "interline_equation", "image_path": "dc3c8a3d403276f2199ce8b764f2f94e3ee15e4f94bd2d840aa84cf9a2d68bb9.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 195, 345, 415, 364.3333333333333 ], "spans": [], "index": 14 }, { "bbox": [ 195, 364.3333333333333, 415, 383.66666666666663 ], "spans": [], "index": 15 }, { "bbox": [ 195, 383.66666666666663, 415, 402.99999999999994 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 106, 406, 504, 428 ], "lines": [ { "bbox": [ 105, 404, 506, 421 ], "spans": [ { "bbox": [ 105, 404, 181, 421 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 182, 407, 194, 417 ], "score": 0.89, "content": "C _ { 0 }", "type": "inline_equation" }, { "bbox": [ 194, 404, 264, 421 ], "score": 1.0, "content": "only depends on", "type": "text" }, { "bbox": [ 265, 406, 293, 419 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 293, 404, 506, 421 ], "score": 1.0, "content": "and its derivatives. We rearrange it to the following", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 417, 129, 429 ], "spans": [ { "bbox": [ 105, 417, 129, 429 ], "score": 1.0, "content": "form", "type": "text" } ], "index": 18 } ], "index": 17.5 }, { "type": "interline_equation", "bbox": [ 229, 426, 382, 451 ], "lines": [ { "bbox": [ 229, 426, 382, 451 ], "spans": [ { "bbox": [ 229, 426, 382, 451 ], "score": 0.93, "content": "\\left\\| \\xi ^ { n + 1 } \\right\\| ^ { 2 } \\leq \\frac { 1 + \\tau } { 1 - \\tau } \\left\\| \\xi ^ { n } \\right\\| ^ { 2 } + \\frac { 2 C _ { 0 } } { 1 - \\tau } \\tau ^ { 5 } .", "type": "interline_equation", "image_path": "d79093492a3f188de216b1fa47cbb62babc1246cfd60cc37dab5c2533a230a28.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 229, 426, 382, 451 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 107, 454, 291, 467 ], "lines": [ { "bbox": [ 106, 453, 292, 468 ], "spans": [ { "bbox": [ 106, 453, 131, 468 ], "score": 1.0, "content": "Since", "type": "text" }, { "bbox": [ 131, 454, 173, 467 ], "score": 0.93, "content": "\\xi ^ { 0 } ( x ) = 0", "type": "inline_equation" }, { "bbox": [ 173, 453, 292, 468 ], "score": 1.0, "content": ", we apply Lemma A.3 to get", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 240, 472, 371, 527 ], "lines": [ { "bbox": [ 240, 472, 371, 527 ], "spans": [ { "bbox": [ 240, 472, 371, 527 ], "score": 0.93, "content": "\\begin{array} { l } { \\displaystyle { \\| \\xi ^ { n } \\| ^ { 2 } \\leq \\frac { 2 C _ { 0 } \\tau ^ { 5 } } { 1 - \\tau } \\cdot \\frac { \\Big ( \\frac { 1 + \\tau } { 1 - \\tau } \\Big ) ^ { n } - 1 } { \\frac { 1 + \\tau } { 1 - \\tau } - 1 } } } \\\\ { \\leq 6 C _ { 0 } t _ { n } \\tau ^ { 4 } . } \\end{array}", "type": "interline_equation", "image_path": "78c7a38680d40e20ec7bb0eeb390242452371725c6e1677ed094ec6bda2fba82.jpg" } ] } ], "index": 21.5, "virtual_lines": [ { "bbox": [ 240, 472, 371, 499.5 ], "spans": [], "index": 21 }, { "bbox": [ 240, 499.5, 371, 527.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 107, 531, 443, 545 ], "lines": [ { "bbox": [ 105, 530, 445, 546 ], "spans": [ { "bbox": [ 105, 530, 155, 546 ], "score": 1.0, "content": "So we have", "type": "text" }, { "bbox": [ 155, 532, 224, 545 ], "score": 0.93, "content": "\\| \\xi ^ { n } \\| \\le C \\sqrt { t _ { n } } \\tau ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 225, 530, 299, 546 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 299, 532, 346, 544 ], "score": 0.93, "content": "C = \\sqrt { 6 C _ { 0 } }", "type": "inline_equation" }, { "bbox": [ 347, 530, 445, 546 ], "score": 1.0, "content": "and we finish the proof.", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 551, 506, 575 ], "lines": [ { "bbox": [ 105, 550, 506, 566 ], "spans": [ { "bbox": [ 105, 550, 192, 566 ], "score": 1.0, "content": "Lemma A.2. Denote", "type": "text" }, { "bbox": [ 193, 552, 294, 564 ], "score": 0.91, "content": "\\eta ^ { n } ( x ) = u ^ { n } ( x ) - u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 294, 550, 323, 566 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 323, 552, 348, 564 ], "score": 0.92, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 349, 550, 506, 566 ], "score": 1.0, "content": "is the Crank-Nicolson time differencing", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 564, 448, 576 ], "spans": [ { "bbox": [ 106, 564, 192, 576 ], "score": 1.0, "content": "discrete solution and", "type": "text" }, { "bbox": [ 192, 564, 222, 576 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 222, 564, 448, 576 ], "score": 1.0, "content": "is the discrete PINN solution, then we have the estimate", "type": "text" } ], "index": 25 } ], "index": 24.5 }, { "type": "interline_equation", "bbox": [ 234, 581, 377, 604 ], "lines": [ { "bbox": [ 234, 581, 377, 604 ], "spans": [ { "bbox": [ 234, 581, 377, 604 ], "score": 0.93, "content": "\\| \\eta ^ { n } \\| \\leq C \\sqrt { t _ { n } } \\big ( \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } \\big ) ,", "type": "interline_equation", "image_path": "18187bbb235f0e6bc56762788ff1532b8c0be083290c8038b7d9ce8c0cfe16e1.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 234, 581, 377, 604 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 105, 618, 505, 642 ], "lines": [ { "bbox": [ 105, 617, 506, 631 ], "spans": [ { "bbox": [ 105, 617, 213, 631 ], "score": 1.0, "content": "Proof. The PINN solution", "type": "text" }, { "bbox": [ 214, 619, 252, 630 ], "score": 0.89, "content": "u _ { \\theta ^ { n + 1 } } ( x )", "type": "inline_equation" }, { "bbox": [ 252, 617, 451, 631 ], "score": 1.0, "content": "is obtained by optimize the physics-informed loss", "type": "text" }, { "bbox": [ 452, 618, 503, 631 ], "score": 0.92, "content": "{ \\mathcal { L } } ^ { n + 1 } ( \\theta ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 503, 617, 506, 631 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 27 }, { "bbox": [ 104, 625, 274, 644 ], "spans": [ { "bbox": [ 104, 625, 220, 644 ], "score": 1.0, "content": "Define the residual function", "type": "text" }, { "bbox": [ 221, 630, 258, 642 ], "score": 0.91, "content": "\\mathcal { R } ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 259, 625, 274, 644 ], "score": 1.0, "content": "by", "type": "text" } ], "index": 28 } ], "index": 27.5 }, { "type": "interline_equation", "bbox": [ 156, 647, 454, 675 ], "lines": [ { "bbox": [ 156, 647, 454, 675 ], "spans": [ { "bbox": [ 156, 647, 454, 675 ], "score": 0.93, "content": "\\mathcal { R } ^ { n + 1 } ( x ) = \\frac { u _ { \\theta ^ { n + 1 } } ( x ) - u _ { \\theta ^ { n } } ( x ) } { \\tau } - \\mathcal { N } \\left[ \\frac { u _ { \\theta ^ { n + 1 } } ( x ) + u _ { \\theta ^ { n } } ( x ) } { 2 } \\right] , \\quad \\forall x \\in \\Omega .", "type": "interline_equation", "image_path": "b07ac21035e453399b662ccf2c5836361eef0890ff3fd424404edef6a91658e9.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 156, 647, 454, 675 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 105, 681, 506, 703 ], "lines": [ { "bbox": [ 104, 679, 506, 696 ], "spans": [ { "bbox": [ 104, 679, 141, 696 ], "score": 1.0, "content": "The loss", "type": "text" }, { "bbox": [ 141, 681, 192, 694 ], "score": 0.93, "content": "{ \\mathcal { L } } ^ { n + 1 } ( \\theta ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 193, 679, 506, 696 ], "score": 1.0, "content": "is partially composed of the residual function on some randomly sampled point,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 694, 119, 704 ], "spans": [ { "bbox": [ 105, 694, 119, 704 ], "score": 1.0, "content": "so", "type": "text" } ], "index": 31 } ], "index": 30.5 }, { "type": "interline_equation", "bbox": [ 251, 701, 360, 735 ], "lines": [ { "bbox": [ 251, 701, 360, 735 ], "spans": [ { "bbox": [ 251, 701, 360, 735 ], "score": 0.95, "content": "\\mathcal { L } ^ { n + 1 } \\geq \\frac { \\lambda _ { r } } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } | \\mathcal { R } ( x _ { r } ^ { i } ) | ^ { 2 } .", "type": "interline_equation", "image_path": "71cd83a80dacdc0e6953476a8258b2369c86632344b59f0407785e2faae0e01a.jpg" } ] } ], "index": 32.5, "virtual_lines": [ { "bbox": [ 251, 701, 360, 718.0 ], "spans": [], "index": 32 }, { "bbox": [ 251, 718.0, 360, 735.0 ], "spans": [], "index": 33 } ] } ], "page_idx": 14, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "15", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 494, 533, 505, 543 ], "lines": [ { "bbox": [ 496, 534, 504, 543 ], "spans": [ { "bbox": [ 496, 534, 504, 543 ], "score": 0.999, "content": "□", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 117 ], "lines": [ { "bbox": [ 105, 81, 505, 96 ], "spans": [ { "bbox": [ 105, 81, 210, 96 ], "score": 1.0, "content": "Proof. Firstly, we replace", "type": "text" }, { "bbox": [ 210, 83, 236, 95 ], "score": 0.92, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 236, 81, 505, 96 ], "score": 1.0, "content": "in the Crank-Nicolson time differencing scheme by the evolutionary", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 92, 506, 106 ], "spans": [ { "bbox": [ 105, 92, 171, 106 ], "score": 1.0, "content": "PDE’s solution", "type": "text" }, { "bbox": [ 171, 94, 204, 106 ], "score": 0.92, "content": "u ( t _ { n } , x )", "type": "inline_equation" }, { "bbox": [ 205, 92, 506, 106 ], "score": 1.0, "content": "and compare the difference. This can be achieved by the standard Taylor", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 444, 119 ], "spans": [ { "bbox": [ 106, 104, 343, 118 ], "score": 1.0, "content": "expansion techniques. We do Taylor expansion at the point", "type": "text" }, { "bbox": [ 344, 104, 417, 119 ], "score": 0.94, "content": "\\begin{array} { r } { t _ { n + \\frac { 1 } { 2 } } = ( n + \\frac { 1 } { 2 } ) \\tau } \\end{array}", "type": "inline_equation" }, { "bbox": [ 417, 104, 444, 118 ], "score": 1.0, "content": "to get", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 105, 81, 506, 119 ] }, { "type": "interline_equation", "bbox": [ 209, 124, 401, 149 ], "lines": [ { "bbox": [ 209, 124, 401, 149 ], "spans": [ { "bbox": [ 209, 124, 401, 149 ], "score": 0.95, "content": "\\frac { u ( t _ { n + 1 } , x ) - u ( t _ { n } , x ) } { \\tau } = u _ { t } ( t _ { n + \\frac { 1 } { 2 } } , x ) + \\mathcal { O } ( \\tau ^ { 2 } ) ,", "type": "interline_equation", "image_path": "78622b632684f69fbc5a0e98a38a4c94791dde4669fe4fb5f8154f9cb9f88d78.jpg" } ] } ], "index": 3, "virtual_lines": [ { "bbox": [ 209, 124, 401, 149 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 106, 154, 124, 165 ], "lines": [ { "bbox": [ 105, 155, 123, 165 ], "spans": [ { "bbox": [ 105, 155, 123, 165 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 4 } ], "index": 4, "bbox_fs": [ 105, 155, 123, 165 ] }, { "type": "interline_equation", "bbox": [ 189, 162, 422, 190 ], "lines": [ { "bbox": [ 189, 162, 422, 190 ], "spans": [ { "bbox": [ 189, 162, 422, 190 ], "score": 0.93, "content": "\\mathcal { N } \\left[ \\frac { u ( t _ { n + 1 } , x ) + u ( t _ { n } , x ) } { 2 } \\right] = \\mathcal { N } \\left[ u ( t _ { n + \\frac { 1 } { 2 } } , x ) \\right] + \\mathcal { O } ( \\tau ^ { 2 } ) .", "type": "interline_equation", "image_path": "a47eadef04a56cc4dcfb26fd0d23f4dd37e4817da3148e9aa00e921a8c28d785.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 189, 162, 422, 190 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 108, 193, 428, 205 ], "lines": [ { "bbox": [ 106, 192, 428, 207 ], "spans": [ { "bbox": [ 106, 192, 161, 207 ], "score": 1.0, "content": "Noticing that", "type": "text" }, { "bbox": [ 162, 193, 189, 205 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 190, 192, 343, 207 ], "score": 1.0, "content": "is satisfied with the evolutionary PDE", "type": "text" }, { "bbox": [ 344, 193, 388, 205 ], "score": 0.92, "content": "u _ { t } = \\mathcal { N } [ u ]", "type": "inline_equation" }, { "bbox": [ 389, 192, 428, 207 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 6 } ], "index": 6, "bbox_fs": [ 106, 192, 428, 207 ] }, { "type": "interline_equation", "bbox": [ 176, 210, 435, 239 ], "lines": [ { "bbox": [ 176, 210, 435, 239 ], "spans": [ { "bbox": [ 176, 210, 435, 239 ], "score": 0.93, "content": "\\frac { u ( t _ { n + 1 } , x ) - u ( t _ { n } , x ) } { \\tau } = \\mathcal { N } \\left[ \\frac { u ( t _ { n + 1 } , x ) + u ( t _ { n } , x ) } { 2 } \\right] + \\mathcal { O } ( \\tau ^ { 2 } ) .", "type": "interline_equation", "image_path": "3853576a90fe2d715164a697760719836d9252c6ff0d661393082cea1db861fd.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 176, 210, 435, 239 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 243, 504, 267 ], "lines": [ { "bbox": [ 105, 240, 506, 258 ], "spans": [ { "bbox": [ 105, 240, 506, 258 ], "score": 1.0, "content": "Now subtracting equation 28 from the Crank-Nicolson scheme, we obtain the relation of the propaga-", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 254, 267, 268 ], "spans": [ { "bbox": [ 105, 254, 146, 268 ], "score": 1.0, "content": "tion error", "type": "text" }, { "bbox": [ 146, 254, 254, 267 ], "score": 0.92, "content": "\\xi ^ { n } ( x ) = \\bar { u } ( t _ { n } , x ) - u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 254, 254, 267, 268 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 9 } ], "index": 8.5, "bbox_fs": [ 105, 240, 506, 268 ] }, { "type": "interline_equation", "bbox": [ 124, 272, 471, 300 ], "lines": [ { "bbox": [ 124, 272, 471, 300 ], "spans": [ { "bbox": [ 124, 272, 471, 300 ], "score": 0.93, "content": "\\frac { \\xi ^ { n + 1 } ( x ) - \\xi ^ { n } ( x ) } { \\tau } = \\mathcal { N } \\left[ \\frac { u ( t _ { n + 1 } , x ) + u ( t _ { n } , x ) } { 2 } \\right] - \\mathcal { N } \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n } ( x ) } { 2 } \\right] + \\mathcal { O } ( \\tau ^ { 2 } ) ,", "type": "interline_equation", "image_path": "4b02e39049024801ee71fa7e2e695e697143cba8408c2f878b9ccbfe03cb7fe5.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 124, 272, 471, 300 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 108, 306, 504, 340 ], "lines": [ { "bbox": [ 106, 306, 505, 319 ], "spans": [ { "bbox": [ 106, 306, 214, 319 ], "score": 1.0, "content": "Secondly, we estimate the", "type": "text" }, { "bbox": [ 215, 306, 227, 317 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 228, 306, 322, 319 ], "score": 1.0, "content": "norm error estimate of", "type": "text" }, { "bbox": [ 323, 307, 347, 319 ], "score": 0.92, "content": "\\xi ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 348, 306, 505, 319 ], "score": 1.0, "content": ". This can be achieved by the standard", "type": "text" } ], "index": 11 }, { "bbox": [ 104, 316, 506, 332 ], "spans": [ { "bbox": [ 104, 316, 368, 332 ], "score": 1.0, "content": "Ho¨der inequality estimate techniques. We multiply equation 29 by", "type": "text" }, { "bbox": [ 368, 317, 450, 331 ], "score": 0.92, "content": "\\begin{array} { r } { \\frac 1 2 ( \\xi ^ { n + 1 } ( x ) + \\xi ^ { n } ( x ) ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 451, 316, 506, 332 ], "score": 1.0, "content": "and integrate", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 329, 351, 341 ], "spans": [ { "bbox": [ 106, 329, 120, 341 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 331, 127, 339 ], "score": 0.76, "content": "x", "type": "inline_equation" }, { "bbox": [ 128, 329, 189, 341 ], "score": 1.0, "content": "on the domain", "type": "text" }, { "bbox": [ 189, 329, 196, 339 ], "score": 0.79, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 197, 329, 351, 341 ], "score": 1.0, "content": ". With Assumption 4.1 holds, we have", "type": "text" } ], "index": 13 } ], "index": 12, "bbox_fs": [ 104, 306, 506, 341 ] }, { "type": "interline_equation", "bbox": [ 195, 345, 415, 403 ], "lines": [ { "bbox": [ 195, 345, 415, 403 ], "spans": [ { "bbox": [ 195, 345, 415, 403 ], "score": 0.94, "content": "\\begin{array} { r l r } { { \\frac { \\| \\xi ^ { n + 1 } \\| ^ { 2 } - \\| \\xi ^ { n } \\| ^ { 2 } } { 2 \\tau } \\leq \\int _ { \\Omega } \\mathcal { O } ( \\tau ^ { 2 } ) \\cdot \\frac { \\xi ^ { n + 1 } ( x ) + \\xi ^ { n } ( x ) } { 2 } } } \\\\ & { } & { \\leq C _ { 0 } \\tau ^ { 4 } + \\frac { 1 } { 2 } \\| \\xi ^ { n + 1 } \\| ^ { 2 } + \\frac { 1 } { 2 } \\| \\xi ^ { n } \\| ^ { 2 } , } \\end{array}", "type": "interline_equation", "image_path": "dc3c8a3d403276f2199ce8b764f2f94e3ee15e4f94bd2d840aa84cf9a2d68bb9.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 195, 345, 415, 364.3333333333333 ], "spans": [], "index": 14 }, { "bbox": [ 195, 364.3333333333333, 415, 383.66666666666663 ], "spans": [], "index": 15 }, { "bbox": [ 195, 383.66666666666663, 415, 402.99999999999994 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 106, 406, 504, 428 ], "lines": [ { "bbox": [ 105, 404, 506, 421 ], "spans": [ { "bbox": [ 105, 404, 181, 421 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 182, 407, 194, 417 ], "score": 0.89, "content": "C _ { 0 }", "type": "inline_equation" }, { "bbox": [ 194, 404, 264, 421 ], "score": 1.0, "content": "only depends on", "type": "text" }, { "bbox": [ 265, 406, 293, 419 ], "score": 0.93, "content": "u ( t , x )", "type": "inline_equation" }, { "bbox": [ 293, 404, 506, 421 ], "score": 1.0, "content": "and its derivatives. We rearrange it to the following", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 417, 129, 429 ], "spans": [ { "bbox": [ 105, 417, 129, 429 ], "score": 1.0, "content": "form", "type": "text" } ], "index": 18 } ], "index": 17.5, "bbox_fs": [ 105, 404, 506, 429 ] }, { "type": "interline_equation", "bbox": [ 229, 426, 382, 451 ], "lines": [ { "bbox": [ 229, 426, 382, 451 ], "spans": [ { "bbox": [ 229, 426, 382, 451 ], "score": 0.93, "content": "\\left\\| \\xi ^ { n + 1 } \\right\\| ^ { 2 } \\leq \\frac { 1 + \\tau } { 1 - \\tau } \\left\\| \\xi ^ { n } \\right\\| ^ { 2 } + \\frac { 2 C _ { 0 } } { 1 - \\tau } \\tau ^ { 5 } .", "type": "interline_equation", "image_path": "d79093492a3f188de216b1fa47cbb62babc1246cfd60cc37dab5c2533a230a28.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 229, 426, 382, 451 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 107, 454, 291, 467 ], "lines": [ { "bbox": [ 106, 453, 292, 468 ], "spans": [ { "bbox": [ 106, 453, 131, 468 ], "score": 1.0, "content": "Since", "type": "text" }, { "bbox": [ 131, 454, 173, 467 ], "score": 0.93, "content": "\\xi ^ { 0 } ( x ) = 0", "type": "inline_equation" }, { "bbox": [ 173, 453, 292, 468 ], "score": 1.0, "content": ", we apply Lemma A.3 to get", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 106, 453, 292, 468 ] }, { "type": "interline_equation", "bbox": [ 240, 472, 371, 527 ], "lines": [ { "bbox": [ 240, 472, 371, 527 ], "spans": [ { "bbox": [ 240, 472, 371, 527 ], "score": 0.93, "content": "\\begin{array} { l } { \\displaystyle { \\| \\xi ^ { n } \\| ^ { 2 } \\leq \\frac { 2 C _ { 0 } \\tau ^ { 5 } } { 1 - \\tau } \\cdot \\frac { \\Big ( \\frac { 1 + \\tau } { 1 - \\tau } \\Big ) ^ { n } - 1 } { \\frac { 1 + \\tau } { 1 - \\tau } - 1 } } } \\\\ { \\leq 6 C _ { 0 } t _ { n } \\tau ^ { 4 } . } \\end{array}", "type": "interline_equation", "image_path": "78c7a38680d40e20ec7bb0eeb390242452371725c6e1677ed094ec6bda2fba82.jpg" } ] } ], "index": 21.5, "virtual_lines": [ { "bbox": [ 240, 472, 371, 499.5 ], "spans": [], "index": 21 }, { "bbox": [ 240, 499.5, 371, 527.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 107, 531, 443, 545 ], "lines": [ { "bbox": [ 105, 530, 445, 546 ], "spans": [ { "bbox": [ 105, 530, 155, 546 ], "score": 1.0, "content": "So we have", "type": "text" }, { "bbox": [ 155, 532, 224, 545 ], "score": 0.93, "content": "\\| \\xi ^ { n } \\| \\le C \\sqrt { t _ { n } } \\tau ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 225, 530, 299, 546 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 299, 532, 346, 544 ], "score": 0.93, "content": "C = \\sqrt { 6 C _ { 0 } }", "type": "inline_equation" }, { "bbox": [ 347, 530, 445, 546 ], "score": 1.0, "content": "and we finish the proof.", "type": "text" } ], "index": 23 } ], "index": 23, "bbox_fs": [ 105, 530, 445, 546 ] }, { "type": "text", "bbox": [ 107, 551, 506, 575 ], "lines": [ { "bbox": [ 105, 550, 506, 566 ], "spans": [ { "bbox": [ 105, 550, 192, 566 ], "score": 1.0, "content": "Lemma A.2. Denote", "type": "text" }, { "bbox": [ 193, 552, 294, 564 ], "score": 0.91, "content": "\\eta ^ { n } ( x ) = u ^ { n } ( x ) - u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 294, 550, 323, 566 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 323, 552, 348, 564 ], "score": 0.92, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 349, 550, 506, 566 ], "score": 1.0, "content": "is the Crank-Nicolson time differencing", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 564, 448, 576 ], "spans": [ { "bbox": [ 106, 564, 192, 576 ], "score": 1.0, "content": "discrete solution and", "type": "text" }, { "bbox": [ 192, 564, 222, 576 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 222, 564, 448, 576 ], "score": 1.0, "content": "is the discrete PINN solution, then we have the estimate", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 105, 550, 506, 576 ] }, { "type": "interline_equation", "bbox": [ 234, 581, 377, 604 ], "lines": [ { "bbox": [ 234, 581, 377, 604 ], "spans": [ { "bbox": [ 234, 581, 377, 604 ], "score": 0.93, "content": "\\| \\eta ^ { n } \\| \\leq C \\sqrt { t _ { n } } \\big ( \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } \\big ) ,", "type": "interline_equation", "image_path": "18187bbb235f0e6bc56762788ff1532b8c0be083290c8038b7d9ce8c0cfe16e1.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 234, 581, 377, 604 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 105, 618, 505, 642 ], "lines": [ { "bbox": [ 105, 617, 506, 631 ], "spans": [ { "bbox": [ 105, 617, 213, 631 ], "score": 1.0, "content": "Proof. The PINN solution", "type": "text" }, { "bbox": [ 214, 619, 252, 630 ], "score": 0.89, "content": "u _ { \\theta ^ { n + 1 } } ( x )", "type": "inline_equation" }, { "bbox": [ 252, 617, 451, 631 ], "score": 1.0, "content": "is obtained by optimize the physics-informed loss", "type": "text" }, { "bbox": [ 452, 618, 503, 631 ], "score": 0.92, "content": "{ \\mathcal { L } } ^ { n + 1 } ( \\theta ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 503, 617, 506, 631 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 27 }, { "bbox": [ 104, 625, 274, 644 ], "spans": [ { "bbox": [ 104, 625, 220, 644 ], "score": 1.0, "content": "Define the residual function", "type": "text" }, { "bbox": [ 221, 630, 258, 642 ], "score": 0.91, "content": "\\mathcal { R } ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 259, 625, 274, 644 ], "score": 1.0, "content": "by", "type": "text" } ], "index": 28 } ], "index": 27.5, "bbox_fs": [ 104, 617, 506, 644 ] }, { "type": "interline_equation", "bbox": [ 156, 647, 454, 675 ], "lines": [ { "bbox": [ 156, 647, 454, 675 ], "spans": [ { "bbox": [ 156, 647, 454, 675 ], "score": 0.93, "content": "\\mathcal { R } ^ { n + 1 } ( x ) = \\frac { u _ { \\theta ^ { n + 1 } } ( x ) - u _ { \\theta ^ { n } } ( x ) } { \\tau } - \\mathcal { N } \\left[ \\frac { u _ { \\theta ^ { n + 1 } } ( x ) + u _ { \\theta ^ { n } } ( x ) } { 2 } \\right] , \\quad \\forall x \\in \\Omega .", "type": "interline_equation", "image_path": "b07ac21035e453399b662ccf2c5836361eef0890ff3fd424404edef6a91658e9.jpg" } ] } ], "index": 29, "virtual_lines": [ { "bbox": [ 156, 647, 454, 675 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 105, 681, 506, 703 ], "lines": [ { "bbox": [ 104, 679, 506, 696 ], "spans": [ { "bbox": [ 104, 679, 141, 696 ], "score": 1.0, "content": "The loss", "type": "text" }, { "bbox": [ 141, 681, 192, 694 ], "score": 0.93, "content": "{ \\mathcal { L } } ^ { n + 1 } ( \\theta ^ { n + 1 } )", "type": "inline_equation" }, { "bbox": [ 193, 679, 506, 696 ], "score": 1.0, "content": "is partially composed of the residual function on some randomly sampled point,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 694, 119, 704 ], "spans": [ { "bbox": [ 105, 694, 119, 704 ], "score": 1.0, "content": "so", "type": "text" } ], "index": 31 } ], "index": 30.5, "bbox_fs": [ 104, 679, 506, 704 ] }, { "type": "interline_equation", "bbox": [ 251, 701, 360, 735 ], "lines": [ { "bbox": [ 251, 701, 360, 735 ], "spans": [ { "bbox": [ 251, 701, 360, 735 ], "score": 0.95, "content": "\\mathcal { L } ^ { n + 1 } \\geq \\frac { \\lambda _ { r } } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } | \\mathcal { R } ( x _ { r } ^ { i } ) | ^ { 2 } .", "type": "interline_equation", "image_path": "71cd83a80dacdc0e6953476a8258b2369c86632344b59f0407785e2faae0e01a.jpg" } ] } ], "index": 32.5, "virtual_lines": [ { "bbox": [ 251, 701, 360, 718.0 ], "spans": [], "index": 32 }, { "bbox": [ 251, 718.0, 360, 735.0 ], "spans": [], "index": 33 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 82, 504, 106 ], "lines": [ { "bbox": [ 105, 82, 504, 95 ], "spans": [ { "bbox": [ 105, 82, 491, 95 ], "score": 1.0, "content": "By the Monte-Carlo quadrature rule in the numerical integration method, we can estimate the", "type": "text" }, { "bbox": [ 491, 82, 504, 92 ], "score": 0.84, "content": "L ^ { 2 }", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 105, 93, 333, 106 ], "spans": [ { "bbox": [ 105, 93, 225, 106 ], "score": 1.0, "content": "norm of the residual function", "type": "text" }, { "bbox": [ 226, 94, 248, 106 ], "score": 0.92, "content": "\\mathcal { R } ( x )", "type": "inline_equation" }, { "bbox": [ 249, 93, 333, 106 ], "score": 1.0, "content": "by the discrete form", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 220, 111, 389, 202 ], "lines": [ { "bbox": [ 220, 111, 389, 202 ], "spans": [ { "bbox": [ 220, 111, 389, 202 ], "score": 0.96, "content": "\\begin{array} { r l r } { { \\| \\mathcal { R } ^ { n + 1 } \\| ^ { 2 } = \\int _ { \\Omega } | \\mathcal { R } ^ { n + 1 } ( x ) | ^ { 2 } d x } } \\\\ & { } & { \\leq \\frac { 1 } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } | \\mathcal { R } ( x _ { r } ^ { i } ) | ^ { 2 } + C _ { 1 } N _ { r } ^ { - \\frac { 1 } { 2 } } } \\\\ & { } & { \\leq \\frac { \\mathcal { L } ^ { n + 1 } } { \\lambda _ { r } } + C _ { 1 } N _ { r } ^ { - \\frac { 1 } { 2 } } , } \\end{array}", "type": "interline_equation", "image_path": "7cb1ba7e5a54c537a87e5edb6f7dfcaff77f80095bc98601a6708e54f8dca42f.jpg" } ] } ], "index": 4.5, "virtual_lines": [ { "bbox": [ 220, 111, 389, 126.16666666666667 ], "spans": [], "index": 2 }, { "bbox": [ 220, 126.16666666666667, 389, 141.33333333333334 ], "spans": [], "index": 3 }, { "bbox": [ 220, 141.33333333333334, 389, 156.5 ], "spans": [], "index": 4 }, { "bbox": [ 220, 156.5, 389, 171.66666666666666 ], "spans": [], "index": 5 }, { "bbox": [ 220, 171.66666666666666, 389, 186.83333333333331 ], "spans": [], "index": 6 }, { "bbox": [ 220, 186.83333333333331, 389, 201.99999999999997 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 206, 422, 219 ], "lines": [ { "bbox": [ 105, 205, 423, 221 ], "spans": [ { "bbox": [ 105, 205, 180, 221 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 180, 208, 192, 218 ], "score": 0.88, "content": "C _ { 1 }", "type": "inline_equation" }, { "bbox": [ 193, 205, 389, 221 ], "score": 1.0, "content": "depends on the regularities of the PINN solution", "type": "text" }, { "bbox": [ 389, 207, 419, 219 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 419, 205, 423, 221 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 106, 223, 506, 269 ], "lines": [ { "bbox": [ 105, 223, 506, 237 ], "spans": [ { "bbox": [ 105, 223, 231, 237 ], "score": 1.0, "content": "Now we turn to estimate the", "type": "text" }, { "bbox": [ 232, 223, 244, 234 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 245, 223, 345, 237 ], "score": 1.0, "content": "norm error estimate of", "type": "text" }, { "bbox": [ 346, 224, 371, 236 ], "score": 0.91, "content": "\\eta ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 371, 223, 450, 237 ], "score": 1.0, "content": ". We first replace", "type": "text" }, { "bbox": [ 450, 224, 475, 236 ], "score": 0.91, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 476, 223, 506, 237 ], "score": 1.0, "content": "in the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 234, 507, 248 ], "spans": [ { "bbox": [ 105, 234, 363, 248 ], "score": 1.0, "content": "Crank-Nicolson time differencing scheme by the PINN solution", "type": "text" }, { "bbox": [ 363, 235, 392, 247 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 393, 234, 507, 248 ], "score": 1.0, "content": "and compare the difference.", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 245, 505, 258 ], "spans": [ { "bbox": [ 105, 245, 505, 258 ], "score": 1.0, "content": "Subtracting equation 31 from the Crank-Nicolson scheme, we obtain the relation of the propagation", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 257, 245, 270 ], "spans": [ { "bbox": [ 105, 257, 128, 270 ], "score": 1.0, "content": "error", "type": "text" }, { "bbox": [ 129, 257, 232, 269 ], "score": 0.91, "content": "\\eta ^ { n } ( x ) \\stackrel { } { = } u ^ { n } ( x ) - u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 232, 257, 245, 270 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 12 } ], "index": 10.5 }, { "type": "interline_equation", "bbox": [ 142, 275, 469, 304 ], "lines": [ { "bbox": [ 142, 275, 469, 304 ], "spans": [ { "bbox": [ 142, 275, 469, 304 ], "score": 0.93, "content": "\\frac { \\eta ^ { n + 1 } - \\eta ^ { n } } { \\tau } - \\left( \\mathcal { N } \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n } ( x ) } { 2 } \\right] - \\mathcal { N } \\left[ \\frac { u _ { \\theta ^ { n + 1 } } ( x ) + u _ { \\theta ^ { n } } ( x ) } { 2 } \\right] \\right) = - \\mathcal { R } ( x )", "type": "interline_equation", "image_path": "62f04c16dc8717a21bc23761b46f2c61dfa7ebed04c5ebc058f90c6298e68b9f.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 142, 275, 469, 284.6666666666667 ], "spans": [], "index": 13 }, { "bbox": [ 142, 284.6666666666667, 469, 294.33333333333337 ], "spans": [], "index": 14 }, { "bbox": [ 142, 294.33333333333337, 469, 304.00000000000006 ], "spans": [], "index": 15 } ] }, { "type": "text", "bbox": [ 104, 309, 505, 334 ], "lines": [ { "bbox": [ 105, 307, 506, 325 ], "spans": [ { "bbox": [ 105, 307, 364, 325 ], "score": 1.0, "content": "Similar to the proof in Lemma A.1, we multiply equation 32 by", "type": "text" }, { "bbox": [ 365, 309, 449, 324 ], "score": 0.92, "content": "\\frac 1 2 ( \\eta ^ { n + 1 } ( x ) + \\eta ^ { n } ( x ) )", "type": "inline_equation" }, { "bbox": [ 449, 307, 506, 325 ], "score": 1.0, "content": "and integrate", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 321, 351, 335 ], "spans": [ { "bbox": [ 105, 321, 120, 335 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 324, 127, 332 ], "score": 0.76, "content": "x", "type": "inline_equation" }, { "bbox": [ 128, 321, 188, 335 ], "score": 1.0, "content": "on the domain", "type": "text" }, { "bbox": [ 189, 322, 196, 332 ], "score": 0.81, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 197, 321, 351, 335 ], "score": 1.0, "content": ". With Assumption 4.1 holds, we have", "type": "text" } ], "index": 17 } ], "index": 16.5 }, { "type": "interline_equation", "bbox": [ 181, 339, 429, 396 ], "lines": [ { "bbox": [ 181, 339, 429, 396 ], "spans": [ { "bbox": [ 181, 339, 429, 396 ], "score": 0.94, "content": "\\begin{array} { r l r } { { \\frac { \\| \\eta ^ { n + 1 } \\| ^ { 2 } - \\| \\eta ^ { n } \\| ^ { 2 } } { 2 \\tau } \\le - \\int _ { \\Omega } \\mathcal { R } ( x ) \\cdot \\frac { \\eta ^ { n + 1 } ( x ) + \\eta ^ { n } ( x ) } { 2 } } } \\\\ & { } & { \\qquad \\le \\displaystyle \\frac { 1 } { 4 } \\| \\mathcal { R } ^ { n + 1 } \\| ^ { 2 } + \\frac { 1 } { 2 } \\| \\eta ^ { n + 1 } \\| ^ { 2 } + \\frac { 1 } { 2 } \\| \\eta ^ { n } \\| ^ { 2 } , } \\end{array}", "type": "interline_equation", "image_path": "d4e221d18829ad970e35d9a8d2415c33f82f1cb665184a8912a409d99a9f1f2b.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 181, 339, 429, 358.0 ], "spans": [], "index": 18 }, { "bbox": [ 181, 358.0, 429, 377.0 ], "spans": [], "index": 19 }, { "bbox": [ 181, 377.0, 429, 396.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 107, 400, 275, 412 ], "lines": [ { "bbox": [ 105, 400, 276, 413 ], "spans": [ { "bbox": [ 105, 400, 276, 413 ], "score": 1.0, "content": "then we rearrange it to the following form", "type": "text" } ], "index": 21 } ], "index": 21 }, { "type": "interline_equation", "bbox": [ 212, 418, 399, 443 ], "lines": [ { "bbox": [ 212, 418, 399, 443 ], "spans": [ { "bbox": [ 212, 418, 399, 443 ], "score": 0.92, "content": "\\left. \\eta ^ { n + 1 } \\right. ^ { 2 } \\leq \\frac { 1 + \\tau } { 1 - \\tau } \\left. \\eta ^ { n } \\right. ^ { 2 } + \\frac { \\tau } { 1 - \\tau } \\left. \\mathcal { R } ^ { n + 1 } \\right. ^ { 2 } .", "type": "interline_equation", "image_path": "1f1510a77ce7bd121decbbbb8b82499e5661536c9d0ac67698b24b2e7ca142f3.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 212, 418, 399, 443 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 106, 449, 240, 461 ], "lines": [ { "bbox": [ 106, 448, 240, 462 ], "spans": [ { "bbox": [ 106, 448, 240, 462 ], "score": 1.0, "content": "then we apply Lemma A.3 to get", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "interline_equation", "bbox": [ 180, 467, 430, 532 ], "lines": [ { "bbox": [ 180, 467, 430, 532 ], "spans": [ { "bbox": [ 180, 467, 430, 532 ], "score": 0.95, "content": "\\begin{array} { r l r } { { \\| \\eta ^ { n } \\| ^ { 2 } \\leq ( \\frac { 1 + \\tau } { 1 - \\tau } ) ^ { n } \\| \\eta ^ { 0 } \\| ^ { 2 } + \\frac { ( \\frac { 1 + \\tau } { 1 - \\tau } ) ^ { n } - 1 } { \\frac { 1 + \\tau } { 1 - \\tau } - 1 } \\cdot \\frac { \\tau \\displaystyle { \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\| \\mathcal { R } ^ { i } \\| ^ { 2 } } } { 1 - \\tau } } } \\\\ & { } & { \\leq ( 1 + 6 t _ { n } ) \\| \\eta ^ { 0 } \\| ^ { 2 } + \\frac { 3 t _ { n } } { 2 } \\displaystyle \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\| \\mathcal { R } ^ { i } \\| ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "c8b69359c18c29e8f2784916d1af56aff427236af63935deb09bb9b1c0fc98a8.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 180, 467, 430, 488.6666666666667 ], "spans": [], "index": 24 }, { "bbox": [ 180, 488.6666666666667, 430, 510.33333333333337 ], "spans": [], "index": 25 }, { "bbox": [ 180, 510.33333333333337, 430, 532.0 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 538, 504, 571 ], "lines": [ { "bbox": [ 105, 538, 506, 560 ], "spans": [ { "bbox": [ 105, 538, 131, 555 ], "score": 1.0, "content": "Since", "type": "text" }, { "bbox": [ 131, 542, 173, 554 ], "score": 0.93, "content": "\\eta ^ { 0 } ( x ) = 0", "type": "inline_equation" }, { "bbox": [ 174, 538, 212, 555 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 212, 538, 351, 560 ], "score": 0.92, "content": "\\| \\eta ^ { n } \\| \\le C \\sqrt { t _ { n } } ( \\operatorname* { m a x } _ { 1 \\le i \\le n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } )", "type": "inline_equation" }, { "bbox": [ 352, 538, 425, 555 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 426, 543, 434, 552 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 435, 538, 506, 555 ], "score": 1.0, "content": "and we finish the", "type": "text" } ], "index": 27 }, { "bbox": [ 103, 558, 502, 572 ], "spans": [ { "bbox": [ 103, 558, 135, 572 ], "score": 1.0, "content": "proof.", "type": "text" }, { "bbox": [ 497, 562, 502, 566 ], "score": 0.0, "content": "", "type": "text" } ], "index": 28 } ], "index": 27.5 }, { "type": "text", "bbox": [ 106, 580, 503, 604 ], "lines": [ { "bbox": [ 104, 579, 506, 596 ], "spans": [ { "bbox": [ 104, 579, 225, 596 ], "score": 1.0, "content": "Lemma A.3. If the sequence", "type": "text" }, { "bbox": [ 225, 581, 262, 593 ], "score": 0.92, "content": "\\{ T _ { n } \\} _ { n = 0 } ^ { \\infty }", "type": "inline_equation" }, { "bbox": [ 263, 579, 506, 596 ], "score": 1.0, "content": "satisfies the following propagation relation for some positive", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 588, 213, 608 ], "spans": [ { "bbox": [ 104, 588, 143, 608 ], "score": 1.0, "content": "constant", "type": "text" }, { "bbox": [ 143, 595, 150, 602 ], "score": 0.53, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 151, 588, 169, 608 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 170, 592, 207, 604 ], "score": 0.92, "content": "\\{ \\beta _ { n } \\} _ { n = 1 } ^ { \\infty }", "type": "inline_equation" }, { "bbox": [ 207, 588, 213, 608 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 30 } ], "index": 29.5 }, { "type": "interline_equation", "bbox": [ 240, 604, 370, 618 ], "lines": [ { "bbox": [ 240, 604, 370, 618 ], "spans": [ { "bbox": [ 240, 604, 370, 618 ], "score": 0.87, "content": "T _ { n + 1 } \\leq \\alpha T _ { n } + \\beta _ { n + 1 } , \\quad n \\geq 0 ,", "type": "interline_equation", "image_path": "211aa00eb2a9edbcf5eff81e1b142c865f0751540c28bd69ee2cd6acd7512c18.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 240, 604, 370, 618 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 106, 622, 160, 633 ], "lines": [ { "bbox": [ 106, 621, 162, 635 ], "spans": [ { "bbox": [ 106, 621, 162, 635 ], "score": 1.0, "content": "then we have", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 220, 632, 391, 657 ], "lines": [ { "bbox": [ 220, 632, 391, 657 ], "spans": [ { "bbox": [ 220, 632, 391, 657 ], "score": 0.91, "content": "T _ { n } \\leq \\alpha ^ { n } T _ { 0 } + \\frac { \\alpha ^ { n } - 1 } { \\alpha - 1 } \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\beta _ { i } , \\quad n \\geq 1 .", "type": "interline_equation", "image_path": "64730d3255e323b68c48a2a5f1fea28ccfe760a5d3c10a93a89c7ff5585c934f.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 220, 632, 391, 657 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 106, 672, 358, 685 ], "lines": [ { "bbox": [ 105, 672, 359, 686 ], "spans": [ { "bbox": [ 105, 672, 359, 686 ], "score": 1.0, "content": "Proof. This is accomplished by a standard recurrence formula.", "type": "text" } ], "index": 34 } ], "index": 34 }, { "type": "title", "bbox": [ 107, 699, 242, 711 ], "lines": [ { "bbox": [ 106, 699, 242, 712 ], "spans": [ { "bbox": [ 106, 699, 242, 712 ], "score": 1.0, "content": "A.4 EXPERIMENTAL DETAILS", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 105, 720, 430, 732 ], "lines": [ { "bbox": [ 105, 719, 431, 733 ], "spans": [ { "bbox": [ 105, 719, 431, 733 ], "score": 1.0, "content": "In this section, we provide the details on the numerical experiments of Section 5.", "type": "text" } ], "index": 36 } ], "index": 36 } ], "page_idx": 15, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "16", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 494, 673, 505, 684 ], "lines": [ { "bbox": [ 496, 675, 505, 685 ], "spans": [ { "bbox": [ 496, 675, 505, 685 ], "score": 0.999, "content": "□", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 82, 504, 106 ], "lines": [ { "bbox": [ 105, 82, 504, 95 ], "spans": [ { "bbox": [ 105, 82, 491, 95 ], "score": 1.0, "content": "By the Monte-Carlo quadrature rule in the numerical integration method, we can estimate the", "type": "text" }, { "bbox": [ 491, 82, 504, 92 ], "score": 0.84, "content": "L ^ { 2 }", "type": "inline_equation" } ], "index": 0 }, { "bbox": [ 105, 93, 333, 106 ], "spans": [ { "bbox": [ 105, 93, 225, 106 ], "score": 1.0, "content": "norm of the residual function", "type": "text" }, { "bbox": [ 226, 94, 248, 106 ], "score": 0.92, "content": "\\mathcal { R } ( x )", "type": "inline_equation" }, { "bbox": [ 249, 93, 333, 106 ], "score": 1.0, "content": "by the discrete form", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 82, 504, 106 ] }, { "type": "interline_equation", "bbox": [ 220, 111, 389, 202 ], "lines": [ { "bbox": [ 220, 111, 389, 202 ], "spans": [ { "bbox": [ 220, 111, 389, 202 ], "score": 0.96, "content": "\\begin{array} { r l r } { { \\| \\mathcal { R } ^ { n + 1 } \\| ^ { 2 } = \\int _ { \\Omega } | \\mathcal { R } ^ { n + 1 } ( x ) | ^ { 2 } d x } } \\\\ & { } & { \\leq \\frac { 1 } { N _ { r } } \\sum _ { i = 1 } ^ { N _ { r } } | \\mathcal { R } ( x _ { r } ^ { i } ) | ^ { 2 } + C _ { 1 } N _ { r } ^ { - \\frac { 1 } { 2 } } } \\\\ & { } & { \\leq \\frac { \\mathcal { L } ^ { n + 1 } } { \\lambda _ { r } } + C _ { 1 } N _ { r } ^ { - \\frac { 1 } { 2 } } , } \\end{array}", "type": "interline_equation", "image_path": "7cb1ba7e5a54c537a87e5edb6f7dfcaff77f80095bc98601a6708e54f8dca42f.jpg" } ] } ], "index": 4.5, "virtual_lines": [ { "bbox": [ 220, 111, 389, 126.16666666666667 ], "spans": [], "index": 2 }, { "bbox": [ 220, 126.16666666666667, 389, 141.33333333333334 ], "spans": [], "index": 3 }, { "bbox": [ 220, 141.33333333333334, 389, 156.5 ], "spans": [], "index": 4 }, { "bbox": [ 220, 156.5, 389, 171.66666666666666 ], "spans": [], "index": 5 }, { "bbox": [ 220, 171.66666666666666, 389, 186.83333333333331 ], "spans": [], "index": 6 }, { "bbox": [ 220, 186.83333333333331, 389, 201.99999999999997 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 206, 422, 219 ], "lines": [ { "bbox": [ 105, 205, 423, 221 ], "spans": [ { "bbox": [ 105, 205, 180, 221 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 180, 208, 192, 218 ], "score": 0.88, "content": "C _ { 1 }", "type": "inline_equation" }, { "bbox": [ 193, 205, 389, 221 ], "score": 1.0, "content": "depends on the regularities of the PINN solution", "type": "text" }, { "bbox": [ 389, 207, 419, 219 ], "score": 0.92, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 419, 205, 423, 221 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 8 } ], "index": 8, "bbox_fs": [ 105, 205, 423, 221 ] }, { "type": "text", "bbox": [ 106, 223, 506, 269 ], "lines": [ { "bbox": [ 105, 223, 506, 237 ], "spans": [ { "bbox": [ 105, 223, 231, 237 ], "score": 1.0, "content": "Now we turn to estimate the", "type": "text" }, { "bbox": [ 232, 223, 244, 234 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 245, 223, 345, 237 ], "score": 1.0, "content": "norm error estimate of", "type": "text" }, { "bbox": [ 346, 224, 371, 236 ], "score": 0.91, "content": "\\eta ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 371, 223, 450, 237 ], "score": 1.0, "content": ". We first replace", "type": "text" }, { "bbox": [ 450, 224, 475, 236 ], "score": 0.91, "content": "u ^ { n } ( x )", "type": "inline_equation" }, { "bbox": [ 476, 223, 506, 237 ], "score": 1.0, "content": "in the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 234, 507, 248 ], "spans": [ { "bbox": [ 105, 234, 363, 248 ], "score": 1.0, "content": "Crank-Nicolson time differencing scheme by the PINN solution", "type": "text" }, { "bbox": [ 363, 235, 392, 247 ], "score": 0.91, "content": "u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 393, 234, 507, 248 ], "score": 1.0, "content": "and compare the difference.", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 245, 505, 258 ], "spans": [ { "bbox": [ 105, 245, 505, 258 ], "score": 1.0, "content": "Subtracting equation 31 from the Crank-Nicolson scheme, we obtain the relation of the propagation", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 257, 245, 270 ], "spans": [ { "bbox": [ 105, 257, 128, 270 ], "score": 1.0, "content": "error", "type": "text" }, { "bbox": [ 129, 257, 232, 269 ], "score": 0.91, "content": "\\eta ^ { n } ( x ) \\stackrel { } { = } u ^ { n } ( x ) - u _ { \\theta ^ { n } } ( x )", "type": "inline_equation" }, { "bbox": [ 232, 257, 245, 270 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 12 } ], "index": 10.5, "bbox_fs": [ 105, 223, 507, 270 ] }, { "type": "interline_equation", "bbox": [ 142, 275, 469, 304 ], "lines": [ { "bbox": [ 142, 275, 469, 304 ], "spans": [ { "bbox": [ 142, 275, 469, 304 ], "score": 0.93, "content": "\\frac { \\eta ^ { n + 1 } - \\eta ^ { n } } { \\tau } - \\left( \\mathcal { N } \\left[ \\frac { u ^ { n + 1 } ( x ) + u ^ { n } ( x ) } { 2 } \\right] - \\mathcal { N } \\left[ \\frac { u _ { \\theta ^ { n + 1 } } ( x ) + u _ { \\theta ^ { n } } ( x ) } { 2 } \\right] \\right) = - \\mathcal { R } ( x )", "type": "interline_equation", "image_path": "62f04c16dc8717a21bc23761b46f2c61dfa7ebed04c5ebc058f90c6298e68b9f.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 142, 275, 469, 284.6666666666667 ], "spans": [], "index": 13 }, { "bbox": [ 142, 284.6666666666667, 469, 294.33333333333337 ], "spans": [], "index": 14 }, { "bbox": [ 142, 294.33333333333337, 469, 304.00000000000006 ], "spans": [], "index": 15 } ] }, { "type": "text", "bbox": [ 104, 309, 505, 334 ], "lines": [ { "bbox": [ 105, 307, 506, 325 ], "spans": [ { "bbox": [ 105, 307, 364, 325 ], "score": 1.0, "content": "Similar to the proof in Lemma A.1, we multiply equation 32 by", "type": "text" }, { "bbox": [ 365, 309, 449, 324 ], "score": 0.92, "content": "\\frac 1 2 ( \\eta ^ { n + 1 } ( x ) + \\eta ^ { n } ( x ) )", "type": "inline_equation" }, { "bbox": [ 449, 307, 506, 325 ], "score": 1.0, "content": "and integrate", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 321, 351, 335 ], "spans": [ { "bbox": [ 105, 321, 120, 335 ], "score": 1.0, "content": "for", "type": "text" }, { "bbox": [ 121, 324, 127, 332 ], "score": 0.76, "content": "x", "type": "inline_equation" }, { "bbox": [ 128, 321, 188, 335 ], "score": 1.0, "content": "on the domain", "type": "text" }, { "bbox": [ 189, 322, 196, 332 ], "score": 0.81, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 197, 321, 351, 335 ], "score": 1.0, "content": ". With Assumption 4.1 holds, we have", "type": "text" } ], "index": 17 } ], "index": 16.5, "bbox_fs": [ 105, 307, 506, 335 ] }, { "type": "interline_equation", "bbox": [ 181, 339, 429, 396 ], "lines": [ { "bbox": [ 181, 339, 429, 396 ], "spans": [ { "bbox": [ 181, 339, 429, 396 ], "score": 0.94, "content": "\\begin{array} { r l r } { { \\frac { \\| \\eta ^ { n + 1 } \\| ^ { 2 } - \\| \\eta ^ { n } \\| ^ { 2 } } { 2 \\tau } \\le - \\int _ { \\Omega } \\mathcal { R } ( x ) \\cdot \\frac { \\eta ^ { n + 1 } ( x ) + \\eta ^ { n } ( x ) } { 2 } } } \\\\ & { } & { \\qquad \\le \\displaystyle \\frac { 1 } { 4 } \\| \\mathcal { R } ^ { n + 1 } \\| ^ { 2 } + \\frac { 1 } { 2 } \\| \\eta ^ { n + 1 } \\| ^ { 2 } + \\frac { 1 } { 2 } \\| \\eta ^ { n } \\| ^ { 2 } , } \\end{array}", "type": "interline_equation", "image_path": "d4e221d18829ad970e35d9a8d2415c33f82f1cb665184a8912a409d99a9f1f2b.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 181, 339, 429, 358.0 ], "spans": [], "index": 18 }, { "bbox": [ 181, 358.0, 429, 377.0 ], "spans": [], "index": 19 }, { "bbox": [ 181, 377.0, 429, 396.0 ], "spans": [], "index": 20 } ] }, { "type": "text", "bbox": [ 107, 400, 275, 412 ], "lines": [ { "bbox": [ 105, 400, 276, 413 ], "spans": [ { "bbox": [ 105, 400, 276, 413 ], "score": 1.0, "content": "then we rearrange it to the following form", "type": "text" } ], "index": 21 } ], "index": 21, "bbox_fs": [ 105, 400, 276, 413 ] }, { "type": "interline_equation", "bbox": [ 212, 418, 399, 443 ], "lines": [ { "bbox": [ 212, 418, 399, 443 ], "spans": [ { "bbox": [ 212, 418, 399, 443 ], "score": 0.92, "content": "\\left. \\eta ^ { n + 1 } \\right. ^ { 2 } \\leq \\frac { 1 + \\tau } { 1 - \\tau } \\left. \\eta ^ { n } \\right. ^ { 2 } + \\frac { \\tau } { 1 - \\tau } \\left. \\mathcal { R } ^ { n + 1 } \\right. ^ { 2 } .", "type": "interline_equation", "image_path": "1f1510a77ce7bd121decbbbb8b82499e5661536c9d0ac67698b24b2e7ca142f3.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 212, 418, 399, 443 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 106, 449, 240, 461 ], "lines": [ { "bbox": [ 106, 448, 240, 462 ], "spans": [ { "bbox": [ 106, 448, 240, 462 ], "score": 1.0, "content": "then we apply Lemma A.3 to get", "type": "text" } ], "index": 23 } ], "index": 23, "bbox_fs": [ 106, 448, 240, 462 ] }, { "type": "interline_equation", "bbox": [ 180, 467, 430, 532 ], "lines": [ { "bbox": [ 180, 467, 430, 532 ], "spans": [ { "bbox": [ 180, 467, 430, 532 ], "score": 0.95, "content": "\\begin{array} { r l r } { { \\| \\eta ^ { n } \\| ^ { 2 } \\leq ( \\frac { 1 + \\tau } { 1 - \\tau } ) ^ { n } \\| \\eta ^ { 0 } \\| ^ { 2 } + \\frac { ( \\frac { 1 + \\tau } { 1 - \\tau } ) ^ { n } - 1 } { \\frac { 1 + \\tau } { 1 - \\tau } - 1 } \\cdot \\frac { \\tau \\displaystyle { \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\| \\mathcal { R } ^ { i } \\| ^ { 2 } } } { 1 - \\tau } } } \\\\ & { } & { \\leq ( 1 + 6 t _ { n } ) \\| \\eta ^ { 0 } \\| ^ { 2 } + \\frac { 3 t _ { n } } { 2 } \\displaystyle \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\| \\mathcal { R } ^ { i } \\| ^ { 2 } . } \\end{array}", "type": "interline_equation", "image_path": "c8b69359c18c29e8f2784916d1af56aff427236af63935deb09bb9b1c0fc98a8.jpg" } ] } ], "index": 25, "virtual_lines": [ { "bbox": [ 180, 467, 430, 488.6666666666667 ], "spans": [], "index": 24 }, { "bbox": [ 180, 488.6666666666667, 430, 510.33333333333337 ], "spans": [], "index": 25 }, { "bbox": [ 180, 510.33333333333337, 430, 532.0 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 538, 504, 571 ], "lines": [ { "bbox": [ 105, 538, 506, 560 ], "spans": [ { "bbox": [ 105, 538, 131, 555 ], "score": 1.0, "content": "Since", "type": "text" }, { "bbox": [ 131, 542, 173, 554 ], "score": 0.93, "content": "\\eta ^ { 0 } ( x ) = 0", "type": "inline_equation" }, { "bbox": [ 174, 538, 212, 555 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 212, 538, 351, 560 ], "score": 0.92, "content": "\\| \\eta ^ { n } \\| \\le C \\sqrt { t _ { n } } ( \\operatorname* { m a x } _ { 1 \\le i \\le n } \\sqrt { \\mathscr { L } ^ { i } } + N _ { r } ^ { \\frac { 1 } { 4 } } )", "type": "inline_equation" }, { "bbox": [ 352, 538, 425, 555 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 426, 543, 434, 552 ], "score": 0.83, "content": "C", "type": "inline_equation" }, { "bbox": [ 435, 538, 506, 555 ], "score": 1.0, "content": "and we finish the", "type": "text" } ], "index": 27 }, { "bbox": [ 103, 558, 502, 572 ], "spans": [ { "bbox": [ 103, 558, 135, 572 ], "score": 1.0, "content": "proof.", "type": "text" }, { "bbox": [ 497, 562, 502, 566 ], "score": 0.0, "content": "", "type": "text" } ], "index": 28 } ], "index": 27.5, "bbox_fs": [ 103, 538, 506, 572 ] }, { "type": "text", "bbox": [ 106, 580, 503, 604 ], "lines": [ { "bbox": [ 104, 579, 506, 596 ], "spans": [ { "bbox": [ 104, 579, 225, 596 ], "score": 1.0, "content": "Lemma A.3. If the sequence", "type": "text" }, { "bbox": [ 225, 581, 262, 593 ], "score": 0.92, "content": "\\{ T _ { n } \\} _ { n = 0 } ^ { \\infty }", "type": "inline_equation" }, { "bbox": [ 263, 579, 506, 596 ], "score": 1.0, "content": "satisfies the following propagation relation for some positive", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 588, 213, 608 ], "spans": [ { "bbox": [ 104, 588, 143, 608 ], "score": 1.0, "content": "constant", "type": "text" }, { "bbox": [ 143, 595, 150, 602 ], "score": 0.53, "content": "\\alpha", "type": "inline_equation" }, { "bbox": [ 151, 588, 169, 608 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 170, 592, 207, 604 ], "score": 0.92, "content": "\\{ \\beta _ { n } \\} _ { n = 1 } ^ { \\infty }", "type": "inline_equation" }, { "bbox": [ 207, 588, 213, 608 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 30 } ], "index": 29.5, "bbox_fs": [ 104, 579, 506, 608 ] }, { "type": "interline_equation", "bbox": [ 240, 604, 370, 618 ], "lines": [ { "bbox": [ 240, 604, 370, 618 ], "spans": [ { "bbox": [ 240, 604, 370, 618 ], "score": 0.87, "content": "T _ { n + 1 } \\leq \\alpha T _ { n } + \\beta _ { n + 1 } , \\quad n \\geq 0 ,", "type": "interline_equation", "image_path": "211aa00eb2a9edbcf5eff81e1b142c865f0751540c28bd69ee2cd6acd7512c18.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 240, 604, 370, 618 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 106, 622, 160, 633 ], "lines": [ { "bbox": [ 106, 621, 162, 635 ], "spans": [ { "bbox": [ 106, 621, 162, 635 ], "score": 1.0, "content": "then we have", "type": "text" } ], "index": 32 } ], "index": 32, "bbox_fs": [ 106, 621, 162, 635 ] }, { "type": "interline_equation", "bbox": [ 220, 632, 391, 657 ], "lines": [ { "bbox": [ 220, 632, 391, 657 ], "spans": [ { "bbox": [ 220, 632, 391, 657 ], "score": 0.91, "content": "T _ { n } \\leq \\alpha ^ { n } T _ { 0 } + \\frac { \\alpha ^ { n } - 1 } { \\alpha - 1 } \\operatorname* { m a x } _ { 1 \\leq i \\leq n } \\beta _ { i } , \\quad n \\geq 1 .", "type": "interline_equation", "image_path": "64730d3255e323b68c48a2a5f1fea28ccfe760a5d3c10a93a89c7ff5585c934f.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 220, 632, 391, 657 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 106, 672, 358, 685 ], "lines": [ { "bbox": [ 105, 672, 359, 686 ], "spans": [ { "bbox": [ 105, 672, 359, 686 ], "score": 1.0, "content": "Proof. This is accomplished by a standard recurrence formula.", "type": "text" } ], "index": 34 } ], "index": 34, "bbox_fs": [ 105, 672, 359, 686 ] }, { "type": "title", "bbox": [ 107, 699, 242, 711 ], "lines": [ { "bbox": [ 106, 699, 242, 712 ], "spans": [ { "bbox": [ 106, 699, 242, 712 ], "score": 1.0, "content": "A.4 EXPERIMENTAL DETAILS", "type": "text" } ], "index": 35 } ], "index": 35 }, { "type": "text", "bbox": [ 105, 720, 430, 732 ], "lines": [ { "bbox": [ 105, 719, 431, 733 ], "spans": [ { "bbox": [ 105, 719, 431, 733 ], "score": 1.0, "content": "In this section, we provide the details on the numerical experiments of Section 5.", "type": "text" } ], "index": 36 } ], "index": 36, "bbox_fs": [ 105, 719, 431, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 101, 505, 146 ], "lines": [ { "bbox": [ 106, 101, 505, 113 ], "spans": [ { "bbox": [ 106, 101, 505, 113 ], "score": 1.0, "content": "We present two practical considerations for the PINN network architecture, which has been applied", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 111, 505, 125 ], "spans": [ { "bbox": [ 105, 111, 505, 125 ], "score": 1.0, "content": "in CausualPINN Wang et al. (2022a) and other PINN frameworks. Although not deemed crucial for", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 124, 505, 136 ], "spans": [ { "bbox": [ 106, 124, 505, 136 ], "score": 1.0, "content": "the successful application of Algorithm 1, we have empirically observed that including them can lead", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 134, 373, 148 ], "spans": [ { "bbox": [ 105, 134, 373, 148 ], "score": 1.0, "content": "to further enhancements in accuracy and computational efficiency.", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "text", "bbox": [ 106, 157, 505, 191 ], "lines": [ { "bbox": [ 105, 156, 506, 171 ], "spans": [ { "bbox": [ 105, 156, 506, 171 ], "score": 1.0, "content": "Fourier Features Embedding. Many researchers have utilized Fourier features embedding to", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 169, 505, 181 ], "spans": [ { "bbox": [ 106, 169, 505, 181 ], "score": 1.0, "content": "enhance the accuracy and generalization Tancik et al. (2020); Wang et al. (2021b). We employ 1-D", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 180, 317, 192 ], "spans": [ { "bbox": [ 106, 180, 317, 192 ], "score": 1.0, "content": "Fourier features embedding in the following format:", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "interline_equation", "bbox": [ 145, 196, 465, 209 ], "lines": [ { "bbox": [ 145, 196, 465, 209 ], "spans": [ { "bbox": [ 145, 196, 465, 209 ], "score": 0.89, "content": "\\gamma ( x ) = [ 1 , \\cos ( \\omega x ) , \\sin ( \\omega x ) , \\cos ( 2 \\omega x ) , \\sin ( 2 \\omega x ) , . . . , \\cos ( M \\omega x ) , \\sin ( M \\omega x ) ] ^ { T }", "type": "interline_equation", "image_path": "5cc18c466f5418207c27cfa4efa4746356967c3fa394a31deec33666667a4937.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 145, 196, 465, 209 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 213, 505, 257 ], "lines": [ { "bbox": [ 106, 213, 506, 226 ], "spans": [ { "bbox": [ 106, 213, 133, 226 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 213, 179, 225 ], "score": 0.93, "content": "\\omega = 2 \\pi / L", "type": "inline_equation" }, { "bbox": [ 179, 213, 198, 226 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 198, 213, 210, 223 ], "score": 0.84, "content": "M", "type": "inline_equation" }, { "bbox": [ 210, 213, 506, 226 ], "score": 1.0, "content": "is a positive integer hyper-parameter. It maps the input data to a higher", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 224, 505, 236 ], "spans": [ { "bbox": [ 106, 224, 505, 236 ], "score": 1.0, "content": "dimensional space by Fourier transforms. The major advantage of this technique is that it improves", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 235, 506, 248 ], "spans": [ { "bbox": [ 105, 235, 506, 248 ], "score": 1.0, "content": "the model’s ability to approximate periodic or oscillatory behavior in the input data. It allows us to", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 246, 277, 258 ], "spans": [ { "bbox": [ 106, 246, 277, 258 ], "score": 1.0, "content": "satisfy the periodic boundary condition as", "type": "text" } ], "index": 11 } ], "index": 9.5 }, { "type": "interline_equation", "bbox": [ 266, 260, 345, 275 ], "lines": [ { "bbox": [ 266, 260, 345, 275 ], "spans": [ { "bbox": [ 266, 260, 345, 275 ], "score": 0.93, "content": "g ( x _ { b } ^ { i } ) = g ( x _ { b } ^ { i } + L )", "type": "interline_equation", "image_path": "671de6d20514855540acfdd9a38a6c42a21498c7eff5ed36f505f824f02ec8c9.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 266, 260, 345, 275 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 104, 278, 504, 301 ], "lines": [ { "bbox": [ 106, 278, 506, 291 ], "spans": [ { "bbox": [ 106, 278, 135, 291 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 279, 144, 288 ], "score": 0.81, "content": "L", "type": "inline_equation" }, { "bbox": [ 144, 278, 506, 291 ], "score": 1.0, "content": "represents the period of the periodic boundary condition. 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In", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 488, 505, 500 ], "spans": [ { "bbox": [ 105, 488, 505, 500 ], "score": 1.0, "content": "such a scenario, the embedding technique can be used to capture the periodic and oscillatory behavior", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 498, 505, 512 ], "spans": [ { "bbox": [ 105, 498, 505, 512 ], "score": 1.0, "content": "of the input data, while the neural network can be trained to satisfy the Dirichlet boundary conditions", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 510, 251, 522 ], "spans": [ { "bbox": [ 106, 510, 251, 522 ], "score": 1.0, "content": "(or Neumann boundary conditions).", "type": "text" } ], "index": 32 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 533, 505, 588 ], "lines": [ { "bbox": [ 105, 532, 506, 547 ], "spans": [ { "bbox": [ 105, 532, 506, 547 ], "score": 1.0, "content": "Modified Multi-layer Perceptrons. In recent researches Wang et al. 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The form of this", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 577, 203, 591 ], "spans": [ { "bbox": [ 105, 577, 203, 591 ], "score": 1.0, "content": "architecture is given as:", "type": "text" } ], "index": 37 } ], "index": 35 }, { "type": "interline_equation", "bbox": [ 158, 591, 426, 663 ], "lines": [ { "bbox": [ 158, 591, 426, 663 ], "spans": [ { "bbox": [ 158, 591, 426, 663 ], "score": 0.9, "content": "\\left\\{ \\begin{array} { l } { U = \\sigma \\big ( X W _ { u } + b _ { u } \\big ) , } \\\\ { V = \\sigma \\big ( X W _ { v } + b _ { v } \\big ) , } \\\\ { H _ { ( 1 ) } = \\sigma \\big ( X W _ { ( 0 ) } + b _ { ( 0 ) } \\big ) , } \\\\ { Z _ { ( n ) } = \\sigma \\big ( H _ { ( n ) } W _ { ( n ) } + b _ { ( n ) } \\big ) , \\quad n = 1 , 2 , . . . , D - 1 . } \\\\ { H _ { ( n + 1 ) } = \\big ( 1 - Z _ { ( n ) } \\big ) \\odot U + Z _ { ( n ) } \\odot V , \\quad n = 1 , 2 , . . . , D - 1 . } \\\\ { u _ { \\theta } ( X ) = H _ { ( D ) } W _ { ( D ) } + b _ { ( D ) } . } \\end{array} \\right.", "type": "interline_equation", "image_path": "517e4a986cb9aeac196a2bcb5955ceed5c1c088508585a49543130cf5ac5862f.jpg" } ] } ], "index": 39, "virtual_lines": [ { "bbox": [ 158, 591, 426, 615.0 ], "spans": [], "index": 38 }, { "bbox": [ 158, 615.0, 426, 639.0 ], "spans": [], "index": 39 }, { "bbox": [ 158, 639.0, 426, 663.0 ], "spans": [], "index": 40 } ] }, { "type": "text", "bbox": [ 106, 665, 506, 732 ], "lines": [ { "bbox": [ 106, 665, 505, 678 ], "spans": [ { "bbox": [ 106, 665, 132, 678 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 132, 666, 150, 678 ], "score": 0.91, "content": "\\sigma ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 150, 665, 269, 678 ], "score": 1.0, "content": "represents activation function", "type": "text" }, { "bbox": [ 269, 666, 302, 678 ], "score": 0.81, "content": "( \\operatorname { t a n h } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 303, 665, 505, 678 ], "score": 1.0, "content": "in this work); the trainable parameters of the neural", "type": "text" } ], "index": 41 }, { "bbox": [ 104, 676, 506, 691 ], "spans": [ { "bbox": [ 104, 676, 214, 691 ], "score": 1.0, "content": "network are indicated by", "type": "text" }, { "bbox": [ 214, 677, 321, 690 ], "score": 0.86, "content": "W _ { u } , W _ { v } , W _ { ( n ) } , b _ { u } , b _ { v } , b _ { ( n ) }", "type": "inline_equation" }, { "bbox": [ 321, 676, 327, 691 ], "score": 1.0, "content": ";", "type": "text" }, { "bbox": [ 327, 677, 337, 687 ], "score": 0.72, "content": "D", "type": "inline_equation" }, { "bbox": [ 337, 676, 506, 691 ], "score": 1.0, "content": "represents the depth of neural network;", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 123, 700 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 689, 133, 698 ], "score": 0.82, "content": "\\odot", "type": "inline_equation" }, { "bbox": [ 133, 688, 505, 700 ], "score": 1.0, "content": "denotes the operation of point-wise multiplication. 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These", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 710, 506, 722 ], "spans": [ { "bbox": [ 105, 710, 506, 722 ], "score": 1.0, "content": "connections enable the network to bypass certain layers and transmit information directly from earlier", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 721, 192, 734 ], "spans": [ { "bbox": [ 105, 721, 192, 734 ], "score": 1.0, "content": "layers to later layers.", "type": "text" } ], "index": 46 } ], "index": 43.5 } ], "page_idx": 16, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 308, 37 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 764 ], "spans": [ { "bbox": [ 299, 750, 312, 764 ], "score": 1.0, "content": "", "type": "text", "height": 14, "width": 13 } ] } ] }, { "type": "discarded", "bbox": [ 107, 82, 296, 94 ], "lines": [ { "bbox": [ 106, 82, 298, 95 ], "spans": [ { "bbox": [ 106, 82, 298, 95 ], "score": 1.0, "content": "A.4.1 NEURAL NETWORK ARCHITECTURE", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 101, 505, 146 ], "lines": [ { "bbox": [ 106, 101, 505, 113 ], "spans": [ { "bbox": [ 106, 101, 505, 113 ], "score": 1.0, "content": "We present two practical considerations for the PINN network architecture, which has been applied", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 111, 505, 125 ], "spans": [ { "bbox": [ 105, 111, 505, 125 ], "score": 1.0, "content": "in CausualPINN Wang et al. (2022a) and other PINN frameworks. Although not deemed crucial for", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 124, 505, 136 ], "spans": [ { "bbox": [ 106, 124, 505, 136 ], "score": 1.0, "content": "the successful application of Algorithm 1, we have empirically observed that including them can lead", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 134, 373, 148 ], "spans": [ { "bbox": [ 105, 134, 373, 148 ], "score": 1.0, "content": "to further enhancements in accuracy and computational efficiency.", "type": "text" } ], "index": 3 } ], "index": 1.5, "bbox_fs": [ 105, 101, 505, 148 ] }, { "type": "text", "bbox": [ 106, 157, 505, 191 ], "lines": [ { "bbox": [ 105, 156, 506, 171 ], "spans": [ { "bbox": [ 105, 156, 506, 171 ], "score": 1.0, "content": "Fourier Features Embedding. Many researchers have utilized Fourier features embedding to", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 169, 505, 181 ], "spans": [ { "bbox": [ 106, 169, 505, 181 ], "score": 1.0, "content": "enhance the accuracy and generalization Tancik et al. (2020); Wang et al. (2021b). We employ 1-D", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 180, 317, 192 ], "spans": [ { "bbox": [ 106, 180, 317, 192 ], "score": 1.0, "content": "Fourier features embedding in the following format:", "type": "text" } ], "index": 6 } ], "index": 5, "bbox_fs": [ 105, 156, 506, 192 ] }, { "type": "interline_equation", "bbox": [ 145, 196, 465, 209 ], "lines": [ { "bbox": [ 145, 196, 465, 209 ], "spans": [ { "bbox": [ 145, 196, 465, 209 ], "score": 0.89, "content": "\\gamma ( x ) = [ 1 , \\cos ( \\omega x ) , \\sin ( \\omega x ) , \\cos ( 2 \\omega x ) , \\sin ( 2 \\omega x ) , . . . , \\cos ( M \\omega x ) , \\sin ( M \\omega x ) ] ^ { T }", "type": "interline_equation", "image_path": "5cc18c466f5418207c27cfa4efa4746356967c3fa394a31deec33666667a4937.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 145, 196, 465, 209 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 213, 505, 257 ], "lines": [ { "bbox": [ 106, 213, 506, 226 ], "spans": [ { "bbox": [ 106, 213, 133, 226 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 213, 179, 225 ], "score": 0.93, "content": "\\omega = 2 \\pi / L", "type": "inline_equation" }, { "bbox": [ 179, 213, 198, 226 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 198, 213, 210, 223 ], "score": 0.84, "content": "M", "type": "inline_equation" }, { "bbox": [ 210, 213, 506, 226 ], "score": 1.0, "content": "is a positive integer hyper-parameter. It maps the input data to a higher", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 224, 505, 236 ], "spans": [ { "bbox": [ 106, 224, 505, 236 ], "score": 1.0, "content": "dimensional space by Fourier transforms. The major advantage of this technique is that it improves", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 235, 506, 248 ], "spans": [ { "bbox": [ 105, 235, 506, 248 ], "score": 1.0, "content": "the model’s ability to approximate periodic or oscillatory behavior in the input data. It allows us to", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 246, 277, 258 ], "spans": [ { "bbox": [ 106, 246, 277, 258 ], "score": 1.0, "content": "satisfy the periodic boundary condition as", "type": "text" } ], "index": 11 } ], "index": 9.5, "bbox_fs": [ 105, 213, 506, 258 ] }, { "type": "interline_equation", "bbox": [ 266, 260, 345, 275 ], "lines": [ { "bbox": [ 266, 260, 345, 275 ], "spans": [ { "bbox": [ 266, 260, 345, 275 ], "score": 0.93, "content": "g ( x _ { b } ^ { i } ) = g ( x _ { b } ^ { i } + L )", "type": "interline_equation", "image_path": "671de6d20514855540acfdd9a38a6c42a21498c7eff5ed36f505f824f02ec8c9.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 266, 260, 345, 275 ], "spans": [], "index": 12 } ] }, { "type": "text", "bbox": [ 104, 278, 504, 301 ], "lines": [ { "bbox": [ 106, 278, 506, 291 ], "spans": [ { "bbox": [ 106, 278, 135, 291 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 279, 144, 288 ], "score": 0.81, "content": "L", "type": "inline_equation" }, { "bbox": [ 144, 278, 506, 291 ], "score": 1.0, "content": "represents the period of the periodic boundary condition. Furthermore, for the two-", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 289, 480, 302 ], "spans": [ { "bbox": [ 106, 289, 480, 302 ], "score": 1.0, "content": "dimensional Navier-Stokes equation, the Fourier feature embedding takes the following form", "type": "text" } ], "index": 14 } ], "index": 13.5, "bbox_fs": [ 106, 278, 506, 302 ] }, { "type": "interline_equation", "bbox": [ 183, 304, 425, 431 ], "lines": [ { "bbox": [ 183, 304, 425, 431 ], "spans": [ { "bbox": [ 183, 304, 425, 431 ], "score": 0.96, "content": "\\gamma ( x ) = { \\left[ \\begin{array} { l } { 1 } \\\\ { \\cos ( \\omega _ { x } x ) , . . . , \\cos ( M \\omega _ { x } x ) } \\\\ { \\cos ( \\omega _ { y } y ) , . . . , \\cos ( M \\omega _ { y } y ) } \\\\ { \\sin ( \\omega _ { x } x ) , . . . , \\sin ( M \\omega _ { x } x ) } \\\\ { \\sin ( \\omega _ { y } y ) , . . . , \\sin ( M \\omega _ { y } y ) } \\\\ { \\cos ( \\omega _ { x } x ) \\cos ( \\omega _ { y } y ) , . . . , \\cos ( M \\omega _ { x } x ) \\cos ( M \\omega _ { y } y ) } \\\\ { \\cos ( \\omega _ { x } x ) \\sin ( \\omega _ { y } y ) , . . . , \\cos ( M \\omega _ { x } x ) \\sin ( M \\omega _ { y } y ) } \\\\ { \\sin ( \\omega _ { x } x ) \\cos ( \\omega _ { y } y ) , . . . , \\sin ( M \\omega _ { x } x ) \\cos ( M \\omega _ { y } y ) } \\\\ { \\sin ( \\omega _ { x } x ) \\sin ( \\omega _ { y } y ) , . . . , \\sin ( M \\omega _ { x } x ) \\sin ( M \\omega _ { y } y ) } \\end{array} \\right] }", "type": "interline_equation", "image_path": "6f991564be2879846ee1b33c9874424ed743ba306057c474d4aff824842c96c6.jpg" } ] } ], "index": 19.5, "virtual_lines": [ { "bbox": [ 183, 304, 425, 316.7 ], "spans": [], "index": 15 }, { "bbox": [ 183, 316.7, 425, 329.4 ], "spans": [], "index": 16 }, { "bbox": [ 183, 329.4, 425, 342.09999999999997 ], "spans": [], "index": 17 }, { "bbox": [ 183, 342.09999999999997, 425, 354.79999999999995 ], "spans": [], "index": 18 }, { "bbox": [ 183, 354.79999999999995, 425, 367.49999999999994 ], "spans": [], "index": 19 }, { "bbox": [ 183, 367.49999999999994, 425, 380.19999999999993 ], "spans": [], "index": 20 }, { "bbox": [ 183, 380.19999999999993, 425, 392.8999999999999 ], "spans": [], "index": 21 }, { "bbox": [ 183, 392.8999999999999, 425, 405.5999999999999 ], "spans": [], "index": 22 }, { "bbox": [ 183, 405.5999999999999, 425, 418.2999999999999 ], "spans": [], "index": 23 }, { "bbox": [ 183, 418.2999999999999, 425, 430.9999999999999 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 106, 432, 505, 522 ], "lines": [ { "bbox": [ 106, 433, 505, 445 ], "spans": [ { "bbox": [ 106, 433, 505, 445 ], "score": 1.0, "content": "Previous studies Lu et al. 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The form of this", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 577, 203, 591 ], "spans": [ { "bbox": [ 105, 577, 203, 591 ], "score": 1.0, "content": "architecture is given as:", "type": "text" } ], "index": 37 } ], "index": 35, "bbox_fs": [ 105, 532, 506, 591 ] }, { "type": "interline_equation", "bbox": [ 158, 591, 426, 663 ], "lines": [ { "bbox": [ 158, 591, 426, 663 ], "spans": [ { "bbox": [ 158, 591, 426, 663 ], "score": 0.9, "content": "\\left\\{ \\begin{array} { l } { U = \\sigma \\big ( X W _ { u } + b _ { u } \\big ) , } \\\\ { V = \\sigma \\big ( X W _ { v } + b _ { v } \\big ) , } \\\\ { H _ { ( 1 ) } = \\sigma \\big ( X W _ { ( 0 ) } + b _ { ( 0 ) } \\big ) , } \\\\ { Z _ { ( n ) } = \\sigma \\big ( H _ { ( n ) } W _ { ( n ) } + b _ { ( n ) } \\big ) , \\quad n = 1 , 2 , . . . , D - 1 . } \\\\ { H _ { ( n + 1 ) } = \\big ( 1 - Z _ { ( n ) } \\big ) \\odot U + Z _ { ( n ) } \\odot V , \\quad n = 1 , 2 , . . . , D - 1 . } \\\\ { u _ { \\theta } ( X ) = H _ { ( D ) } W _ { ( D ) } + b _ { ( D ) } . } \\end{array} \\right.", "type": "interline_equation", "image_path": "517e4a986cb9aeac196a2bcb5955ceed5c1c088508585a49543130cf5ac5862f.jpg" } ] } ], "index": 39, "virtual_lines": [ { "bbox": [ 158, 591, 426, 615.0 ], "spans": [], "index": 38 }, { "bbox": [ 158, 615.0, 426, 639.0 ], "spans": [], "index": 39 }, { "bbox": [ 158, 639.0, 426, 663.0 ], "spans": [], "index": 40 } ] }, { "type": "text", "bbox": [ 106, 665, 506, 732 ], "lines": [ { "bbox": [ 106, 665, 505, 678 ], "spans": [ { "bbox": [ 106, 665, 132, 678 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 132, 666, 150, 678 ], "score": 0.91, "content": "\\sigma ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 150, 665, 269, 678 ], "score": 1.0, "content": "represents activation function", "type": "text" }, { "bbox": [ 269, 666, 302, 678 ], "score": 0.81, "content": "( \\operatorname { t a n h } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 303, 665, 505, 678 ], "score": 1.0, "content": "in this work); the trainable parameters of the neural", "type": "text" } ], "index": 41 }, { "bbox": [ 104, 676, 506, 691 ], "spans": [ { "bbox": [ 104, 676, 214, 691 ], "score": 1.0, "content": "network are indicated by", "type": "text" }, { "bbox": [ 214, 677, 321, 690 ], "score": 0.86, "content": "W _ { u } , W _ { v } , W _ { ( n ) } , b _ { u } , b _ { v } , b _ { ( n ) }", "type": "inline_equation" }, { "bbox": [ 321, 676, 327, 691 ], "score": 1.0, "content": ";", "type": "text" }, { "bbox": [ 327, 677, 337, 687 ], "score": 0.72, "content": "D", "type": "inline_equation" }, { "bbox": [ 337, 676, 506, 691 ], "score": 1.0, "content": "represents the depth of neural network;", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 688, 505, 700 ], "spans": [ { "bbox": [ 106, 688, 123, 700 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 124, 689, 133, 698 ], "score": 0.82, "content": "\\odot", "type": "inline_equation" }, { "bbox": [ 133, 688, 505, 700 ], "score": 1.0, "content": "denotes the operation of point-wise multiplication. 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This is because the hidden function", "type": "text" }, { "bbox": [ 430, 126, 453, 138 ], "score": 0.92, "content": "k _ { i } ( x )", "type": "inline_equation" }, { "bbox": [ 453, 127, 505, 139 ], "score": 1.0, "content": "can differ in", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 135, 506, 151 ], "spans": [ { "bbox": [ 104, 135, 201, 151 ], "score": 1.0, "content": "scale from the solution", "type": "text" }, { "bbox": [ 201, 137, 237, 149 ], "score": 0.9, "content": "u ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 237, 135, 306, 151 ], "score": 1.0, "content": ". Instead, we use", "type": "text" }, { "bbox": [ 306, 138, 329, 149 ], "score": 0.91, "content": "q + 1", "type": "inline_equation" }, { "bbox": [ 330, 135, 506, 151 ], "score": 1.0, "content": "neural networks to separately approximate", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 147, 507, 161 ], "spans": [ { "bbox": [ 106, 148, 245, 160 ], "score": 0.89, "content": "k _ { 1 } ( x ) , k _ { 2 } ( x ) , \\cdot \\cdot \\cdot , k _ { q } ( x ) , u ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 245, 147, 507, 161 ], "score": 1.0, "content": ". 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Multiple neural networks are used in our TL-DPINN2 method", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 295, 505, 307 ], "spans": [ { "bbox": [ 106, 295, 505, 307 ], "score": 1.0, "content": "while a single neural network is used in our TL-DPINN1 method. Adam optimizer with an initial", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 305, 506, 318 ], "spans": [ { "bbox": [ 105, 305, 506, 318 ], "score": 1.0, "content": "learning rate of 0.001 and exponential rate decay is used. More details about the hyper-parameters of", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 316, 439, 329 ], "spans": [ { "bbox": [ 105, 316, 439, 329 ], "score": 1.0, "content": "neural networks and the hyper-parameters of Algorithm 1 are presented in Table 7.", "type": "text" } ], "index": 18 } ], "index": 16 }, { "type": "table", "bbox": [ 119, 357, 493, 437 ], "blocks": [ { "type": "table_caption", "bbox": [ 199, 337, 411, 349 ], "group_id": 0, "lines": [ { "bbox": [ 199, 335, 411, 350 ], "spans": [ { "bbox": [ 199, 335, 411, 350 ], "score": 1.0, "content": "Table 7: Detailed experimental settings of Section 5.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "table_body", "bbox": [ 119, 357, 493, 437 ], "group_id": 0, "lines": [ { "bbox": [ 119, 357, 493, 437 ], "spans": [ { "bbox": [ 119, 357, 493, 437 ], "score": 0.982, "html": "
EquationsDepthWidthFeatures MNtNrIterations (Mo,M1)E
RD412810200512(10000,1000)1e-9
AC412810200512(10000,2000)1e-10
KS(regular)32565250500(10000,3000)1e-8
KS(chaotic)81285250500(10000,7000)1e-10
NS41285100100(10000,5000)1e-5
", "type": "table", "image_path": "3d465d6e5962e9152a6067575f4cdf4f024c2188341e5ab50faab5f91e2b4ccb.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 119, 357, 493, 383.6666666666667 ], "spans": [], "index": 20 }, { "bbox": [ 119, 383.6666666666667, 493, 410.33333333333337 ], "spans": [], "index": 21 }, { "bbox": [ 119, 410.33333333333337, 493, 437.00000000000006 ], "spans": [], "index": 22 } ] } ], "index": 20.0 }, { "type": "text", "bbox": [ 106, 447, 505, 568 ], "lines": [ { "bbox": [ 106, 447, 506, 460 ], "spans": [ { "bbox": [ 106, 447, 506, 460 ], "score": 1.0, "content": "For the configuration of other five baselines: 1) original PINN Raissi et al. (2019); 2) adaptive sam-", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 457, 506, 471 ], "spans": [ { "bbox": [ 105, 457, 506, 471 ], "score": 1.0, "content": "pling L. Wight & Zhao (2021); 3) self-attention McClenny & Braga-Neto (2023); 4) time marching", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 468, 506, 482 ], "spans": [ { "bbox": [ 105, 468, 506, 482 ], "score": 1.0, "content": "Mattey & Ghosh (2022) and 5) causal PINN Wang et al. (2022a), all of them have a neural network", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 480, 505, 492 ], "spans": [ { "bbox": [ 106, 480, 505, 492 ], "score": 1.0, "content": "size with the same width and 1 deeper depth than that in Table 7. The collocation points number", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 490, 505, 504 ], "spans": [ { "bbox": [ 105, 490, 275, 504 ], "score": 1.0, "content": "for all five baselines are configured to be", "type": "text" }, { "bbox": [ 275, 491, 312, 502 ], "score": 0.92, "content": "N _ { t } \\times N _ { r }", "type": "inline_equation" }, { "bbox": [ 312, 490, 505, 504 ], "score": 1.0, "content": "in Table 7. For example, a continuous original", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 501, 506, 514 ], "spans": [ { "bbox": [ 105, 501, 307, 514 ], "score": 1.0, "content": "PINN has size [2, 128, 128, 128, 128, 128, 1] and", "type": "text" }, { "bbox": [ 307, 502, 352, 513 ], "score": 0.9, "content": "2 0 0 \\times 5 1 2", "type": "inline_equation" }, { "bbox": [ 352, 501, 506, 514 ], "score": 1.0, "content": "collocation points on the space-time", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 512, 505, 525 ], "spans": [ { "bbox": [ 105, 512, 505, 525 ], "score": 1.0, "content": "domain to compute the loss, then each discrete PINN has the smaller size [1, 128, 128, 128, 128, 1]", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 524, 505, 536 ], "spans": [ { "bbox": [ 106, 524, 505, 536 ], "score": 1.0, "content": "and much smaller collocation points 512 on space domain. The total parameters and computation", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 534, 505, 547 ], "spans": [ { "bbox": [ 106, 534, 505, 547 ], "score": 1.0, "content": "of 200 discrete PINNs and the computation on the loss calculation are about the same with a single", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 546, 505, 558 ], "spans": [ { "bbox": [ 105, 546, 505, 558 ], "score": 1.0, "content": "continuous PINN. In this configuration, we can sure that the comparison between our TL-DPINNs", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 557, 497, 569 ], "spans": [ { "bbox": [ 106, 557, 497, 569 ], "score": 1.0, "content": "and other five baselines is fair, showing the discrete PINNs are efficient for practical applications.", "type": "text" } ], "index": 33 } ], "index": 28 }, { "type": "title", "bbox": [ 108, 579, 402, 591 ], "lines": [ { "bbox": [ 106, 579, 403, 592 ], "spans": [ { "bbox": [ 106, 579, 403, 592 ], "score": 1.0, "content": "A.4.3 ADDITIONAL RESULTS FOR REACTION-DIFFUSION EQUATION", "type": "text" } ], "index": 34 } ], "index": 34 }, { "type": "text", "bbox": [ 106, 598, 505, 676 ], "lines": [ { "bbox": [ 105, 598, 506, 612 ], "spans": [ { "bbox": [ 105, 598, 220, 612 ], "score": 1.0, "content": "Figure 6 (a) depicts how the", "type": "text" }, { "bbox": [ 221, 598, 233, 609 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 233, 598, 430, 612 ], "score": 1.0, "content": "error changes as time goes on, as we can see, the", "type": "text" }, { "bbox": [ 430, 598, 443, 609 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 443, 598, 506, 612 ], "score": 1.0, "content": "error increases", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 610, 505, 622 ], "spans": [ { "bbox": [ 105, 610, 357, 622 ], "score": 1.0, "content": "in the early training steps and is kept at a stable level between", "type": "text" }, { "bbox": [ 358, 610, 403, 621 ], "score": 0.89, "content": "1 . 0 0 e \\mathrm { ~ - ~ } 0 5", "type": "inline_equation" }, { "bbox": [ 404, 610, 421, 622 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 421, 610, 468, 621 ], "score": 0.86, "content": "5 . 0 0 e - 0 5", "type": "inline_equation" }, { "bbox": [ 468, 610, 505, 622 ], "score": 1.0, "content": "later. As", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 620, 505, 633 ], "spans": [ { "bbox": [ 105, 620, 505, 633 ], "score": 1.0, "content": "shown in Figure 6 (b), based on the trainable parameters of the preceding time stamp, only a few", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 631, 506, 645 ], "spans": [ { "bbox": [ 105, 631, 506, 645 ], "score": 1.0, "content": "hundred steps of training are required for each time stamp to satisfy the early stopping criterion,", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 643, 506, 655 ], "spans": [ { "bbox": [ 106, 643, 506, 655 ], "score": 1.0, "content": "and then move to the training of the next time stamp. Figure 8 shows the training loss at different", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 654, 506, 666 ], "spans": [ { "bbox": [ 105, 654, 506, 666 ], "score": 1.0, "content": "time steps. Figure 7 compares the predicted and reference solutions at different time instants. The", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 665, 395, 677 ], "spans": [ { "bbox": [ 106, 665, 395, 677 ], "score": 1.0, "content": "predictions given by our method are identical to the reference solutions.", "type": "text" } ], "index": 41 } ], "index": 38 }, { "type": "title", "bbox": [ 107, 689, 357, 700 ], "lines": [ { "bbox": [ 106, 689, 358, 702 ], "spans": [ { "bbox": [ 106, 689, 358, 702 ], "score": 1.0, "content": "A.5 ADDITIONAL RESULTS FOR ALLEN-CAHN EQUATION", "type": "text" } ], "index": 42 } ], "index": 42 }, { "type": "text", "bbox": [ 106, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Figure 9 shows the predicted solution against the reference solution, our proposed method achieves a", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 719, 505, 733 ], "spans": [ { "bbox": [ 105, 719, 139, 733 ], "score": 1.0, "content": "relative", "type": "text" }, { "bbox": [ 139, 720, 152, 730 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 152, 719, 187, 733 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 187, 721, 233, 731 ], "score": 0.83, "content": "5 . 9 2 e \\mathrm { ~ - ~ } 0 4", "type": "inline_equation" }, { "bbox": [ 233, 719, 505, 733 ], "score": 1.0, "content": ". Figure 10 presents the comparison between the reference and the", "type": "text" } ], "index": 44 } ], "index": 43.5 } ], "page_idx": 17, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 308, 37 ], "lines": [ { "bbox": [ 107, 25, 308, 38 ], "spans": [ { "bbox": [ 107, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2024", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 310, 760 ], "lines": [ { "bbox": [ 299, 750, 312, 763 ], "spans": [ { "bbox": [ 299, 750, 312, 763 ], "score": 1.0, "content": "18", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 506, 171 ], "lines": [ { "bbox": [ 105, 81, 506, 96 ], "spans": [ { "bbox": [ 105, 81, 506, 96 ], "score": 1.0, "content": "Multiple Neural Networks. For PINN with backward Euler or Crank-Nicolson time differencing,", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 295, 106 ], "score": 1.0, "content": "the neural network has the form of single input", "type": "text" }, { "bbox": [ 296, 96, 303, 104 ], "score": 0.74, "content": "x", "type": "inline_equation" }, { "bbox": [ 303, 93, 375, 106 ], "score": 1.0, "content": "and single output", "type": "text" }, { "bbox": [ 375, 93, 399, 105 ], "score": 0.9, "content": "u _ { \\theta } ( x )", "type": "inline_equation" }, { "bbox": [ 400, 93, 506, 106 ], "score": 1.0, "content": ". However, for the general", "type": "text" } ], "index": 1 }, { "bbox": [ 104, 103, 506, 118 ], "spans": [ { "bbox": [ 104, 103, 216, 118 ], "score": 1.0, "content": "form of Runge-Kutta with", "type": "text" }, { "bbox": [ 216, 106, 223, 116 ], "score": 0.8, "content": "q", "type": "inline_equation" }, { "bbox": [ 223, 103, 360, 118 ], "score": 1.0, "content": "stages, we have multiple outputs", "type": "text" }, { "bbox": [ 360, 104, 502, 117 ], "score": 0.91, "content": "\\left[ k _ { 1 } ( x ) , k _ { 2 } ( x ) , \\cdot \\cdot \\cdot , k _ { q } ( x ) , u ^ { \\bar { n + 1 } } ( x ) \\right]", "type": "inline_equation" }, { "bbox": [ 503, 103, 506, 118 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 506, 128 ], "spans": [ { "bbox": [ 105, 114, 506, 128 ], "score": 1.0, "content": "While it is possible to use a single neural network with multiple outputs for the PINN approximation,", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 126, 505, 139 ], "spans": [ { "bbox": [ 106, 127, 429, 139 ], "score": 1.0, "content": "this approach may lead to slow convergence. This is because the hidden function", "type": "text" }, { "bbox": [ 430, 126, 453, 138 ], "score": 0.92, "content": "k _ { i } ( x )", "type": "inline_equation" }, { "bbox": [ 453, 127, 505, 139 ], "score": 1.0, "content": "can differ in", "type": "text" } ], "index": 4 }, { "bbox": [ 104, 135, 506, 151 ], "spans": [ { "bbox": [ 104, 135, 201, 151 ], "score": 1.0, "content": "scale from the solution", "type": "text" }, { "bbox": [ 201, 137, 237, 149 ], "score": 0.9, "content": "u ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 237, 135, 306, 151 ], "score": 1.0, "content": ". Instead, we use", "type": "text" }, { "bbox": [ 306, 138, 329, 149 ], "score": 0.91, "content": "q + 1", "type": "inline_equation" }, { "bbox": [ 330, 135, 506, 151 ], "score": 1.0, "content": "neural networks to separately approximate", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 147, 507, 161 ], "spans": [ { "bbox": [ 106, 148, 245, 160 ], "score": 0.89, "content": "k _ { 1 } ( x ) , k _ { 2 } ( x ) , \\cdot \\cdot \\cdot , k _ { q } ( x ) , u ^ { n + 1 } ( x )", "type": "inline_equation" }, { "bbox": [ 245, 147, 507, 161 ], "score": 1.0, "content": ". Although this approach leads to an increase in the number of", "type": "text" } ], "index": 6 }, { "bbox": [ 104, 158, 461, 173 ], "spans": [ { "bbox": [ 104, 158, 461, 173 ], "score": 1.0, "content": "neural network parameters, it greatly enhances both the training efficiency and accuracy.", "type": "text" } ], "index": 7 } ], "index": 3.5, "bbox_fs": [ 104, 81, 507, 173 ] }, { "type": "title", "bbox": [ 108, 182, 274, 194 ], "lines": [ { "bbox": [ 106, 181, 275, 194 ], "spans": [ { "bbox": [ 106, 181, 275, 194 ], "score": 1.0, "content": "A.4.2 CONFIGURATION OF TRAINING", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 107, 201, 504, 224 ], "lines": [ { "bbox": [ 105, 200, 506, 214 ], "spans": [ { "bbox": [ 105, 200, 433, 214 ], "score": 1.0, "content": "Error metric To quantify the performance of our methods, we apply a relative", "type": "text" }, { "bbox": [ 433, 201, 445, 211 ], "score": 0.85, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 445, 200, 506, 214 ], "score": 1.0, "content": "norm over the", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 212, 209, 225 ], "spans": [ { "bbox": [ 105, 212, 209, 225 ], "score": 1.0, "content": "spatial-temporal domain:", "type": "text" } ], "index": 10 } ], "index": 9.5, "bbox_fs": [ 105, 200, 506, 225 ] }, { "type": "interline_equation", "bbox": [ 186, 226, 425, 266 ], "lines": [ { "bbox": [ 186, 226, 425, 266 ], "spans": [ { "bbox": [ 186, 226, 425, 266 ], "score": 0.92, "content": "\\mathrm { r e l a t i v e } \\ L ^ { 2 } \\mathrm { e r r o r } = \\sqrt { \\frac { \\sum _ { n = 1 } ^ { N _ { t } } \\sum _ { i = 1 } ^ { N _ { r } } | u _ { \\theta ^ { n } } ( x _ { i } ) - u ( t _ { n } , x _ { i } ) | ^ { 2 } } { \\sum _ { n = 1 } ^ { N _ { t } } \\sum _ { i = 1 } ^ { N _ { r } } u ( t _ { n } , x _ { i } ) ^ { 2 } } }", "type": "interline_equation", "image_path": "ed42ed6bb277a213249641d29949305ad560a4048b69704e4449a7cc768ba2cb.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 186, 226, 425, 239.33333333333334 ], "spans": [], "index": 11 }, { "bbox": [ 186, 239.33333333333334, 425, 252.66666666666669 ], "spans": [], "index": 12 }, { "bbox": [ 186, 252.66666666666669, 425, 266.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 107, 273, 505, 329 ], "lines": [ { "bbox": [ 106, 273, 505, 285 ], "spans": [ { "bbox": [ 106, 273, 505, 285 ], "score": 1.0, "content": "Neural networks and training parameters In all examples, the Fourier feature embedding is", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 283, 505, 297 ], "spans": [ { "bbox": [ 105, 283, 505, 297 ], "score": 1.0, "content": "applied and the modified MLP is used. Multiple neural networks are used in our TL-DPINN2 method", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 295, 505, 307 ], "spans": [ { "bbox": [ 106, 295, 505, 307 ], "score": 1.0, "content": "while a single neural network is used in our TL-DPINN1 method. Adam optimizer with an initial", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 305, 506, 318 ], "spans": [ { "bbox": [ 105, 305, 506, 318 ], "score": 1.0, "content": "learning rate of 0.001 and exponential rate decay is used. More details about the hyper-parameters of", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 316, 439, 329 ], "spans": [ { "bbox": [ 105, 316, 439, 329 ], "score": 1.0, "content": "neural networks and the hyper-parameters of Algorithm 1 are presented in Table 7.", "type": "text" } ], "index": 18 } ], "index": 16, "bbox_fs": [ 105, 273, 506, 329 ] }, { "type": "table", "bbox": [ 119, 357, 493, 437 ], "blocks": [ { "type": "table_caption", "bbox": [ 199, 337, 411, 349 ], "group_id": 0, "lines": [ { "bbox": [ 199, 335, 411, 350 ], "spans": [ { "bbox": [ 199, 335, 411, 350 ], "score": 1.0, "content": "Table 7: Detailed experimental settings of Section 5.", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "table_body", "bbox": [ 119, 357, 493, 437 ], "group_id": 0, "lines": [ { "bbox": [ 119, 357, 493, 437 ], "spans": [ { "bbox": [ 119, 357, 493, 437 ], "score": 0.982, "html": "
EquationsDepthWidthFeatures MNtNrIterations (Mo,M1)E
RD412810200512(10000,1000)1e-9
AC412810200512(10000,2000)1e-10
KS(regular)32565250500(10000,3000)1e-8
KS(chaotic)81285250500(10000,7000)1e-10
NS41285100100(10000,5000)1e-5
", "type": "table", "image_path": "3d465d6e5962e9152a6067575f4cdf4f024c2188341e5ab50faab5f91e2b4ccb.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 119, 357, 493, 383.6666666666667 ], "spans": [], "index": 20 }, { "bbox": [ 119, 383.6666666666667, 493, 410.33333333333337 ], "spans": [], "index": 21 }, { "bbox": [ 119, 410.33333333333337, 493, 437.00000000000006 ], "spans": [], "index": 22 } ] } ], "index": 20.0 }, { "type": "text", "bbox": [ 106, 447, 505, 568 ], "lines": [ { "bbox": [ 106, 447, 506, 460 ], "spans": [ { "bbox": [ 106, 447, 506, 460 ], "score": 1.0, "content": "For the configuration of other five baselines: 1) original PINN Raissi et al. (2019); 2) adaptive sam-", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 457, 506, 471 ], "spans": [ { "bbox": [ 105, 457, 506, 471 ], "score": 1.0, "content": "pling L. Wight & Zhao (2021); 3) self-attention McClenny & Braga-Neto (2023); 4) time marching", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 468, 506, 482 ], "spans": [ { "bbox": [ 105, 468, 506, 482 ], "score": 1.0, "content": "Mattey & Ghosh (2022) and 5) causal PINN Wang et al. (2022a), all of them have a neural network", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 480, 505, 492 ], "spans": [ { "bbox": [ 106, 480, 505, 492 ], "score": 1.0, "content": "size with the same width and 1 deeper depth than that in Table 7. The collocation points number", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 490, 505, 504 ], "spans": [ { "bbox": [ 105, 490, 275, 504 ], "score": 1.0, "content": "for all five baselines are configured to be", "type": "text" }, { "bbox": [ 275, 491, 312, 502 ], "score": 0.92, "content": "N _ { t } \\times N _ { r }", "type": "inline_equation" }, { "bbox": [ 312, 490, 505, 504 ], "score": 1.0, "content": "in Table 7. For example, a continuous original", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 501, 506, 514 ], "spans": [ { "bbox": [ 105, 501, 307, 514 ], "score": 1.0, "content": "PINN has size [2, 128, 128, 128, 128, 128, 1] and", "type": "text" }, { "bbox": [ 307, 502, 352, 513 ], "score": 0.9, "content": "2 0 0 \\times 5 1 2", "type": "inline_equation" }, { "bbox": [ 352, 501, 506, 514 ], "score": 1.0, "content": "collocation points on the space-time", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 512, 505, 525 ], "spans": [ { "bbox": [ 105, 512, 505, 525 ], "score": 1.0, "content": "domain to compute the loss, then each discrete PINN has the smaller size [1, 128, 128, 128, 128, 1]", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 524, 505, 536 ], "spans": [ { "bbox": [ 106, 524, 505, 536 ], "score": 1.0, "content": "and much smaller collocation points 512 on space domain. The total parameters and computation", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 534, 505, 547 ], "spans": [ { "bbox": [ 106, 534, 505, 547 ], "score": 1.0, "content": "of 200 discrete PINNs and the computation on the loss calculation are about the same with a single", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 546, 505, 558 ], "spans": [ { "bbox": [ 105, 546, 505, 558 ], "score": 1.0, "content": "continuous PINN. In this configuration, we can sure that the comparison between our TL-DPINNs", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 557, 497, 569 ], "spans": [ { "bbox": [ 106, 557, 497, 569 ], "score": 1.0, "content": "and other five baselines is fair, showing the discrete PINNs are efficient for practical applications.", "type": "text" } ], "index": 33 } ], "index": 28, "bbox_fs": [ 105, 447, 506, 569 ] }, { "type": "title", "bbox": [ 108, 579, 402, 591 ], "lines": [ { "bbox": [ 106, 579, 403, 592 ], "spans": [ { "bbox": [ 106, 579, 403, 592 ], "score": 1.0, "content": "A.4.3 ADDITIONAL RESULTS FOR REACTION-DIFFUSION EQUATION", "type": "text" } ], "index": 34 } ], "index": 34 }, { "type": "text", "bbox": [ 106, 598, 505, 676 ], "lines": [ { "bbox": [ 105, 598, 506, 612 ], "spans": [ { "bbox": [ 105, 598, 220, 612 ], "score": 1.0, "content": "Figure 6 (a) depicts how the", "type": "text" }, { "bbox": [ 221, 598, 233, 609 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 233, 598, 430, 612 ], "score": 1.0, "content": "error changes as time goes on, as we can see, the", "type": "text" }, { "bbox": [ 430, 598, 443, 609 ], "score": 0.88, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 443, 598, 506, 612 ], "score": 1.0, "content": "error increases", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 610, 505, 622 ], "spans": [ { "bbox": [ 105, 610, 357, 622 ], "score": 1.0, "content": "in the early training steps and is kept at a stable level between", "type": "text" }, { "bbox": [ 358, 610, 403, 621 ], "score": 0.89, "content": "1 . 0 0 e \\mathrm { ~ - ~ } 0 5", "type": "inline_equation" }, { "bbox": [ 404, 610, 421, 622 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 421, 610, 468, 621 ], "score": 0.86, "content": "5 . 0 0 e - 0 5", "type": "inline_equation" }, { "bbox": [ 468, 610, 505, 622 ], "score": 1.0, "content": "later. As", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 620, 505, 633 ], "spans": [ { "bbox": [ 105, 620, 505, 633 ], "score": 1.0, "content": "shown in Figure 6 (b), based on the trainable parameters of the preceding time stamp, only a few", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 631, 506, 645 ], "spans": [ { "bbox": [ 105, 631, 506, 645 ], "score": 1.0, "content": "hundred steps of training are required for each time stamp to satisfy the early stopping criterion,", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 643, 506, 655 ], "spans": [ { "bbox": [ 106, 643, 506, 655 ], "score": 1.0, "content": "and then move to the training of the next time stamp. Figure 8 shows the training loss at different", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 654, 506, 666 ], "spans": [ { "bbox": [ 105, 654, 506, 666 ], "score": 1.0, "content": "time steps. Figure 7 compares the predicted and reference solutions at different time instants. The", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 665, 395, 677 ], "spans": [ { "bbox": [ 106, 665, 395, 677 ], "score": 1.0, "content": "predictions given by our method are identical to the reference solutions.", "type": "text" } ], "index": 41 } ], "index": 38, "bbox_fs": [ 105, 598, 506, 677 ] }, { "type": "title", "bbox": [ 107, 689, 357, 700 ], "lines": [ { "bbox": [ 106, 689, 358, 702 ], "spans": [ { "bbox": [ 106, 689, 358, 702 ], "score": 1.0, "content": "A.5 ADDITIONAL RESULTS FOR ALLEN-CAHN EQUATION", "type": "text" } ], "index": 42 } ], "index": 42 }, { "type": "text", "bbox": [ 106, 709, 504, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "Figure 9 shows the predicted solution against the reference solution, our proposed method achieves a", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 719, 505, 733 ], "spans": [ { "bbox": [ 105, 719, 139, 733 ], "score": 1.0, "content": "relative", "type": "text" }, { "bbox": [ 139, 720, 152, 730 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 152, 719, 187, 733 ], "score": 1.0, "content": "error of", "type": "text" }, { "bbox": [ 187, 721, 233, 731 ], "score": 0.83, "content": "5 . 9 2 e \\mathrm { ~ - ~ } 0 4", "type": "inline_equation" }, { "bbox": [ 233, 719, 505, 733 ], "score": 1.0, "content": ". 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As time passes, both the absolute", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 640, 505, 654 ], "spans": [ { "bbox": [ 105, 640, 160, 654 ], "score": 1.0, "content": "error and the", "type": "text" }, { "bbox": [ 160, 641, 173, 651 ], "score": 0.87, "content": "L ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 173, 640, 344, 654 ], "score": 1.0, "content": "error between the reference and predicted", "type": "text" }, { "bbox": [ 344, 641, 383, 653 ], "score": 0.93, "content": "w ( t , x , y )", "type": "inline_equation" }, { "bbox": [ 384, 640, 505, 654 ], "score": 1.0, "content": "increase gradually. 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