diff --git "a/parse/train/blfSjHeFM_e/blfSjHeFM_e_middle.json" "b/parse/train/blfSjHeFM_e/blfSjHeFM_e_middle.json" new file mode 100644--- /dev/null +++ "b/parse/train/blfSjHeFM_e/blfSjHeFM_e_middle.json" @@ -0,0 +1,58008 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 79, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 79, + 504, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 504, + 98 + ], + "score": 1.0, + "content": "MALI: A MEMORY EFFICIENT AND REVERSE ACCU-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 398, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 398, + 117 + ], + "score": 1.0, + "content": "RATE INTEGRATOR FOR NEURAL ODES", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 512, + 168 + ], + "lines": [ + { + "bbox": [ + 111, + 134, + 429, + 148 + ], + "spans": [ + { + "bbox": [ + 111, + 134, + 429, + 148 + ], + "score": 1.0, + "content": "Juntang Zhuang; Nicha C. 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The theory", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "of dynamical systems has been applied to analyze the properties of neural networks or guide the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "design of networks (Weinan, 2017; Ruthotto & Haber, 2019; Lu et al., 2018). 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We validate MALI in various", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 335, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 470, + 347 + ], + "score": 1.0, + "content": "tasks: on image recognition tasks, to our knowledge, MALI is the first to en-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 346, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 346, + 470, + 359 + ], + "score": 1.0, + "content": "able feasible training of a Neural ODE on ImageNet and outperform a well-tuned", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 357, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 469, + 370 + ], + "score": 1.0, + "content": "ResNet, while existing methods fail due to either heavy memory burden or in-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 367, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 367, + 470, + 381 + ], + "score": 1.0, + "content": "accuracy; for time series modeling, MALI significantly outperforms the adjoint", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 142, + 379, + 469, + 391 + ], + "spans": [ + { + "bbox": [ + 142, + 379, + 469, + 391 + ], + "score": 1.0, + "content": "method; and for continuous generative models, MALI achieves new state-of-the-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 390, + 468, + 402 + ], + "spans": [ + { + "bbox": [ + 142, + 390, + 468, + 402 + ], + "score": 1.0, + "content": "art performance.We provide a pypi package: https://jzkay12.github.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 401, + 252, + 413 + ], + "spans": [ + { + "bbox": [ + 142, + 401, + 252, + 413 + ], + "score": 1.0, + "content": "io/TorchDiffEqPack", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14, + "bbox_fs": [ + 141, + 225, + 470, + 413 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 440, + 206, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 439, + 208, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 208, + 456 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "Recent research builds the connection between continuous models and neural networks. The theory", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "of dynamical systems has been applied to analyze the properties of neural networks or guide the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "design of networks (Weinan, 2017; Ruthotto & Haber, 2019; Lu et al., 2018). In these works, a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "residual block (He et al., 2016) is typically viewed as a one-step Euler discretization of an ODE;", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 503, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 503, + 524 + ], + "score": 1.0, + "content": "instead of directly analyzing the discretized neural network, it might be easier to analyze the ODE.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 467, + 506, + 524 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 504, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "Another direction is the neural ordinary differential equation (Neural ODE) (Chen et al., 2018),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "which takes a continuous depth instead of discretized depth. The dynamics of a Neural ODE is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 562 + ], + "score": 1.0, + "content": "typically approximated by numerical integration with adaptive ODE solvers. Neural ODEs have", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "been applied in irregularly sampled time-series (Rubanova et al., 2019), free-form continuous gen-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "erative models (Grathwohl et al., 2018; Finlay et al., 2020), mean-field games (Ruthotto et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "2020), stochastic differential equations (Li et al., 2020) and physically informed modeling (Sanchez-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 594, + 277, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 277, + 606 + ], + "score": 1.0, + "content": "Gonzalez et al., 2019; Zhong et al., 2019).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 528, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "Though the Neural ODE has been widely applied in practice, how to train it is not extensively stud-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "ied. The naive method directly backpropagates through an ODE solver, but tracking a continuous", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "trajectory requires a huge memory. Chen et al. (2018) proposed to use the adjoint method to deter-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 416, + 657 + ], + "score": 1.0, + "content": "mine the gradient in continuous cases, which achieves constant memory cost", + "type": "text" + }, + { + "bbox": [ + 417, + 645, + 435, + 654 + ], + "score": 0.41, + "content": "w . r . t", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "integration time;", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "however, as pointed out by Zhuang et al. (2020), the adjoint method suffers from numerical errors", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "due to the inaccuracy in reverse-time trajectory. Zhuang et al. (2020) proposed the adaptive check-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "point adjoint (ACA) method to achieve accuracy in gradient estimation at a much smaller memory", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "cost compared to the naive method, yet the memory consumption of ACA still grows linearly with", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "integration time. 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With the ALF integrator, each numerical step forward in time is reversible.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "Therefore, with MALI, we delete the trajectory and only keep the end-time states, hence achieve", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "constant memory cost w.r.t integration time; using the reversibility, we can accurately reconstruct", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 497, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 497, + 160 + ], + "score": 1.0, + "content": "the trajectory from the end-time value, hence achieve accuracy in gradient. 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(b) In time-series modeling, MALI achieves comparable or better results", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 260, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 118, + 260, + 505, + 271 + ], + "score": 1.0, + "content": "than other methods. 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Algorithm1:Numerical Integration
Input initial state x,start timeto,end time T,error tolerance etol,initial
stepsize h. Initialize z(O) = x,t = to
Whilet<T error_est=
While error_est>etol h←h×DecayFactor
,error_est=yh(t,z) If error_est<etol h←h×IncreaseFactor
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We provide", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 118, + 190, + 201, + 202 + ], + "spans": [ + { + "bbox": [ + 118, + 190, + 201, + 202 + ], + "score": 1.0, + "content": "theoretical analysis.", + "type": "text" + } + ], + "index": 9, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 203, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 218 + ], + "score": 1.0, + "content": "2. We validate our method with extensive experiments: (a) for image classification tasks, MALI", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 118, + 216, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 118, + 216, + 505, + 227 + ], + "score": 1.0, + "content": "enables a Neural ODE to achieve better accuracy than a well-tuned ResNet with the same number", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 118, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 118, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "of parameters; to our knowledge, MALI is the first method to enable training of Neural ODEs on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 118, + 236, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 118, + 236, + 505, + 251 + ], + "score": 1.0, + "content": "a large-scale dataset such as ImageNet, while existing methods fail due to either heavy memory", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 118, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 118, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "burden or inaccuracy. 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(c) For generative modeling, a FFJORD model trained with MALI achieves", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 117, + 270, + 326, + 282 + ], + "spans": [ + { + "bbox": [ + 117, + 270, + 326, + 282 + ], + "score": 1.0, + "content": "new state-of-the-art results on MNIST and Cifar10.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 167, + 506, + 282 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 297, + 208, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 209, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 209, + 312 + ], + "score": 1.0, + "content": "2 PRELIMINARIES", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 321, + 293, + 333 + ], + "lines": [ + { + "bbox": [ + 106, + 321, + 294, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 294, + 334 + ], + "score": 1.0, + "content": "2.1 NUMERICAL INTEGRATION METHODS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 367, + 354 + ], + "lines": [ + { + "bbox": [ + 106, + 341, + 367, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 367, + 356 + ], + "score": 1.0, + "content": "An ordinary differential equation (ODE) typically takes the form", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 341, + 367, + 356 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 356, + 453, + 380 + ], + "lines": [ + { + "bbox": [ + 157, + 356, + 453, + 380 + ], + "spans": [ + { + "bbox": [ + 157, + 356, + 453, + 380 + ], + "score": 0.91, + "content": "\\frac { \\mathrm { d } z ( t ) } { \\mathrm { d } t } = f _ { \\theta } ( t , z ( t ) ) \\quad s . t . \\quad z ( t _ { 0 } ) = x , t \\in [ t _ { 0 } , T ] , \\quad L o s s = L ( z ( T ) , y )", + "type": "interline_equation", + "image_path": "6a5c4ae84297367035d0efae4d9569c9195552ff7643b5f8b5982f3cc1aaacec.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 157, + 356, + 453, + 380 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 504, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 134, + 396 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 383, + 152, + 395 + ], + "score": 0.91, + "content": "z ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 381, + 318, + 396 + ], + "score": 1.0, + "content": "is the hidden state evolving with time,", + "type": "text" + }, + { + "bbox": [ + 319, + 383, + 328, + 393 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 381, + 398, + 396 + ], + "score": 1.0, + "content": "is the end time,", + "type": "text" + }, + { + "bbox": [ + 399, + 384, + 409, + 394 + ], + "score": 0.87, + "content": "t _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "is the start time (typi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 144, + 406 + ], + "score": 1.0, + "content": "cally 0),", + "type": "text" + }, + { + "bbox": [ + 144, + 396, + 151, + 404 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 393, + 306, + 406 + ], + "score": 1.0, + "content": "is the initial state. 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Eq. 1 is called the initial value problem (IVP) because only", + "type": "text" + }, + { + "bbox": [ + 431, + 417, + 453, + 428 + ], + "score": 0.9, + "content": "z ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 416, + 504, + 428 + ], + "score": 1.0, + "content": "is specified.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 381, + 506, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 316, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 317, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 317, + 451 + ], + "score": 1.0, + "content": "Notations We summarize the notations following", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 448, + 193, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 193, + 461 + ], + "score": 1.0, + "content": "Zhuang et al. 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Algorithm1:Numerical Integration
Input initial state x,start timeto,end time T,error tolerance etol,initial
stepsize h. Initialize z(O) = x,t = to
Whilet<T error_est=
While error_est>etol h←h×DecayFactor
,error_est=yh(t,z) If error_est<etol h←h×IncreaseFactor
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Different methods typically use different", + "type": "text" + }, + { + "bbox": [ + 378, + 654, + 386, + 665 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 654, + 505, + 667 + ], + "score": 1.0, + "content": ", for example different orders", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 663, + 280, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 280, + 678 + ], + "score": 1.0, + "content": "of the Runge-Kutta method (Runge, 1895).", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 587, + 317, + 611 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 676 + ], + "lines": [], + "index": 51.5, + "bbox_fs": [ + 105, + 610, + 505, + 678 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 690, + 372, + 700 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 374, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 374, + 702 + ], + "score": 1.0, + "content": "2.2 ANALYTICAL FORM OF GRADIENT IN CONTINUOUS CASE", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 55 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 502, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 504, + 722 + ], + "score": 1.0, + "content": "We first briefly introduce the analytical form of the gradient in the continuous case, then we compare", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "different numerical implementations in the literature to estimate the gradient. The analytical form", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 56.5, + "bbox_fs": [ + 106, + 708, + 504, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 114, + 510, + 158 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 81, + 505, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 505, + 92 + ], + "score": 1.0, + "content": "Table 1: Comparison between different methods for gradient estimation in continuous case. MALI achieves", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 505, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 235, + 102 + ], + "score": 1.0, + "content": "reverse accuracy, constant memory", + "type": "text" + }, + { + "bbox": [ + 235, + 92, + 252, + 100 + ], + "score": 0.53, + "content": "w . r . t", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 91, + 505, + 102 + ], + "score": 1.0, + "content": "number of solver steps in integration, shallow computation graph and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 101, + 187, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 187, + 112 + ], + "score": 1.0, + "content": "low computation cost.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 114, + 510, + 158 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 114, + 510, + 158 + ], + "spans": [ + { + "bbox": [ + 108, + 114, + 510, + 158 + ], + "score": 0.956, + "html": "
NaiveAdjointACAMALI
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MemoryNzNf ×Nt × mNzNfNz(Nf+Nt)Nz(Nf +1)
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Reverse accuracyX
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Naive,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 321, + 312, + 502, + 324 + ], + "spans": [ + { + "bbox": [ + 321, + 312, + 502, + 324 + ], + "score": 1.0, + "content": "ACA and MALI track the forward-time trajectory,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 321, + 322, + 502, + 333 + ], + "spans": [ + { + "bbox": [ + 321, + 322, + 502, + 333 + ], + "score": 1.0, + "content": "hence are accurate. ACA and MALI only back-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 320, + 332, + 503, + 344 + ], + "spans": [ + { + "bbox": [ + 320, + 332, + 503, + 344 + ], + "score": 1.0, + "content": "propagate through the accepted step, while naive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 321, + 342, + 502, + 354 + ], + "spans": [ + { + "bbox": [ + 321, + 342, + 502, + 354 + ], + "score": 1.0, + "content": "method backpropagates through the search pro-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 322, + 352, + 478, + 363 + ], + "spans": [ + { + "bbox": [ + 322, + 352, + 478, + 363 + ], + "score": 1.0, + "content": "cess hence has deeper computation graphs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 376, + 266, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 375, + 267, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 267, + 388 + ], + "score": 1.0, + "content": "of the gradient in the continuous case is", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 392, + 378, + 420 + ], + "lines": [ + { + "bbox": [ + 233, + 392, + 378, + 420 + ], + "spans": [ + { + "bbox": [ + 233, + 392, + 378, + 420 + ], + "score": 0.95, + "content": "\\frac { \\mathrm { d } L } { \\mathrm { d } \\theta } = - \\int _ { T } ^ { 0 } \\boldsymbol { a } ( t ) ^ { \\top } \\frac { \\partial f ( \\boldsymbol { z } ( t ) , t , \\theta ) } { \\partial \\theta } d t", + "type": "interline_equation", + "image_path": "09a861031a46726205354e1ae7c2c32061e5653fc732ddda7398f8c2ad014a38.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 233, + 392, + 378, + 406.0 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 233, + 406.0, + 378, + 420.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 424, + 442, + 452 + ], + "lines": [ + { + "bbox": [ + 170, + 424, + 442, + 452 + ], + "spans": [ + { + "bbox": [ + 170, + 424, + 442, + 452 + ], + "score": 0.93, + "content": "\\frac { \\mathrm { d } a ( t ) } { d t } + \\bigg ( \\frac { \\partial f ( z ( t ) , t , \\theta ) } { \\partial z ( t ) } \\bigg ) ^ { \\top } a ( t ) = 0 \\ \\forall t \\in ( 0 , T ) , a ( T ) = \\frac { \\partial L } { \\partial z ( T ) }", + "type": "interline_equation", + "image_path": "dbf61bf404c8a76ac14a5d9008ebd8e992523ad71c90019c2085435d9df50182.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 170, + 424, + 442, + 433.3333333333333 + ], + "spans": [], + "index": 41 + }, + { + "bbox": [ + 170, + 433.3333333333333, + 442, + 442.66666666666663 + ], + "spans": [], + "index": 42 + }, + { + "bbox": [ + 170, + 442.66666666666663, + 442, + 451.99999999999994 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 506, + 477 + ], + "lines": [ + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 133, + 466 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 454, + 151, + 466 + ], + "score": 0.92, + "content": "a ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "is the “adjoint state”. Detailed proof is given in (Pontryagin, 1962). In the next section", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 465, + 398, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 398, + 477 + ], + "score": 1.0, + "content": "we compare different numerical implementations of this analytical form.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + }, + { + "type": "title", + "bbox": [ + 106, + 490, + 481, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 481, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 481, + 502 + ], + "score": 1.0, + "content": "2.3 NUMERICAL IMPLEMENTATIONS IN THE LITERATURE FOR THE ANALYTICAL FORM", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 522 + ], + "score": 1.0, + "content": "We compare different numerical implementations of the analytical form in this section. The forward-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 522, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 504, + 533 + ], + "score": 1.0, + "content": "pass and backward-pass of different methods are demonstrated in Fig. 1 and Fig. 2 respectively.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "Forward-pass is similar for different methods. The comparison of backward-pass among different", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 544, + 431, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 431, + 555 + ], + "score": 1.0, + "content": "methods are summarized in Table. 1. We explain methods in the literature below.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5 + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "Naive method The naive method saves all of the computation graph (including search for optimal", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "stepsize, green curve in Fig. 2) in memory, and backpropagates through it. Hence the memory cost is", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 107, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 582, + 175, + 595 + ], + "score": 0.92, + "content": "N _ { z } N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 582, + 319, + 595 + ], + "score": 1.0, + "content": "and depth of computation graph are", + "type": "text" + }, + { + "bbox": [ + 319, + 583, + 375, + 595 + ], + "score": 0.92, + "content": "N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 582, + 506, + 595 + ], + "score": 1.0, + "content": ", and the computation is doubled", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "considering both forward and backward passes. Besides the large memory and computation, the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 603, + 476, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 476, + 618 + ], + "score": 1.0, + "content": "deep computation graph might cause vanishing or exploding gradient (Pascanu et al., 2013).", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "Adjoint method Note that we use “adjoint state equation” to refer to the analytical form in Eq. 2", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "and 3, while we use “adjoint method” to refer to the numerical implementation by Chen et al. (2018).", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "As in Fig. 1 and 2, the adjoint method forgets forward-time trajectory (blue curve) to achieve", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 161, + 667 + ], + "score": 1.0, + "content": "memory cost", + "type": "text" + }, + { + "bbox": [ + 162, + 654, + 188, + 666 + ], + "score": 0.91, + "content": "N _ { z } N _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "which is constant to integration time; it takes the end-time state (derived from", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 666, + 501, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 501, + 677 + ], + "score": 1.0, + "content": "forward-time integration) as the initial state, and solves a separate IVP (red curve) in reverse-time.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 58 + }, + { + "type": "text", + "bbox": [ + 106, + 679, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 101, + 677, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 101, + 684, + 360, + 712 + ], + "score": 1.0, + "content": "the reconstructed initial value by the adjoint method is", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 429, + 693 + ], + "score": 1.0, + "content": "Theorem 2.1. (Zhuang et al., 2020) For an ODE solver of order", + "type": "text" + }, + { + "bbox": [ + 361, + 691, + 505, + 706 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { k = 0 } ^ { N - 1 } \\left[ h _ { k } ^ { p + 1 } D \\Phi _ { t _ { k } } ^ { T } ( z _ { k } ) l ( t _ { k } , z _ { k } ) \\right. + } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 682, + 436, + 691 + ], + "score": 0.47, + "content": "p _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 677, + 507, + 693 + ], + "score": 1.0, + "content": ", the error of", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 704, + 507, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 277, + 719 + ], + "score": 0.91, + "content": "( - h _ { k } ) ^ { p + 1 } D \\Phi _ { T } ^ { t _ { k } } ( { \\overline { { z _ { k } } } } ) \\overline { { l ( t _ { k } , { \\overline { { z _ { k } } } } ) } } ] + O ( h ^ { p + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 704, + 309, + 720 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 310, + 707, + 319, + 717 + ], + "score": 0.79, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 704, + 409, + 720 + ], + "score": 1.0, + "content": "is the ideal solution,", + "type": "text" + }, + { + "bbox": [ + 409, + 707, + 426, + 717 + ], + "score": 0.73, + "content": "D \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 704, + 507, + 720 + ], + "score": 1.0, + "content": "is the Jacobian of", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 107, + 718, + 496, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 720, + 144, + 732 + ], + "score": 0.8, + "content": "\\Phi , l ( t , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 718, + 163, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 720, + 187, + 732 + ], + "score": 0.91, + "content": "\\overline { { l ( t , z ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 718, + 496, + 734 + ], + "score": 1.0, + "content": "are the local error in forward-time and reverse-time integration respectively.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 62 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "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": "table", + "bbox": [ + 108, + 114, + 510, + 158 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 81, + 505, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 80, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 505, + 92 + ], + "score": 1.0, + "content": "Table 1: Comparison between different methods for gradient estimation in continuous case. 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NaiveAdjointACAMALI
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MemoryNzNf ×Nt × mNzNfNz(Nf+Nt)Nz(Nf +1)
Computation graph depthNf×Nt × mNf × NrNf×NtNf×Nt
Reverse accuracyX
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Detailed proof is given in (Pontryagin, 1962). In the next section", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 465, + 398, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 398, + 477 + ], + "score": 1.0, + "content": "we compare different numerical implementations of this analytical form.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 454, + 505, + 477 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 490, + 481, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 481, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 481, + 502 + ], + "score": 1.0, + "content": "2.3 NUMERICAL IMPLEMENTATIONS IN THE LITERATURE FOR THE ANALYTICAL FORM", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 504, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 504, + 522 + ], + "score": 1.0, + "content": "We compare different numerical implementations of the analytical form in this section. The forward-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 522, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 504, + 533 + ], + "score": 1.0, + "content": "pass and backward-pass of different methods are demonstrated in Fig. 1 and Fig. 2 respectively.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "Forward-pass is similar for different methods. The comparison of backward-pass among different", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 544, + 431, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 431, + 555 + ], + "score": 1.0, + "content": "methods are summarized in Table. 1. We explain methods in the literature below.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 510, + 505, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "Naive method The naive method saves all of the computation graph (including search for optimal", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "stepsize, green curve in Fig. 2) in memory, and backpropagates through it. Hence the memory cost is", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 107, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 107, + 582, + 175, + 595 + ], + "score": 0.92, + "content": "N _ { z } N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 582, + 319, + 595 + ], + "score": 1.0, + "content": "and depth of computation graph are", + "type": "text" + }, + { + "bbox": [ + 319, + 583, + 375, + 595 + ], + "score": 0.92, + "content": "N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 582, + 506, + 595 + ], + "score": 1.0, + "content": ", and the computation is doubled", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 606 + ], + "score": 1.0, + "content": "considering both forward and backward passes. Besides the large memory and computation, the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 603, + 476, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 476, + 618 + ], + "score": 1.0, + "content": "deep computation graph might cause vanishing or exploding gradient (Pascanu et al., 2013).", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 53, + "bbox_fs": [ + 105, + 559, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "Adjoint method Note that we use “adjoint state equation” to refer to the analytical form in Eq. 2", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "and 3, while we use “adjoint method” to refer to the numerical implementation by Chen et al. (2018).", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 106, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "As in Fig. 1 and 2, the adjoint method forgets forward-time trajectory (blue curve) to achieve", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 161, + 667 + ], + "score": 1.0, + "content": "memory cost", + "type": "text" + }, + { + "bbox": [ + 162, + 654, + 188, + 666 + ], + "score": 0.91, + "content": "N _ { z } N _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "which is constant to integration time; it takes the end-time state (derived from", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 666, + 501, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 501, + 677 + ], + "score": 1.0, + "content": "forward-time integration) as the initial state, and solves a separate IVP (red curve) in reverse-time.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 58, + "bbox_fs": [ + 105, + 621, + 505, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 679, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 101, + 677, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 101, + 684, + 360, + 712 + ], + "score": 1.0, + "content": "the reconstructed initial value by the adjoint method is", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 429, + 693 + ], + "score": 1.0, + "content": "Theorem 2.1. (Zhuang et al., 2020) For an ODE solver of order", + "type": "text" + }, + { + "bbox": [ + 361, + 691, + 505, + 706 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { k = 0 } ^ { N - 1 } \\left[ h _ { k } ^ { p + 1 } D \\Phi _ { t _ { k } } ^ { T } ( z _ { k } ) l ( t _ { k } , z _ { k } ) \\right. + } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 682, + 436, + 691 + ], + "score": 0.47, + "content": "p _ { ; }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 677, + 507, + 693 + ], + "score": 1.0, + "content": ", the error of", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 704, + 507, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 705, + 277, + 719 + ], + "score": 0.91, + "content": "( - h _ { k } ) ^ { p + 1 } D \\Phi _ { T } ^ { t _ { k } } ( { \\overline { { z _ { k } } } } ) \\overline { { l ( t _ { k } , { \\overline { { z _ { k } } } } ) } } ] + O ( h ^ { p + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 704, + 309, + 720 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 310, + 707, + 319, + 717 + ], + "score": 0.79, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 704, + 409, + 720 + ], + "score": 1.0, + "content": "is the ideal solution,", + "type": "text" + }, + { + "bbox": [ + 409, + 707, + 426, + 717 + ], + "score": 0.73, + "content": "D \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 704, + 507, + 720 + ], + "score": 1.0, + "content": "is the Jacobian of", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 107, + 718, + 496, + 734 + ], + "spans": [ + { + "bbox": [ + 107, + 720, + 144, + 732 + ], + "score": 0.8, + "content": "\\Phi , l ( t , z )", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 718, + 163, + 734 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 720, + 187, + 732 + ], + "score": 0.91, + "content": "\\overline { { l ( t , z ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 718, + 496, + 734 + ], + "score": 1.0, + "content": "are the local error in forward-time and reverse-time integration respectively.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 62, + "bbox_fs": [ + 101, + 677, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Theorem 2.1 is stated as Theorem 3.2 in Zhuang et al. (2020); please see reference paper for de-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "tailed proof. To summarize, due to inevitable errors with numerical ODE solvers, the reverse-time", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 194, + 118 + ], + "score": 1.0, + "content": "trajectory (red curve,", + "type": "text" + }, + { + "bbox": [ + 194, + 104, + 213, + 117 + ], + "score": 0.89, + "content": "\\overline { { z } } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 104, + 437, + 118 + ], + "score": 1.0, + "content": ") cannot match the forward-time trajectory (blue curve,", + "type": "text" + }, + { + "bbox": [ + 437, + 104, + 455, + 117 + ], + "score": 0.86, + "content": "z ( t ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 104, + 506, + 118 + ], + "score": 1.0, + "content": ") accurately.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 474, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 156, + 133 + ], + "score": 1.0, + "content": "The error in", + "type": "text" + }, + { + "bbox": [ + 156, + 118, + 163, + 127 + ], + "score": 0.75, + "content": "\\overline { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 113, + 220, + 133 + ], + "score": 1.0, + "content": "propagates to", + "type": "text" + }, + { + "bbox": [ + 221, + 115, + 233, + 130 + ], + "score": 0.9, + "content": "\\textstyle { \\frac { \\mathrm { d } L } { \\mathrm { d } \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 113, + 474, + 133 + ], + "score": 1.0, + "content": "by Eq. 2, hence affects the accuracy in gradient estimation.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "Adaptive checkpoint adjoint (ACA) To solve the inaccuracy of adjoint method, Zhuang et al.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 145, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 505, + 157 + ], + "score": 1.0, + "content": "(2020) proposed ACA: ACA stores forward-time trajectory in memory for backward-pass, hence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "guarantees accuracy; ACA deletes the search process (green curve in Fig. 2), and only back-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "propagates through the accepted step (blue curve in Fig. 2), hence has a shallower computation graph", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 108, + 176, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 108, + 179, + 148, + 191 + ], + "score": 0.9, + "content": "( N _ { f } \\times N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 176, + 199, + 195 + ], + "score": 1.0, + "content": "for ACA vs", + "type": "text" + }, + { + "bbox": [ + 200, + 179, + 259, + 191 + ], + "score": 0.91, + "content": "N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 176, + 409, + 195 + ], + "score": 1.0, + "content": "for naive method). ACA only stores", + "type": "text" + }, + { + "bbox": [ + 410, + 177, + 453, + 191 + ], + "score": 0.93, + "content": "\\{ z ( t _ { i } ) \\} _ { i = 1 } ^ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 176, + 507, + 195 + ], + "score": 1.0, + "content": ", and deletes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 102, + 187, + 508, + 210 + ], + "spans": [ + { + "bbox": [ + 102, + 187, + 214, + 210 + ], + "score": 1.0, + "content": "the computation graph for", + "type": "text" + }, + { + "bbox": [ + 214, + 190, + 284, + 205 + ], + "score": 0.93, + "content": "\\{ f ( z ( t _ { i } ) , t _ { i } ) \\} _ { i = 1 } ^ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 187, + 393, + 210 + ], + "score": 1.0, + "content": ", hence the memory cost is", + "type": "text" + }, + { + "bbox": [ + 393, + 192, + 451, + 204 + ], + "score": 0.9, + "content": "N _ { z } ( N _ { f } + N _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 187, + 508, + 210 + ], + "score": 1.0, + "content": ". Though the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 202, + 410, + 215 + ], + "score": 1.0, + "content": "memory cost is much smaller than the naive method, it grows linearly with", + "type": "text" + }, + { + "bbox": [ + 410, + 204, + 423, + 214 + ], + "score": 0.87, + "content": "N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 202, + 506, + 215 + ], + "score": 1.0, + "content": ", and can not handle", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "very high dimensional models. In the following sections, we propose a method to overcome all these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 225, + 248, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 248, + 237 + ], + "score": 1.0, + "content": "disadvantages of existing methods.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 254, + 178, + 267 + ], + "lines": [ + { + "bbox": [ + 104, + 252, + 181, + 269 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 181, + 269 + ], + "score": 1.0, + "content": "3 METHODS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 280, + 311, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 312, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 312, + 293 + ], + "score": 1.0, + "content": "3.1 ASYNCHRONOUS LEAPFROG INTEGRATOR", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "In this section we give a brief introduction to the asynchronous leapfrog (ALF) method (Mutze,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "2013), and we provide theoretical analysis which is missing in Mutze (2013). For general first-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 322, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 284, + 337 + ], + "score": 1.0, + "content": "order ODEs in the form of Eq. 1, the tuple", + "type": "text" + }, + { + "bbox": [ + 284, + 323, + 305, + 335 + ], + "score": 0.9, + "content": "( z , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 322, + 506, + 337 + ], + "score": 1.0, + "content": "is sufficient for most ODE solvers to take a step", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 279, + 347 + ], + "score": 1.0, + "content": "numerically. For ALF, the required tuple is", + "type": "text" + }, + { + "bbox": [ + 279, + 335, + 310, + 346 + ], + "score": 0.91, + "content": "( z , v , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 334, + 340, + 347 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 341, + 336, + 347, + 344 + ], + "score": 0.72, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "is the “approximated derivative”. Most", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 439, + 357 + ], + "score": 1.0, + "content": "numerical ODE solvers such as the Runge-Kutta method (Runge, 1895) track state", + "type": "text" + }, + { + "bbox": [ + 440, + 347, + 447, + 355 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "evolving with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 302, + 369 + ], + "score": 1.0, + "content": "time, while ALF tracks the “augmented state”", + "type": "text" + }, + { + "bbox": [ + 302, + 356, + 325, + 368 + ], + "score": 0.92, + "content": "( z , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 356, + 505, + 369 + ], + "score": 1.0, + "content": ". We explain the details of ALF as below.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 373, + 250, + 384 + ], + "lines": [ + { + "bbox": [ + 107, + 372, + 252, + 386 + ], + "spans": [ + { + "bbox": [ + 107, + 372, + 210, + 386 + ], + "score": 1.0, + "content": "Algorithm 2: Forward of", + "type": "text" + }, + { + "bbox": [ + 211, + 374, + 219, + 384 + ], + "score": 0.77, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 372, + 252, + 386 + ], + "score": 1.0, + "content": "in ALF", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 105, + 382, + 297, + 500 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 105, + 382, + 297, + 500 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 382, + 297, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 297, + 500 + ], + "score": 0.473, + "html": "
Input (zin, Uin, Sin,h) where Sin is current time, Zin and Uin are correponding values at time Sin,h is stepsize.
Forward S1 = Sin +h/2
k1 = Zin + Uin × h/2
u1 = f(k1,S1)
Uout = Uin + 2(u1 - Uin)
Zout = k1 + Uout × h/2
Sout = S1 +h/2 Output (Zout,Vout, Sout,h)
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Input (zout, Uout, Sout,h) where Sout is current time, Zout and vout are corresponding values at Sout,h is stepsize. Inverse S1 = Sout -h/2 k1 = Zout - Uout X h/2 u1 = f(k1,S1)
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As in", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 353, + 691 + ], + "score": 1.0, + "content": "Fig. 3, we can reconstruct the entire trajectory given the state", + "type": "text" + }, + { + "bbox": [ + 353, + 677, + 384, + 689 + ], + "score": 0.91, + "content": "( z _ { j } , v _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 676, + 414, + 691 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 415, + 678, + 424, + 690 + ], + "score": 0.85, + "content": "t _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 676, + 506, + 691 + ], + "score": 1.0, + "content": ", and the discretized", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 688, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 156, + 705 + ], + "score": 1.0, + "content": "time points", + "type": "text" + }, + { + "bbox": [ + 157, + 690, + 202, + 702 + ], + "score": 0.93, + "content": "\\left\\{ { t } _ { 0 } , . . . t _ { N _ { t } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 688, + 295, + 705 + ], + "score": 1.0, + "content": ". 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(2020); please see reference paper for de-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "tailed proof. To summarize, due to inevitable errors with numerical ODE solvers, the reverse-time", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 194, + 118 + ], + "score": 1.0, + "content": "trajectory (red curve,", + "type": "text" + }, + { + "bbox": [ + 194, + 104, + 213, + 117 + ], + "score": 0.89, + "content": "\\overline { { z } } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 104, + 437, + 118 + ], + "score": 1.0, + "content": ") cannot match the forward-time trajectory (blue curve,", + "type": "text" + }, + { + "bbox": [ + 437, + 104, + 455, + 117 + ], + "score": 0.86, + "content": "z ( t ) .", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 104, + 506, + 118 + ], + "score": 1.0, + "content": ") accurately.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 474, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 156, + 133 + ], + "score": 1.0, + "content": "The error in", + "type": "text" + }, + { + "bbox": [ + 156, + 118, + 163, + 127 + ], + "score": 0.75, + "content": "\\overline { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 113, + 220, + 133 + ], + "score": 1.0, + "content": "propagates to", + "type": "text" + }, + { + "bbox": [ + 221, + 115, + 233, + 130 + ], + "score": 0.9, + "content": "\\textstyle { \\frac { \\mathrm { d } L } { \\mathrm { d } \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 113, + 474, + 133 + ], + "score": 1.0, + "content": "by Eq. 2, hence affects the accuracy in gradient estimation.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 81, + 506, + 133 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 146 + ], + "score": 1.0, + "content": "Adaptive checkpoint adjoint (ACA) To solve the inaccuracy of adjoint method, Zhuang et al.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 145, + 505, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 145, + 505, + 157 + ], + "score": 1.0, + "content": "(2020) proposed ACA: ACA stores forward-time trajectory in memory for backward-pass, hence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 155, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 104, + 155, + 505, + 168 + ], + "score": 1.0, + "content": "guarantees accuracy; ACA deletes the search process (green curve in Fig. 2), and only back-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 166, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 104, + 166, + 505, + 180 + ], + "score": 1.0, + "content": "propagates through the accepted step (blue curve in Fig. 2), hence has a shallower computation graph", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 108, + 176, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 108, + 179, + 148, + 191 + ], + "score": 0.9, + "content": "( N _ { f } \\times N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 176, + 199, + 195 + ], + "score": 1.0, + "content": "for ACA vs", + "type": "text" + }, + { + "bbox": [ + 200, + 179, + 259, + 191 + ], + "score": 0.91, + "content": "N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 176, + 409, + 195 + ], + "score": 1.0, + "content": "for naive method). ACA only stores", + "type": "text" + }, + { + "bbox": [ + 410, + 177, + 453, + 191 + ], + "score": 0.93, + "content": "\\{ z ( t _ { i } ) \\} _ { i = 1 } ^ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 176, + 507, + 195 + ], + "score": 1.0, + "content": ", and deletes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 102, + 187, + 508, + 210 + ], + "spans": [ + { + "bbox": [ + 102, + 187, + 214, + 210 + ], + "score": 1.0, + "content": "the computation graph for", + "type": "text" + }, + { + "bbox": [ + 214, + 190, + 284, + 205 + ], + "score": 0.93, + "content": "\\{ f ( z ( t _ { i } ) , t _ { i } ) \\} _ { i = 1 } ^ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 187, + 393, + 210 + ], + "score": 1.0, + "content": ", hence the memory cost is", + "type": "text" + }, + { + "bbox": [ + 393, + 192, + 451, + 204 + ], + "score": 0.9, + "content": "N _ { z } ( N _ { f } + N _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 187, + 508, + 210 + ], + "score": 1.0, + "content": ". Though the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 202, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 202, + 410, + 215 + ], + "score": 1.0, + "content": "memory cost is much smaller than the naive method, it grows linearly with", + "type": "text" + }, + { + "bbox": [ + 410, + 204, + 423, + 214 + ], + "score": 0.87, + "content": "N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 202, + 506, + 215 + ], + "score": 1.0, + "content": ", and can not handle", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "very high dimensional models. In the following sections, we propose a method to overcome all these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 225, + 248, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 248, + 237 + ], + "score": 1.0, + "content": "disadvantages of existing methods.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8, + "bbox_fs": [ + 102, + 132, + 508, + 237 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 254, + 178, + 267 + ], + "lines": [ + { + "bbox": [ + 104, + 252, + 181, + 269 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 181, + 269 + ], + "score": 1.0, + "content": "3 METHODS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 280, + 311, + 291 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 312, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 312, + 293 + ], + "score": 1.0, + "content": "3.1 ASYNCHRONOUS LEAPFROG INTEGRATOR", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 368 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "In this section we give a brief introduction to the asynchronous leapfrog (ALF) method (Mutze,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "2013), and we provide theoretical analysis which is missing in Mutze (2013). For general first-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 322, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 284, + 337 + ], + "score": 1.0, + "content": "order ODEs in the form of Eq. 1, the tuple", + "type": "text" + }, + { + "bbox": [ + 284, + 323, + 305, + 335 + ], + "score": 0.9, + "content": "( z , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 322, + 506, + 337 + ], + "score": 1.0, + "content": "is sufficient for most ODE solvers to take a step", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 279, + 347 + ], + "score": 1.0, + "content": "numerically. For ALF, the required tuple is", + "type": "text" + }, + { + "bbox": [ + 279, + 335, + 310, + 346 + ], + "score": 0.91, + "content": "( z , v , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 334, + 340, + 347 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 341, + 336, + 347, + 344 + ], + "score": 0.72, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "is the “approximated derivative”. Most", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 439, + 357 + ], + "score": 1.0, + "content": "numerical ODE solvers such as the Runge-Kutta method (Runge, 1895) track state", + "type": "text" + }, + { + "bbox": [ + 440, + 347, + 447, + 355 + ], + "score": 0.75, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "evolving with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 302, + 369 + ], + "score": 1.0, + "content": "time, while ALF tracks the “augmented state”", + "type": "text" + }, + { + "bbox": [ + 302, + 356, + 325, + 368 + ], + "score": 0.92, + "content": "( z , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 356, + 505, + 369 + ], + "score": 1.0, + "content": ". We explain the details of ALF as below.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 300, + 506, + 369 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 373, + 250, + 384 + ], + "lines": [ + { + "bbox": [ + 107, + 372, + 252, + 386 + ], + "spans": [ + { + "bbox": [ + 107, + 372, + 210, + 386 + ], + "score": 1.0, + "content": "Algorithm 2: Forward of", + "type": "text" + }, + { + "bbox": [ + 211, + 374, + 219, + 384 + ], + "score": 0.77, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 372, + 252, + 386 + ], + "score": 1.0, + "content": "in ALF", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "table", + "bbox": [ + 105, + 382, + 297, + 500 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 105, + 382, + 297, + 500 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 382, + 297, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 297, + 500 + ], + "score": 0.473, + "html": "
Input (zin, Uin, Sin,h) where Sin is current time, Zin and Uin are correponding values at time Sin,h is stepsize.
Forward S1 = Sin +h/2
k1 = Zin + Uin × h/2
u1 = f(k1,S1)
Uout = Uin + 2(u1 - Uin)
Zout = k1 + Uout × h/2
Sout = S1 +h/2 Output (Zout,Vout, Sout,h)
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Input (zout, Uout, Sout,h) where Sout is current time, Zout and vout are corresponding values at Sout,h is stepsize. Inverse S1 = Sout -h/2 k1 = Zout - Uout X h/2 u1 = f(k1,S1)
", + "type": "table", + "image_path": "99846493f981a74f718160a7953067984dc07282ad40d1f977ccd473b5d53e33.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 312, + 387, + 494, + 401.125 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 312, + 401.125, + 494, + 415.25 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 312, + 415.25, + 494, + 429.375 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 312, + 429.375, + 494, + 443.5 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 312, + 443.5, + 494, + 457.625 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 312, + 457.625, + 494, + 471.75 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 312, + 471.75, + 494, + 485.875 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 312, + 485.875, + 494, + 500.0 + ], + "spans": [], + "index": 38 + } + ] + } + ], + "index": 32.25 + }, + { + "type": "text", + "bbox": [ + 106, + 508, + 324, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 508, + 324, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 324, + 520 + ], + "score": 1.0, + "content": "Procedure of ALF Different ODE solvers have dif-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 519, + 325, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 133, + 532 + ], + "score": 1.0, + "content": "ferent", + "type": "text" + }, + { + "bbox": [ + 133, + 520, + 141, + 531 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 519, + 300, + 532 + ], + "score": 1.0, + "content": "in Algo. 1, hence we only summarize", + "type": "text" + }, + { + "bbox": [ + 300, + 520, + 309, + 531 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 519, + 325, + 532 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 529, + 325, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 325, + 542 + ], + "score": 1.0, + "content": "ALF in Algo. 2. Note that for a complete algorithm", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 541, + 325, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 325, + 553 + ], + "score": 1.0, + "content": "of integration for ALF, we need to plug Algo. 2 into", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 551, + 325, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 325, + 564 + ], + "score": 1.0, + "content": "Algo. 1. The forward-pass is summarized in Algo. 2.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 562, + 326, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 169, + 577 + ], + "score": 1.0, + "content": "Given stepsize", + "type": "text" + }, + { + "bbox": [ + 169, + 564, + 176, + 574 + ], + "score": 0.76, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 562, + 228, + 577 + ], + "score": 1.0, + "content": ", with input", + "type": "text" + }, + { + "bbox": [ + 228, + 564, + 284, + 576 + ], + "score": 0.91, + "content": "( z _ { i n } , v _ { i n } , s _ { i n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 562, + 326, + 577 + ], + "score": 1.0, + "content": ", a single", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 573, + 262, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 189, + 588 + ], + "score": 1.0, + "content": "step of ALF outputs", + "type": "text" + }, + { + "bbox": [ + 190, + 575, + 256, + 586 + ], + "score": 0.87, + "content": "( z _ { o u t } , v _ { o u t } , s _ { o u t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 573, + 262, + 588 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 508, + 326, + 588 + ] + }, + { + "type": "image", + "bbox": [ + 332, + 520, + 503, + 569 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 332, + 520, + 503, + 569 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 332, + 520, + 503, + 569 + ], + "spans": [ + { + "bbox": [ + 332, + 520, + 503, + 569 + ], + "score": 0.955, + "type": "image", + "image_path": "da4dde8b53890c9e783c1ad01b8224fb995c8dfef6b9f8608c5ca38f19fdbd37.jpg" + } + ] + } + ], + "index": 47, + "virtual_lines": [ + { + "bbox": [ + 332, + 520, + 503, + 536.3333333333334 + ], + "spans": [], + "index": 46 + }, + { + "bbox": [ + 332, + 536.3333333333334, + 503, + 552.6666666666667 + ], + "spans": [], + "index": 47 + }, + { + "bbox": [ + 332, + 552.6666666666667, + 503, + 569.0000000000001 + ], + "spans": [], + "index": 48 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 332, + 572, + 505, + 614 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 333, + 570, + 501, + 604 + ], + "spans": [ + { + "bbox": [ + 333, + 583, + 372, + 594 + ], + "score": 0.92, + "content": "( z _ { j } , v _ { j } , t _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 570, + 472, + 604 + ], + "score": 1.0, + "content": "With ALF method, given and discretized time points", + "type": "text" + }, + { + "bbox": [ + 473, + 582, + 501, + 594 + ], + "score": 0.91, + "content": "\\{ t _ { i } \\} _ { i = 1 } ^ { N _ { t } }", + "type": "inline_equation" + } + ], + "index": 49 + }, + { + "bbox": [ + 331, + 603, + 468, + 614 + ], + "spans": [ + { + "bbox": [ + 331, + 603, + 468, + 614 + ], + "score": 1.0, + "content": "rately due to the reversibility of ALF.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5 + } + ], + "index": 48.75 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 323, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 590, + 324, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 183, + 604 + ], + "score": 1.0, + "content": "As in Fig. 3, given", + "type": "text" + }, + { + "bbox": [ + 183, + 591, + 226, + 603 + ], + "score": 0.92, + "content": "( z _ { 0 } , v _ { 0 } , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 590, + 324, + 604 + ], + "score": 1.0, + "content": ", the numerical forward-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 602, + 271, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 271, + 615 + ], + "score": 1.0, + "content": "time integration calls Algo. 2 iteratively:", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 590, + 324, + 615 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 135, + 620, + 296, + 650 + ], + "lines": [ + { + "bbox": [ + 135, + 620, + 296, + 650 + ], + "spans": [ + { + "bbox": [ + 135, + 620, + 296, + 650 + ], + "score": 0.89, + "content": "\\begin{array} { r } { ( z _ { i } , v _ { i } , t _ { i } , h _ { i } ) = \\psi ( z _ { i - 1 } , v _ { i - 1 } , t _ { i - 1 } , h _ { i } ) } \\\\ { s . t . \\ h _ { i } = t _ { i } - t _ { i - 1 } , \\ i = 1 , 2 , . . . N _ { t } } \\end{array}", + "type": "interline_equation", + "image_path": "92f888d797f9370c77e60493e4445e7d8161e37ec7ac441ffda3bb432ee24bf2.jpg" + } + ] + } + ], + "index": 53.5, + "virtual_lines": [ + { + "bbox": [ + 135, + 620, + 296, + 635.0 + ], + "spans": [], + "index": 53 + }, + { + "bbox": [ + 135, + 635.0, + 296, + 650.0 + ], + "spans": [], + "index": 54 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 655, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 343, + 667 + ], + "score": 1.0, + "content": "Invertibility of ALF An interesting property of ALF is that", + "type": "text" + }, + { + "bbox": [ + 343, + 656, + 351, + 666 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "defines a bijective mapping; therefore,", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 186, + 680 + ], + "score": 1.0, + "content": "we can reconstruct", + "type": "text" + }, + { + "bbox": [ + 187, + 666, + 253, + 678 + ], + "score": 0.92, + "content": "( z _ { i n } , v _ { i n } , s _ { i n } , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 665, + 279, + 680 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 279, + 666, + 356, + 678 + ], + "score": 0.88, + "content": "( z _ { o u t } , v _ { o u t } , s _ { o u t } , h )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 665, + 506, + 680 + ], + "score": 1.0, + "content": ", as demonstrated in Algo. 7. As in", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 105, + 676, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 353, + 691 + ], + "score": 1.0, + "content": "Fig. 3, we can reconstruct the entire trajectory given the state", + "type": "text" + }, + { + "bbox": [ + 353, + 677, + 384, + 689 + ], + "score": 0.91, + "content": "( z _ { j } , v _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 676, + 414, + 691 + ], + "score": 1.0, + "content": "at time", + "type": "text" + }, + { + "bbox": [ + 415, + 678, + 424, + 690 + ], + "score": 0.85, + "content": "t _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 676, + 506, + 691 + ], + "score": 1.0, + "content": ", and the discretized", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 688, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 156, + 705 + ], + "score": 1.0, + "content": "time points", + "type": "text" + }, + { + "bbox": [ + 157, + 690, + 202, + 702 + ], + "score": 0.93, + "content": "\\left\\{ { t } _ { 0 } , . . . t _ { N _ { t } } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 688, + 295, + 705 + ], + "score": 1.0, + "content": ". For example, given", + "type": "text" + }, + { + "bbox": [ + 295, + 690, + 337, + 703 + ], + "score": 0.92, + "content": "\\left( z _ { N _ { t } } , v _ { N _ { t } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 688, + 359, + 705 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 359, + 689, + 390, + 703 + ], + "score": 0.91, + "content": "\\{ t _ { i } \\} _ { i = 0 } ^ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 688, + 506, + 705 + ], + "score": 1.0, + "content": ", the trajectory for Eq. 4 is", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 701, + 165, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 165, + 712 + ], + "score": 1.0, + "content": "reconstructed:", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 57, + "bbox_fs": [ + 105, + 655, + 506, + 712 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 718, + 478, + 733 + ], + "lines": [ + { + "bbox": [ + 133, + 718, + 478, + 733 + ], + "spans": [ + { + "bbox": [ + 133, + 718, + 478, + 733 + ], + "score": 0.86, + "content": "( z _ { i - 1 } , v _ { i - 1 } , t _ { i - 1 } , h _ { i } ) = \\psi ^ { - 1 } ( z _ { i } , v _ { i } , t _ { i } , h _ { i } ) { \\ s . t . \\ h } _ { i } = t _ { i } - t _ { i - 1 } , { \\ i = N _ { t } , N _ { t } - 1 , . . . , 1 }", + "type": "interline_equation", + "image_path": "0835135a6b9bbe11f99000f66eb0143f1f94c46c1f21146b15c67c533316da64.jpg" + } + ] + } + ], + "index": 60, + "virtual_lines": [ + { + "bbox": [ + 133, + 718, + 478, + 733 + ], + "spans": [], + "index": 60 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "In the following sections, we will show the invertibility of ALF is the key to maintain accuracy at a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "constant memory cost to train Neural ODEs. Note that “inverse” refers to reconstructing the input", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 478, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 478, + 117 + ], + "score": 1.0, + "content": "from the output without computing the gradient, hence is different from “back-propagation”.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 104, + 121, + 503, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 399, + 135 + ], + "score": 1.0, + "content": "Initial value For an initial value problem (IVP) such as Eq. 1, typically", + "type": "text" + }, + { + "bbox": [ + 399, + 121, + 445, + 133 + ], + "score": 0.93, + "content": "z _ { 0 } = z ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "is given while", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 497, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 117, + 144 + ], + "score": 0.82, + "content": "v _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 131, + 259, + 146 + ], + "score": 1.0, + "content": "is undetermined. We can construct", + "type": "text" + }, + { + "bbox": [ + 259, + 132, + 329, + 144 + ], + "score": 0.93, + "content": "v _ { 0 } = f ( z ( t _ { 0 } ) , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 131, + 461, + 146 + ], + "score": 1.0, + "content": ", so the initial augmented state is", + "type": "text" + }, + { + "bbox": [ + 462, + 132, + 492, + 144 + ], + "score": 0.92, + "content": "( z _ { 0 } , v _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 131, + 497, + 146 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "Difference from midpoint integrator The midpoint integrator (Suli & Mayers, 2003) is similar ¨", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 256, + 174 + ], + "score": 1.0, + "content": "to Algo. 2, except that it recomputes", + "type": "text" + }, + { + "bbox": [ + 256, + 160, + 329, + 172 + ], + "score": 0.92, + "content": "v _ { i n } = f ( z _ { i n } , s _ { i n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 159, + 506, + 174 + ], + "score": 1.0, + "content": "for every step, while ALF directly uses the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 446, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 129, + 184 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 129, + 173, + 144, + 182 + ], + "score": 0.88, + "content": "v _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 171, + 446, + 184 + ], + "score": 1.0, + "content": ". Therefore, the midpoint method does not have an explicit form of inverse.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 187, + 504, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "Local truncation error Theorem 3.1 indicates that the local truncation error of ALF is of order", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 432, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 133, + 212 + ], + "score": 0.91, + "content": "O ( h ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 198, + 259, + 211 + ], + "score": 1.0, + "content": "; this implies the global error is", + "type": "text" + }, + { + "bbox": [ + 259, + 199, + 286, + 211 + ], + "score": 0.92, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 198, + 432, + 211 + ], + "score": 1.0, + "content": ". Detailed proof is in Appendix A.3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 504, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 317, + 226 + ], + "score": 1.0, + "content": "Theorem 3.1. For a single step in ALF with stepsize", + "type": "text" + }, + { + "bbox": [ + 317, + 214, + 324, + 224 + ], + "score": 0.64, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 212, + 441, + 226 + ], + "score": 1.0, + "content": ", the local truncation error of", + "type": "text" + }, + { + "bbox": [ + 441, + 216, + 447, + 223 + ], + "score": 0.68, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 212, + 457, + 226 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 457, + 213, + 483, + 226 + ], + "score": 0.91, + "content": "O ( h ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 212, + 506, + 226 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 268, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 237, + 236 + ], + "score": 1.0, + "content": "the local truncation error of v is", + "type": "text" + }, + { + "bbox": [ + 238, + 224, + 264, + 236 + ], + "score": 0.91, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 223, + 268, + 236 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "A-Stability The ALF solver has a limited stability region, but this can be solved with damping. The", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 251, + 269 + ], + "score": 1.0, + "content": "damped ALF replaces the update of", + "type": "text" + }, + { + "bbox": [ + 252, + 257, + 269, + 267 + ], + "score": 0.88, + "content": "v _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 255, + 333, + 269 + ], + "score": 1.0, + "content": "in Algo. 2 with", + "type": "text" + }, + { + "bbox": [ + 334, + 256, + 443, + 268 + ], + "score": 0.91, + "content": "v _ { o u t } = v _ { i n } + 2 \\eta ( u _ { 1 } - v _ { i n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 255, + 473, + 269 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 258, + 480, + 268 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 498, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 498, + 279 + ], + "score": 1.0, + "content": "“damping coefficient” between 0 and 1. We have the following theorem on its numerical stability.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 502, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 346, + 294 + ], + "score": 1.0, + "content": "Theorem 3.2. For the damped ALF integrator with stepsize", + "type": "text" + }, + { + "bbox": [ + 347, + 282, + 353, + 291 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 279, + 383, + 294 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 384, + 282, + 394, + 292 + ], + "score": 0.83, + "content": "\\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 279, + 418, + 294 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 419, + 282, + 423, + 291 + ], + "score": 0.39, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "-th eigenvalue of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 291, + 502, + 314 + ], + "spans": [ + { + "bbox": [ + 103, + 291, + 145, + 314 + ], + "score": 1.0, + "content": "Jacobian", + "type": "text" + }, + { + "bbox": [ + 146, + 294, + 158, + 310 + ], + "score": 0.89, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 291, + 231, + 314 + ], + "score": 1.0, + "content": ", then the solver is", + "type": "text" + }, + { + "bbox": [ + 232, + 297, + 239, + 307 + ], + "score": 0.36, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 291, + 270, + 314 + ], + "score": 1.0, + "content": "-stable", + "type": "text" + }, + { + "bbox": [ + 270, + 292, + 482, + 313 + ], + "score": 0.86, + "content": "\\left. \\dot { \\tau } f \\right| 1 + \\eta ( h \\sigma _ { i } - 1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\Big | < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 291, + 502, + 314 + ], + "score": 1.0, + "content": ", ∀i", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 424, + 333 + ], + "score": 1.0, + "content": "Proof is in Appendix A.4 and A.5. Theorem 3.2 implies the following: when", + "type": "text" + }, + { + "bbox": [ + 424, + 321, + 451, + 332 + ], + "score": 0.9, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 320, + 505, + 333 + ], + "score": 1.0, + "content": ", the damped", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 351, + 344 + ], + "score": 1.0, + "content": "ALF reduces to ALF, and the stability region is empty; when", + "type": "text" + }, + { + "bbox": [ + 352, + 332, + 394, + 343 + ], + "score": 0.92, + "content": "0 < \\eta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 331, + 505, + 344 + ], + "score": 1.0, + "content": ", the stability region is non-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 337, + 356 + ], + "score": 1.0, + "content": "empty. However, stability describes the behaviour when", + "type": "text" + }, + { + "bbox": [ + 337, + 343, + 346, + 352 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "goes to infinity; in practice we always", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 454, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 165, + 366 + ], + "score": 1.0, + "content": "use a bounded", + "type": "text" + }, + { + "bbox": [ + 166, + 354, + 174, + 363 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 353, + 454, + 366 + ], + "score": 1.0, + "content": "and ALF performs well. Inverse of damped ALF is in Appendix A.5.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 106, + 374, + 458, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 459, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 459, + 388 + ], + "score": 1.0, + "content": "3.2 MEMORY-EFFICIENT ALF INTEGRATOR (MALI) FOR GRADIENT ESTIMATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "An ideal solver for Neural ODEs should achieve two goals: accuracy in gradient estimation and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "constant memory cost w.r.t integration time. Yet none of the existing methods can achieve both", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "goals. We propose a method based on the ALF solver, which to our knowledge is the first method to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 259, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 259, + 438 + ], + "score": 1.0, + "content": "achieve the two goals simultaneously.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "table", + "bbox": [ + 105, + 450, + 501, + 634 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 105, + 450, + 501, + 634 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 450, + 501, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 501, + 634 + ], + "score": 0.884, + "html": "
Algorithm 4: MALI to acheive accuracy at a constant memory cost w.r.t integration time
Input Initial state zo, start time to, end time T
Forward Apply the numerical integration in Algo.1, with the function defined by Algo. 2.
Delete computation graph on the fly, only keep end-time state (zNt, UNt）
Keep accepted discretized time points {ti}N=0 (ignore processto search for optimal stepsize) Backward
8L by Eq.3,initialize dL =0
Initialize a(T) = (T) d
For i in {Nt,Nt -1,..,2,1}:
Reconstruct (zi-1, Ui-1) from (zi,Ui) by Algo.7 Local forward (zi,Ui,ti,hi)= γ(zi-1,Ui-1,ti-1,hi)
Local backward, get Of(zi-1,ti-1,0) and f（2i-1ti-1,0)
dzi-1 80 dL
Update a(t) and byEq.2 and Eq. 3 discretized at time points ti-1 and ti de
Delete local computation graph
Output the adjoint state a(to) (gradient w.r.t input zo) and parameter gradient
", + "type": "table", + "image_path": "0390ae9d88e50f6cc6ea18c563fd4647053ff00ab939ddf36ad670f57afdf6d5.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 105, + 450, + 501, + 511.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 105, + 511.3333333333333, + 501, + 572.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 105, + 572.6666666666666, + 501, + 634.0 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 504, + 661 + ], + "score": 1.0, + "content": "Procedure of MALI Details of MALI are summarized in Algo. 4. For the forward-pass, we only", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 204, + 673 + ], + "score": 1.0, + "content": "keep the end-time state", + "type": "text" + }, + { + "bbox": [ + 204, + 660, + 246, + 672 + ], + "score": 0.92, + "content": "\\left( z _ { N _ { t } } , v _ { N _ { t } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "and the accepted discretized time points (blue curves in Fig. 1", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 669, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 683 + ], + "score": 1.0, + "content": "and 2). We ignore the search process for optimal stepsize (green curve in Fig. 1 and 2), and delete", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "other variables to save memory. During the backward pass, we can reconstruct the forward-time", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 692, + 490, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 490, + 705 + ], + "score": 1.0, + "content": "trajectory as in Eq. 5, then calculate the gradient by numerical discretization of Eq. 2 and Eq. 3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Constant memory cost w.r.t number of solver steps in integration We delete the computation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 421, + 734 + ], + "score": 1.0, + "content": "graph and only keep the end-time state to save memory. The memory cost is", + "type": "text" + }, + { + "bbox": [ + 421, + 720, + 473, + 733 + ], + "score": 0.93, + "content": "N _ { z } ( N _ { f } + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 95 + ], + "score": 1.0, + "content": "In the following sections, we will show the invertibility of ALF is the key to maintain accuracy at a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "constant memory cost to train Neural ODEs. Note that “inverse” refers to reconstructing the input", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 478, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 478, + 117 + ], + "score": 1.0, + "content": "from the output without computing the gradient, hence is different from “back-propagation”.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 83, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 121, + 503, + 145 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 399, + 135 + ], + "score": 1.0, + "content": "Initial value For an initial value problem (IVP) such as Eq. 1, typically", + "type": "text" + }, + { + "bbox": [ + 399, + 121, + 445, + 133 + ], + "score": 0.93, + "content": "z _ { 0 } = z ( t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "is given while", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 497, + 146 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 117, + 144 + ], + "score": 0.82, + "content": "v _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 131, + 259, + 146 + ], + "score": 1.0, + "content": "is undetermined. We can construct", + "type": "text" + }, + { + "bbox": [ + 259, + 132, + 329, + 144 + ], + "score": 0.93, + "content": "v _ { 0 } = f ( z ( t _ { 0 } ) , t _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 131, + 461, + 146 + ], + "score": 1.0, + "content": ", so the initial augmented state is", + "type": "text" + }, + { + "bbox": [ + 462, + 132, + 492, + 144 + ], + "score": 0.92, + "content": "( z _ { 0 } , v _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 131, + 497, + 146 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 120, + 505, + 146 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 149, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "Difference from midpoint integrator The midpoint integrator (Suli & Mayers, 2003) is similar ¨", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 256, + 174 + ], + "score": 1.0, + "content": "to Algo. 2, except that it recomputes", + "type": "text" + }, + { + "bbox": [ + 256, + 160, + 329, + 172 + ], + "score": 0.92, + "content": "v _ { i n } = f ( z _ { i n } , s _ { i n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 159, + 506, + 174 + ], + "score": 1.0, + "content": "for every step, while ALF directly uses the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 171, + 446, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 129, + 184 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 129, + 173, + 144, + 182 + ], + "score": 0.88, + "content": "v _ { i n }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 171, + 446, + 184 + ], + "score": 1.0, + "content": ". Therefore, the midpoint method does not have an explicit form of inverse.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 149, + 506, + 184 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 187, + 504, + 211 + ], + "lines": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "Local truncation error Theorem 3.1 indicates that the local truncation error of ALF is of order", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 198, + 432, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 133, + 212 + ], + "score": 0.91, + "content": "O ( h ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 198, + 259, + 211 + ], + "score": 1.0, + "content": "; this implies the global error is", + "type": "text" + }, + { + "bbox": [ + 259, + 199, + 286, + 211 + ], + "score": 0.92, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 198, + 432, + 211 + ], + "score": 1.0, + "content": ". Detailed proof is in Appendix A.3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 106, + 187, + 505, + 212 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 504, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 317, + 226 + ], + "score": 1.0, + "content": "Theorem 3.1. For a single step in ALF with stepsize", + "type": "text" + }, + { + "bbox": [ + 317, + 214, + 324, + 224 + ], + "score": 0.64, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 212, + 441, + 226 + ], + "score": 1.0, + "content": ", the local truncation error of", + "type": "text" + }, + { + "bbox": [ + 441, + 216, + 447, + 223 + ], + "score": 0.68, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 212, + 457, + 226 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 457, + 213, + 483, + 226 + ], + "score": 0.91, + "content": "O ( h ^ { 3 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 212, + 506, + 226 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 268, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 237, + 236 + ], + "score": 1.0, + "content": "the local truncation error of v is", + "type": "text" + }, + { + "bbox": [ + 238, + 224, + 264, + 236 + ], + "score": 0.91, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 223, + 268, + 236 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 212, + 506, + 236 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 244, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "A-Stability The ALF solver has a limited stability region, but this can be solved with damping. The", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 251, + 269 + ], + "score": 1.0, + "content": "damped ALF replaces the update of", + "type": "text" + }, + { + "bbox": [ + 252, + 257, + 269, + 267 + ], + "score": 0.88, + "content": "v _ { o u t }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 255, + 333, + 269 + ], + "score": 1.0, + "content": "in Algo. 2 with", + "type": "text" + }, + { + "bbox": [ + 334, + 256, + 443, + 268 + ], + "score": 0.91, + "content": "v _ { o u t } = v _ { i n } + 2 \\eta ( u _ { 1 } - v _ { i n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 255, + 473, + 269 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 474, + 258, + 480, + 268 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 255, + 505, + 269 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 498, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 498, + 279 + ], + "score": 1.0, + "content": "“damping coefficient” between 0 and 1. We have the following theorem on its numerical stability.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 245, + 505, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 502, + 312 + ], + "lines": [ + { + "bbox": [ + 105, + 279, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 346, + 294 + ], + "score": 1.0, + "content": "Theorem 3.2. For the damped ALF integrator with stepsize", + "type": "text" + }, + { + "bbox": [ + 347, + 282, + 353, + 291 + ], + "score": 0.74, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 279, + 383, + 294 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 384, + 282, + 394, + 292 + ], + "score": 0.83, + "content": "\\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 279, + 418, + 294 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 419, + 282, + 423, + 291 + ], + "score": 0.39, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 279, + 505, + 294 + ], + "score": 1.0, + "content": "-th eigenvalue of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 291, + 502, + 314 + ], + "spans": [ + { + "bbox": [ + 103, + 291, + 145, + 314 + ], + "score": 1.0, + "content": "Jacobian", + "type": "text" + }, + { + "bbox": [ + 146, + 294, + 158, + 310 + ], + "score": 0.89, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 291, + 231, + 314 + ], + "score": 1.0, + "content": ", then the solver is", + "type": "text" + }, + { + "bbox": [ + 232, + 297, + 239, + 307 + ], + "score": 0.36, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 291, + 270, + 314 + ], + "score": 1.0, + "content": "-stable", + "type": "text" + }, + { + "bbox": [ + 270, + 292, + 482, + 313 + ], + "score": 0.86, + "content": "\\left. \\dot { \\tau } f \\right| 1 + \\eta ( h \\sigma _ { i } - 1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\Big | < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 291, + 502, + 314 + ], + "score": 1.0, + "content": ", ∀i", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 103, + 279, + 505, + 314 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 320, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 424, + 333 + ], + "score": 1.0, + "content": "Proof is in Appendix A.4 and A.5. Theorem 3.2 implies the following: when", + "type": "text" + }, + { + "bbox": [ + 424, + 321, + 451, + 332 + ], + "score": 0.9, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 320, + 505, + 333 + ], + "score": 1.0, + "content": ", the damped", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 351, + 344 + ], + "score": 1.0, + "content": "ALF reduces to ALF, and the stability region is empty; when", + "type": "text" + }, + { + "bbox": [ + 352, + 332, + 394, + 343 + ], + "score": 0.92, + "content": "0 < \\eta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 331, + 505, + 344 + ], + "score": 1.0, + "content": ", the stability region is non-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 337, + 356 + ], + "score": 1.0, + "content": "empty. However, stability describes the behaviour when", + "type": "text" + }, + { + "bbox": [ + 337, + 343, + 346, + 352 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "goes to infinity; in practice we always", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 353, + 454, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 165, + 366 + ], + "score": 1.0, + "content": "use a bounded", + "type": "text" + }, + { + "bbox": [ + 166, + 354, + 174, + 363 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 353, + 454, + 366 + ], + "score": 1.0, + "content": "and ALF performs well. Inverse of damped ALF is in Appendix A.5.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 320, + 505, + 366 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 374, + 458, + 387 + ], + "lines": [ + { + "bbox": [ + 105, + 375, + 459, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 459, + 388 + ], + "score": 1.0, + "content": "3.2 MEMORY-EFFICIENT ALF INTEGRATOR (MALI) FOR GRADIENT ESTIMATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "An ideal solver for Neural ODEs should achieve two goals: accuracy in gradient estimation and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "constant memory cost w.r.t integration time. Yet none of the existing methods can achieve both", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "goals. We propose a method based on the ALF solver, which to our knowledge is the first method to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 259, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 259, + 438 + ], + "score": 1.0, + "content": "achieve the two goals simultaneously.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 392, + 505, + 438 + ] + }, + { + "type": "table", + "bbox": [ + 105, + 450, + 501, + 634 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 105, + 450, + 501, + 634 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 450, + 501, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 501, + 634 + ], + "score": 0.884, + "html": "
Algorithm 4: MALI to acheive accuracy at a constant memory cost w.r.t integration time
Input Initial state zo, start time to, end time T
Forward Apply the numerical integration in Algo.1, with the function defined by Algo. 2.
Delete computation graph on the fly, only keep end-time state (zNt, UNt）
Keep accepted discretized time points {ti}N=0 (ignore processto search for optimal stepsize) Backward
8L by Eq.3,initialize dL =0
Initialize a(T) = (T) d
For i in {Nt,Nt -1,..,2,1}:
Reconstruct (zi-1, Ui-1) from (zi,Ui) by Algo.7 Local forward (zi,Ui,ti,hi)= γ(zi-1,Ui-1,ti-1,hi)
Local backward, get Of(zi-1,ti-1,0) and f（2i-1ti-1,0)
dzi-1 80 dL
Update a(t) and byEq.2 and Eq. 3 discretized at time points ti-1 and ti de
Delete local computation graph
Output the adjoint state a(to) (gradient w.r.t input zo) and parameter gradient
", + "type": "table", + "image_path": "0390ae9d88e50f6cc6ea18c563fd4647053ff00ab939ddf36ad670f57afdf6d5.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 105, + 450, + 501, + 511.3333333333333 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 105, + 511.3333333333333, + 501, + 572.6666666666666 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 105, + 572.6666666666666, + 501, + 634.0 + ], + "spans": [], + "index": 28 + } + ] + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 504, + 661 + ], + "score": 1.0, + "content": "Procedure of MALI Details of MALI are summarized in Algo. 4. For the forward-pass, we only", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 204, + 673 + ], + "score": 1.0, + "content": "keep the end-time state", + "type": "text" + }, + { + "bbox": [ + 204, + 660, + 246, + 672 + ], + "score": 0.92, + "content": "\\left( z _ { N _ { t } } , v _ { N _ { t } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "and the accepted discretized time points (blue curves in Fig. 1", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 669, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 683 + ], + "score": 1.0, + "content": "and 2). We ignore the search process for optimal stepsize (green curve in Fig. 1 and 2), and delete", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "other variables to save memory. During the backward pass, we can reconstruct the forward-time", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 692, + 490, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 490, + 705 + ], + "score": 1.0, + "content": "trajectory as in Eq. 5, then calculate the gradient by numerical discretization of Eq. 2 and Eq. 3.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 648, + 505, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Constant memory cost w.r.t number of solver steps in integration We delete the computation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 421, + 734 + ], + "score": 1.0, + "content": "graph and only keep the end-time state to save memory. The memory cost is", + "type": "text" + }, + { + "bbox": [ + 421, + 720, + 473, + 733 + ], + "score": 0.93, + "content": "N _ { z } ( N _ { f } + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 720, + 505, + 734 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 81, + 497, + 173 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 81, + 497, + 173 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 81, + 497, + 173 + ], + "spans": [ + { + "bbox": [ + 112, + 81, + 497, + 173 + ], + "score": 0.781, + "type": "image", + "image_path": "72f602b837ed8e185f46ef9611847d89781d838c60ee6b4f4fd38f08381e9dc4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 81, + 497, + 111.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 111.66666666666667, + 497, + 142.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 142.33333333333334, + 497, + 173.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 114, + 175, + 490, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 168, + 494, + 194 + ], + "spans": [ + { + "bbox": [ + 114, + 168, + 351, + 194 + ], + "score": 1.0, + "content": "Figure 4: Comparison of error in gradient in Eq. 6. (a) error in", + "type": "text" + }, + { + "bbox": [ + 351, + 174, + 365, + 189 + ], + "score": 0.91, + "content": "\\scriptstyle { \\frac { \\mathrm { d } L } { \\mathrm { d } z _ { 0 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 168, + 411, + 194 + ], + "score": 1.0, + "content": ". (b) error in", + "type": "text" + }, + { + "bbox": [ + 411, + 174, + 423, + 188 + ], + "score": 0.9, + "content": "\\textstyle { \\frac { \\mathrm { d } L } { \\mathrm { d } \\alpha } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 168, + 494, + 194 + ], + "score": 1.0, + "content": ". (c) memory cost.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "image", + "bbox": [ + 110, + 195, + 499, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 195, + 499, + 283 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 110, + 195, + 499, + 283 + ], + "spans": [ + { + "bbox": [ + 110, + 195, + 499, + 283 + ], + "score": 0.889, + "type": "image", + "image_path": "a5790d97491493f8fedf10284fe335bda1b1750debbcc4560d4977556c8c12ca.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 110, + 195, + 499, + 224.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 110, + 224.33333333333334, + 499, + 253.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 110, + 253.66666666666669, + 499, + 283.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 288, + 504, + 318 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 299 + ], + "score": 1.0, + "content": "Figure 5: Results on Cifar10. From left to right: (1) box plot of test accuracy (first 4 columns are Neural", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 298, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 252, + 309 + ], + "score": 1.0, + "content": "ODEs, last is ResNet); (2) test accuracy", + "type": "text" + }, + { + "bbox": [ + 252, + 299, + 274, + 308 + ], + "score": 0.73, + "content": "\\pm s t d", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 298, + 468, + 309 + ], + "score": 1.0, + "content": "v.s. training epoch for Neural ODE; (3) test accuracy", + "type": "text" + }, + { + "bbox": [ + 468, + 299, + 490, + 308 + ], + "score": 0.64, + "content": "\\pm s t d", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 298, + 506, + 309 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 308, + 265, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 265, + 319 + ], + "score": 1.0, + "content": "training time of 90 epochs for Neural ODE.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 366 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 133, + 344 + ], + "score": 0.91, + "content": "N _ { z } N _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 331, + 218, + 345 + ], + "score": 1.0, + "content": "is due to evaluating", + "type": "text" + }, + { + "bbox": [ + 219, + 332, + 246, + 344 + ], + "score": 0.93, + "content": "f ( \\boldsymbol { z } , t )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 331, + 506, + 345 + ], + "score": 1.0, + "content": "and is irreducible for all methods. Compared with the adjoint", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 247, + 356 + ], + "score": 1.0, + "content": "method, MALI only requires extra", + "type": "text" + }, + { + "bbox": [ + 247, + 344, + 261, + 354 + ], + "score": 0.89, + "content": "N _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 343, + 336, + 356 + ], + "score": 1.0, + "content": "memory to record", + "type": "text" + }, + { + "bbox": [ + 336, + 345, + 352, + 355 + ], + "score": 0.88, + "content": "v _ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 343, + 505, + 356 + ], + "score": 1.0, + "content": ", and also has a constant memory cost", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 354, + 318, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 125, + 365 + ], + "score": 0.57, + "content": "w . r . t", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 354, + 164, + 366 + ], + "score": 1.0, + "content": "time step", + "type": "text" + }, + { + "bbox": [ + 164, + 354, + 177, + 365 + ], + "score": 0.86, + "content": "N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 354, + 263, + 366 + ], + "score": 1.0, + "content": ". The memory cost is", + "type": "text" + }, + { + "bbox": [ + 263, + 354, + 314, + 366 + ], + "score": 0.93, + "content": "N _ { z } ( N _ { f } + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 354, + 318, + 366 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "Accuracy Our method guarantees the accuracy of reverse-time trajectory (e.g. blue curve in Fig. 2", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 440, + 394 + ], + "score": 1.0, + "content": "matches the blue curve in Fig. 1), because ALF is explicitly invertible for free-form", + "type": "text" + }, + { + "bbox": [ + 440, + 382, + 448, + 393 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "(see Algo. 7).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 392, + 482, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 482, + 405 + ], + "score": 1.0, + "content": "Therefore, the gradient estimation in MALI is more accurate compared to the adjoint method.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 316, + 422 + ], + "score": 1.0, + "content": "Computation cost Recall that on average it takes", + "type": "text" + }, + { + "bbox": [ + 316, + 412, + 326, + 420 + ], + "score": 0.62, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "steps to find an acceptable stepsize, whose", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "error estimate is below tolerance. Therefore, the forward-pass with search process has computation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 430, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 136, + 446 + ], + "score": 1.0, + "content": "burden", + "type": "text" + }, + { + "bbox": [ + 137, + 432, + 219, + 444 + ], + "score": 0.93, + "content": "N _ { z } \\times N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 430, + 505, + 446 + ], + "score": 1.0, + "content": ". Note that we only reconstruct and backprop through the accepted step", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 319, + 455 + ], + "score": 1.0, + "content": "and ignore the search process, hence it takes another", + "type": "text" + }, + { + "bbox": [ + 319, + 443, + 399, + 455 + ], + "score": 0.92, + "content": "N _ { z } \\times N _ { f } \\times N _ { t } \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "computation. The overall", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 453, + 352, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 198, + 466 + ], + "score": 1.0, + "content": "computation burden is", + "type": "text" + }, + { + "bbox": [ + 198, + 453, + 294, + 466 + ], + "score": 0.93, + "content": "N _ { z } N _ { f } \\times N _ { t } \\times ( m + 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 454, + 352, + 466 + ], + "score": 1.0, + "content": "as in Table 1.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "Shallow computation graph Similar to ACA, MALI only backpropagates through the accepted", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "step (blue curve in Fig. 2) and ignores the search process (green curve in Fig. 2), hence the depth of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 193, + 504 + ], + "score": 1.0, + "content": "computation graph is", + "type": "text" + }, + { + "bbox": [ + 194, + 493, + 232, + 505 + ], + "score": 0.92, + "content": "N _ { f } \\times N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 493, + 505, + 504 + ], + "score": 1.0, + "content": ". The computation graph of MALI is much shallower than the naive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 504, + 466, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 466, + 516 + ], + "score": 1.0, + "content": "method, hence is more robust to vanishing and exploding gradients (Pascanu et al., 2013).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "Summary The adjoint method suffers from inaccuracy in reverse-time trajectory, the naive method", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "suffers from exploding or vanishing gradient caused by deep computation graph, and ACA finds a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "balance but the memory grows linearly with integration time. MALI achieves accuracy in reverse-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 554, + 464, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 240, + 565 + ], + "score": 1.0, + "content": "time trajectory, constant memory", + "type": "text" + }, + { + "bbox": [ + 240, + 554, + 259, + 564 + ], + "score": 0.81, + "content": "w . r . t", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 554, + 464, + 565 + ], + "score": 1.0, + "content": "integration time, and a shallow computation graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 108, + 581, + 200, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 201, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 201, + 595 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 603, + 272, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 274, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 274, + 615 + ], + "score": 1.0, + "content": "4.1 VALIDATION ON A TOY EXAMPLE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 425, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 426, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 426, + 639 + ], + "score": 1.0, + "content": "We compare the performance of different methods on a toy example, defined as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 640, + 416, + 655 + ], + "lines": [ + { + "bbox": [ + 195, + 640, + 416, + 655 + ], + "spans": [ + { + "bbox": [ + 195, + 640, + 416, + 655 + ], + "score": 0.87, + "content": "L ( z ( T ) ) = z ( T ) ^ { 2 } \\ s . t . \\ z ( 0 ) = z _ { 0 } , \\ \\mathrm { d } z ( t ) / \\mathrm { d } t = \\alpha z ( t )", + "type": "interline_equation", + "image_path": "7dbd97a564adced1b1ef06e873e32c97c55a0823aa7394c5b5f867f7c9530324.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 195, + 640, + 416, + 655 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 209, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 210, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 210, + 673 + ], + "score": 1.0, + "content": "The analytical solution is", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 677, + 454, + 693 + ], + "lines": [ + { + "bbox": [ + 157, + 677, + 454, + 693 + ], + "spans": [ + { + "bbox": [ + 157, + 677, + 454, + 693 + ], + "score": 0.89, + "content": "z ( t ) = z _ { 0 } e ^ { \\alpha t } , L = z _ { 0 } ^ { 2 } e ^ { 2 \\alpha T } , \\mathrm { d } L / \\mathrm { d } z _ { 0 } = 2 z _ { 0 } e ^ { 2 \\alpha T } , \\mathrm { d } L / d \\alpha = 2 T z _ { 0 } ^ { 2 } e ^ { 2 \\alpha T }", + "type": "interline_equation", + "image_path": "ea86e99f56579157eb696faf168523db9bd21f68f35d9ec106eeddedb03d396f.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 157, + 677, + 454, + 693 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 495, + 711 + ], + "score": 1.0, + "content": "We plot the amplitude of error between numerical solution and analytical solution varying with", + "type": "text" + }, + { + "bbox": [ + 496, + 700, + 504, + 709 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 282, + 722 + ], + "score": 1.0, + "content": "(integrated under the same error tolerance,", + "type": "text" + }, + { + "bbox": [ + 282, + 709, + 392, + 721 + ], + "score": 0.7, + "content": "\\mathrm { r t o l } = 1 0 ^ { - 5 } , \\mathrm { a t o l } = 1 0 ^ { - 6 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "in Fig 4. ACA and MALI", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "have similar errors, both outperforming other methods. We also plot the memory consumption for", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 81, + 497, + 173 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 81, + 497, + 173 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 81, + 497, + 173 + ], + "spans": [ + { + "bbox": [ + 112, + 81, + 497, + 173 + ], + "score": 0.781, + "type": "image", + "image_path": "72f602b837ed8e185f46ef9611847d89781d838c60ee6b4f4fd38f08381e9dc4.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 81, + 497, + 111.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 111.66666666666667, + 497, + 142.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 142.33333333333334, + 497, + 173.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 114, + 175, + 490, + 189 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 168, + 494, + 194 + ], + "spans": [ + { + "bbox": [ + 114, + 168, + 351, + 194 + ], + "score": 1.0, + "content": "Figure 4: Comparison of error in gradient in Eq. 6. (a) error in", + "type": "text" + }, + { + "bbox": [ + 351, + 174, + 365, + 189 + ], + "score": 0.91, + "content": "\\scriptstyle { \\frac { \\mathrm { d } L } { \\mathrm { d } z _ { 0 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 168, + 411, + 194 + ], + "score": 1.0, + "content": ". (b) error in", + "type": "text" + }, + { + "bbox": [ + 411, + 174, + 423, + 188 + ], + "score": 0.9, + "content": "\\textstyle { \\frac { \\mathrm { d } L } { \\mathrm { d } \\alpha } }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 168, + 494, + 194 + ], + "score": 1.0, + "content": ". 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Compared with the adjoint", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 247, + 356 + ], + "score": 1.0, + "content": "method, MALI only requires extra", + "type": "text" + }, + { + "bbox": [ + 247, + 344, + 261, + 354 + ], + "score": 0.89, + "content": "N _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 343, + 336, + 356 + ], + "score": 1.0, + "content": "memory to record", + "type": "text" + }, + { + "bbox": [ + 336, + 345, + 352, + 355 + ], + "score": 0.88, + "content": "v _ { N _ { t } }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 343, + 505, + 356 + ], + "score": 1.0, + "content": ", and also has a constant memory cost", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 354, + 318, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 125, + 365 + ], + "score": 0.57, + "content": "w . r . t", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 354, + 164, + 366 + ], + "score": 1.0, + "content": "time step", + "type": "text" + }, + { + "bbox": [ + 164, + 354, + 177, + 365 + ], + "score": 0.86, + "content": "N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 354, + 263, + 366 + ], + "score": 1.0, + "content": ". The memory cost is", + "type": "text" + }, + { + "bbox": [ + 263, + 354, + 314, + 366 + ], + "score": 0.93, + "content": "N _ { z } ( N _ { f } + 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 354, + 318, + 366 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 331, + 506, + 366 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 506, + 384 + ], + "score": 1.0, + "content": "Accuracy Our method guarantees the accuracy of reverse-time trajectory (e.g. blue curve in Fig. 2", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 440, + 394 + ], + "score": 1.0, + "content": "matches the blue curve in Fig. 1), because ALF is explicitly invertible for free-form", + "type": "text" + }, + { + "bbox": [ + 440, + 382, + 448, + 393 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "(see Algo. 7).", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 392, + 482, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 482, + 405 + ], + "score": 1.0, + "content": "Therefore, the gradient estimation in MALI is more accurate compared to the adjoint method.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 370, + 506, + 405 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 316, + 422 + ], + "score": 1.0, + "content": "Computation cost Recall that on average it takes", + "type": "text" + }, + { + "bbox": [ + 316, + 412, + 326, + 420 + ], + "score": 0.62, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "steps to find an acceptable stepsize, whose", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "error estimate is below tolerance. Therefore, the forward-pass with search process has computation", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 430, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 136, + 446 + ], + "score": 1.0, + "content": "burden", + "type": "text" + }, + { + "bbox": [ + 137, + 432, + 219, + 444 + ], + "score": 0.93, + "content": "N _ { z } \\times N _ { f } \\times N _ { t } \\times m", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 430, + 505, + 446 + ], + "score": 1.0, + "content": ". Note that we only reconstruct and backprop through the accepted step", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 319, + 455 + ], + "score": 1.0, + "content": "and ignore the search process, hence it takes another", + "type": "text" + }, + { + "bbox": [ + 319, + 443, + 399, + 455 + ], + "score": 0.92, + "content": "N _ { z } \\times N _ { f } \\times N _ { t } \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "computation. The overall", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 453, + 352, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 198, + 466 + ], + "score": 1.0, + "content": "computation burden is", + "type": "text" + }, + { + "bbox": [ + 198, + 453, + 294, + 466 + ], + "score": 0.93, + "content": "N _ { z } N _ { f } \\times N _ { t } \\times ( m + 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 454, + 352, + 466 + ], + "score": 1.0, + "content": "as in Table 1.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 409, + 505, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "Shallow computation graph Similar to ACA, MALI only backpropagates through the accepted", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "step (blue curve in Fig. 2) and ignores the search process (green curve in Fig. 2), hence the depth of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 193, + 504 + ], + "score": 1.0, + "content": "computation graph is", + "type": "text" + }, + { + "bbox": [ + 194, + 493, + 232, + 505 + ], + "score": 0.92, + "content": "N _ { f } \\times N _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 493, + 505, + 504 + ], + "score": 1.0, + "content": ". The computation graph of MALI is much shallower than the naive", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 504, + 466, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 466, + 516 + ], + "score": 1.0, + "content": "method, hence is more robust to vanishing and exploding gradients (Pascanu et al., 2013).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 470, + 506, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "Summary The adjoint method suffers from inaccuracy in reverse-time trajectory, the naive method", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "suffers from exploding or vanishing gradient caused by deep computation graph, and ACA finds a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 555 + ], + "score": 1.0, + "content": "balance but the memory grows linearly with integration time. MALI achieves accuracy in reverse-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 554, + 464, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 240, + 565 + ], + "score": 1.0, + "content": "time trajectory, constant memory", + "type": "text" + }, + { + "bbox": [ + 240, + 554, + 259, + 564 + ], + "score": 0.81, + "content": "w . r . t", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 554, + 464, + 565 + ], + "score": 1.0, + "content": "integration time, and a shallow computation graph.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 520, + 506, + 565 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 581, + 200, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 201, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 201, + 595 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 603, + 272, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 274, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 274, + 615 + ], + "score": 1.0, + "content": "4.1 VALIDATION ON A TOY EXAMPLE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 624, + 425, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 426, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 426, + 639 + ], + "score": 1.0, + "content": "We compare the performance of different methods on a toy example, defined as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 623, + 426, + 639 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 640, + 416, + 655 + ], + "lines": [ + { + "bbox": [ + 195, + 640, + 416, + 655 + ], + "spans": [ + { + "bbox": [ + 195, + 640, + 416, + 655 + ], + "score": 0.87, + "content": "L ( z ( T ) ) = z ( T ) ^ { 2 } \\ s . t . \\ z ( 0 ) = z _ { 0 } , \\ \\mathrm { d } z ( t ) / \\mathrm { d } t = \\alpha z ( t )", + "type": "interline_equation", + "image_path": "7dbd97a564adced1b1ef06e873e32c97c55a0823aa7394c5b5f867f7c9530324.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 195, + 640, + 416, + 655 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 661, + 209, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 210, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 210, + 673 + ], + "score": 1.0, + "content": "The analytical solution is", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 660, + 210, + 673 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 677, + 454, + 693 + ], + "lines": [ + { + "bbox": [ + 157, + 677, + 454, + 693 + ], + "spans": [ + { + "bbox": [ + 157, + 677, + 454, + 693 + ], + "score": 0.89, + "content": "z ( t ) = z _ { 0 } e ^ { \\alpha t } , L = z _ { 0 } ^ { 2 } e ^ { 2 \\alpha T } , \\mathrm { d } L / \\mathrm { d } z _ { 0 } = 2 z _ { 0 } e ^ { 2 \\alpha T } , \\mathrm { d } L / d \\alpha = 2 T z _ { 0 } ^ { 2 } e ^ { 2 \\alpha T }", + "type": "interline_equation", + "image_path": "ea86e99f56579157eb696faf168523db9bd21f68f35d9ec106eeddedb03d396f.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 157, + 677, + 454, + 693 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 495, + 711 + ], + "score": 1.0, + "content": "We plot the amplitude of error between numerical solution and analytical solution varying with", + "type": "text" + }, + { + "bbox": [ + 496, + 700, + 504, + 709 + ], + "score": 0.8, + "content": "T", + "type": "inline_equation" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 282, + 722 + ], + "score": 1.0, + "content": "(integrated under the same error tolerance,", + "type": "text" + }, + { + "bbox": [ + 282, + 709, + 392, + 721 + ], + "score": 0.7, + "content": "\\mathrm { r t o l } = 1 0 ^ { - 5 } , \\mathrm { a t o l } = 1 0 ^ { - 6 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "in Fig 4. ACA and MALI", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "have similar errors, both outperforming other methods. We also plot the memory consumption for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "different methods on a Neural ODE with the same input in Fig. 4. As the error tolerance decreases,", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "the solver evaluates more steps, hence the naive method and ACA increase memory consumption,", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "while MALI and the adjoint method have a constant memory cost. These results validate our analysis", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 306, + 457, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 457, + 318 + ], + "score": 1.0, + "content": "in Sec. 3.2 and Table 1, and shows MALI achieves accuracy at a constant memory cost.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 699, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 140, + 105, + 469, + 162 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 81, + 504, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 92 + ], + "score": 1.0, + "content": "Table 2: Top-1 test accuracy of Neural ODE and ResNet on ImageNet. Neural ODE is trained with MALI,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 89, + 495, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 89, + 495, + 104 + ], + "score": 1.0, + "content": "and ResNet is trained as the original model; Neural ODE is tested using different solvers without retraining.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 140, + 105, + 469, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 105, + 469, + 162 + ], + "spans": [ + { + "bbox": [ + 140, + 105, + 469, + 162 + ], + "score": 0.972, + "html": "
Fixed-stepsize solvers of various stepsizesAdaptive-stepsize solver of various tolerances
Stepsize10.50.250.150.1Tolerance1.00E+001.00E-011.00E-02
Neural ODEMALI42.3366.469.5970.1769.94MALI62.5669.8969.87
Euler21.9461.2567.3868.6970.02Heun-Euler68.4869.8769.88
RK242.336969.7270.1469.92RK2350.7769.8969.93
RK412.669.9969.9170.2169.96 70.09Dopri552.368.5869.71
ResNet
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∈=1/255∈=2/255
MALIHeun-EulerRK23Dopri5MALIHeun-EulerRK23Dopri5
Neural ODEMALI14.6914.7214.7715.7110.3810.4610.6210.62
Heun-Euler14.7714.7514.8015.7410.6310.4710.4410.49
RK2314.8214.7714.7915.6910.7810.5310.4810.56
Dopri514.8214.7814.7915.1510.7610.4910.4810.51
ResNet13.029.57
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As the error tolerance decreases,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 297 + ], + "score": 1.0, + "content": "the solver evaluates more steps, hence the naive method and ACA increase memory consumption,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "while MALI and the adjoint method have a constant memory cost. These results validate our analysis", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 306, + 457, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 457, + 318 + ], + "score": 1.0, + "content": "in Sec. 3.2 and Table 1, and shows MALI achieves accuracy at a constant memory cost.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 337, + 312, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 337, + 313, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 313, + 350 + ], + "score": 1.0, + "content": "4.2 IMAGE RECOGNITION WITH NEURAL ODE", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 360, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "We validate MALI on image recognition tasks using Cifar10 and ImageNet datasets. Similar to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "Zhuang et al. (2020), we modify a ResNet18 into its corresponding Neural ODE: the forward func-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 133, + 399 + ], + "score": 1.0, + "content": "tion is", + "type": "text" + }, + { + "bbox": [ + 134, + 384, + 194, + 397 + ], + "score": 0.91, + "content": "y = x + f _ { \\theta } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 383, + 212, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 382, + 295, + 398 + ], + "score": 0.93, + "content": "\\begin{array} { r } { y = \\overset { \\cdot } { x } + \\int _ { 0 } ^ { T } f _ { \\theta } ( z ) \\mathrm { d } t } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 383, + 506, + 399 + ], + "score": 1.0, + "content": "for the residual block and Neural ODE respectively,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 396, + 502, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 171, + 408 + ], + "score": 1.0, + "content": "where the same", + "type": "text" + }, + { + "bbox": [ + 171, + 397, + 181, + 407 + ], + "score": 0.87, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 396, + 502, + 408 + ], + "score": 1.0, + "content": "is shared. We compare MALI with the naive method, adjoint method and ACA.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "Results on Cifar10 Results of 5 independent runs on Cifar10 are summarized in Fig. 5. MALI", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "score": 1.0, + "content": "achieves comparable accuracy to ACA, and both significantly outperform the naive and the adjoint", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "method. Furthermore, the training speed of MALI is similar to ACA, and both are almost two times", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "faster than the adjoint memthod, and three times faster than the naive method. This validates our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 335, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 335, + 469 + ], + "score": 1.0, + "content": "analysis on accuracy and computation burden in Table 1.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 356, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 356, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 356, + 491 + ], + "score": 1.0, + "content": "Accuracy on ImageNet Due to the heavy memory burden", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 356, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 356, + 501 + ], + "score": 1.0, + "content": "caused by large images, the naive method and ACA are unable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 357, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 357, + 513 + ], + "score": 1.0, + "content": "to train a Neural ODE on ImageNet with 4 GPUs; only MALI", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 357, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 357, + 523 + ], + "score": 1.0, + "content": "and the adjoint method are feasible due to the constant mem-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 356, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 356, + 534 + ], + "score": 1.0, + "content": "ory. We also compare the Neural ODE to a standard ResNet.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 356, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 356, + 545 + ], + "score": 1.0, + "content": "As shown in Fig. 6, the accuracy of the Neural ODE trained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 544, + 357, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 357, + 557 + ], + "score": 1.0, + "content": "with MALI closely follows ResNet, and significantly outper-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 346, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 281, + 567 + ], + "score": 1.0, + "content": "forms the adjoint method (top-1 validation:", + "type": "text" + }, + { + "bbox": [ + 282, + 555, + 301, + 566 + ], + "score": 0.85, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 555, + 318, + 567 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 318, + 555, + 338, + 566 + ], + "score": 0.85, + "content": "63 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 555, + 346, + 567 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5 + }, + { + "type": "image", + "bbox": [ + 366, + 493, + 502, + 593 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 366, + 493, + 502, + 593 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 366, + 493, + 502, + 593 + ], + "spans": [ + { + "bbox": [ + 366, + 493, + 502, + 593 + ], + "score": 0.967, + "type": "image", + "image_path": "5baeedcaa97100e0df410b2d664c7ea69138b65803c881e78de14fc713349129.jpg" + } + ] + } + ], + "index": 34.0, + "virtual_lines": [ + { + "bbox": [ + 366, + 493, + 502, + 543.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 366, + 543.0, + 502, + 593.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 363, + 596, + 503, + 617 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 363, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 363, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "Figure 6: Top-1 accuracy on Ima-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 363, + 606, + 456, + 617 + ], + "spans": [ + { + "bbox": [ + 363, + 606, + 456, + 617 + ], + "score": 1.0, + "content": "geNet validation dataset.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "index": 36.75 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 357, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 356, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 356, + 584 + ], + "score": 1.0, + "content": "Invariance to discretization scheme A continuous model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 357, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 357, + 595 + ], + "score": 1.0, + "content": "should be invariant to discretization schemes (e.g. different", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 357, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 357, + 605 + ], + "score": 1.0, + "content": "types of ODE solvers) as long as the discretization is suf-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 357, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 357, + 617 + ], + "score": 1.0, + "content": "ficiently accurate. We test the Neural ODE using different", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 616, + 357, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 357, + 628 + ], + "score": 1.0, + "content": "solvers without re-training; since ResNet is often viewed as", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 672 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "a one-step Euler discretization of an ODE (Haber & Ruthotto, 2017), we perform similar exper-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 441, + 650 + ], + "score": 1.0, + "content": "iments. As shown in Table 2, Neural ODE consistently achieves high accuracy", + "type": "text" + }, + { + "bbox": [ + 442, + 638, + 474, + 649 + ], + "score": 0.87, + "content": "( \\sim 7 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 639, + 505, + 650 + ], + "score": 1.0, + "content": ", while", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 245, + 661 + ], + "score": 1.0, + "content": "ResNet drops to random guessing", + "type": "text" + }, + { + "bbox": [ + 246, + 649, + 281, + 660 + ], + "score": 0.89, + "content": "( \\sim 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "because ResNet as a one-step Euler discretization fails", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 353, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 353, + 673 + ], + "score": 1.0, + "content": "to be a meaningful dynamical system (Queiruga et al., 2020).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "Robustness to adversarial attack Hanshu et al. (2019) demonstrated that Neural ODE is more", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "robust to adversarial attack than ResNet on small-scale datasets such as Cifar10. We validate this", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "result on the large-scale ImageNet dataset. The top-1 accuracy of Neural ODE and ResNet under", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "FGSM attack (Goodfellow et al., 2014) are summarized in Table 3. For Neural ODE, due to its", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "invariance to discretization scheme, we derive the gradient for attack using a certain solver (row in", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 140, + 105, + 469, + 162 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 81, + 504, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 92 + ], + "score": 1.0, + "content": "Table 2: Top-1 test accuracy of Neural ODE and ResNet on ImageNet. Neural ODE is trained with MALI,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 89, + 495, + 104 + ], + "spans": [ + { + "bbox": [ + 105, + 89, + 495, + 104 + ], + "score": 1.0, + "content": "and ResNet is trained as the original model; Neural ODE is tested using different solvers without retraining.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 140, + 105, + 469, + 162 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 105, + 469, + 162 + ], + "spans": [ + { + "bbox": [ + 140, + 105, + 469, + 162 + ], + "score": 0.972, + "html": "
Fixed-stepsize solvers of various stepsizesAdaptive-stepsize solver of various tolerances
Stepsize10.50.250.150.1Tolerance1.00E+001.00E-011.00E-02
Neural ODEMALI42.3366.469.5970.1769.94MALI62.5669.8969.87
Euler21.9461.2567.3868.6970.02Heun-Euler68.4869.8769.88
RK242.336969.7270.1469.92RK2350.7769.8969.93
RK412.669.9969.9170.2169.96 70.09Dopri552.368.5869.71
ResNet
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∈=1/255∈=2/255
MALIHeun-EulerRK23Dopri5MALIHeun-EulerRK23Dopri5
Neural ODEMALI14.6914.7214.7715.7110.3810.4610.6210.62
Heun-Euler14.7714.7514.8015.7410.6310.4710.4410.49
RK2314.8214.7714.7915.6910.7810.5310.4810.56
Dopri514.8214.7814.7915.1510.7610.4910.4810.51
ResNet13.029.57
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Similar to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "Zhuang et al. (2020), we modify a ResNet18 into its corresponding Neural ODE: the forward func-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 382, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 133, + 399 + ], + "score": 1.0, + "content": "tion is", + "type": "text" + }, + { + "bbox": [ + 134, + 384, + 194, + 397 + ], + "score": 0.91, + "content": "y = x + f _ { \\theta } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 383, + 212, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 212, + 382, + 295, + 398 + ], + "score": 0.93, + "content": "\\begin{array} { r } { y = \\overset { \\cdot } { x } + \\int _ { 0 } ^ { T } f _ { \\theta } ( z ) \\mathrm { d } t } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 383, + 506, + 399 + ], + "score": 1.0, + "content": "for the residual block and Neural ODE respectively,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 396, + 502, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 171, + 408 + ], + "score": 1.0, + "content": "where the same", + "type": "text" + }, + { + "bbox": [ + 171, + 397, + 181, + 407 + ], + "score": 0.87, + "content": "f _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 396, + 502, + 408 + ], + "score": 1.0, + "content": "is shared. We compare MALI with the naive method, adjoint method and ACA.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 360, + 506, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "Results on Cifar10 Results of 5 independent runs on Cifar10 are summarized in Fig. 5. MALI", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 435 + ], + "score": 1.0, + "content": "achieves comparable accuracy to ACA, and both significantly outperform the naive and the adjoint", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "method. Furthermore, the training speed of MALI is similar to ACA, and both are almost two times", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "faster than the adjoint memthod, and three times faster than the naive method. This validates our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 335, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 335, + 469 + ], + "score": 1.0, + "content": "analysis on accuracy and computation burden in Table 1.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 413, + 506, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 356, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 356, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 356, + 491 + ], + "score": 1.0, + "content": "Accuracy on ImageNet Due to the heavy memory burden", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 356, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 356, + 501 + ], + "score": 1.0, + "content": "caused by large images, the naive method and ACA are unable", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 357, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 357, + 513 + ], + "score": 1.0, + "content": "to train a Neural ODE on ImageNet with 4 GPUs; only MALI", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 511, + 357, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 357, + 523 + ], + "score": 1.0, + "content": "and the adjoint method are feasible due to the constant mem-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 356, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 356, + 534 + ], + "score": 1.0, + "content": "ory. We also compare the Neural ODE to a standard ResNet.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 356, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 356, + 545 + ], + "score": 1.0, + "content": "As shown in Fig. 6, the accuracy of the Neural ODE trained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 544, + 357, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 357, + 557 + ], + "score": 1.0, + "content": "with MALI closely follows ResNet, and significantly outper-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 346, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 281, + 567 + ], + "score": 1.0, + "content": "forms the adjoint method (top-1 validation:", + "type": "text" + }, + { + "bbox": [ + 282, + 555, + 301, + 566 + ], + "score": 0.85, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 555, + 318, + 567 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 318, + 555, + 338, + 566 + ], + "score": 0.85, + "content": "63 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 555, + 346, + 567 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 478, + 357, + 567 + ] + }, + { + "type": "image", + "bbox": [ + 366, + 493, + 502, + 593 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 366, + 493, + 502, + 593 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 366, + 493, + 502, + 593 + ], + "spans": [ + { + "bbox": [ + 366, + 493, + 502, + 593 + ], + "score": 0.967, + "type": "image", + "image_path": "5baeedcaa97100e0df410b2d664c7ea69138b65803c881e78de14fc713349129.jpg" + } + ] + } + ], + "index": 34.0, + "virtual_lines": [ + { + "bbox": [ + 366, + 493, + 502, + 543.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 366, + 543.0, + 502, + 593.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 363, + 596, + 503, + 617 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 363, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 363, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "Figure 6: Top-1 accuracy on Ima-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 363, + 606, + 456, + 617 + ], + "spans": [ + { + "bbox": [ + 363, + 606, + 456, + 617 + ], + "score": 1.0, + "content": "geNet validation dataset.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "index": 36.75 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 357, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 356, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 356, + 584 + ], + "score": 1.0, + "content": "Invariance to discretization scheme A continuous model", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 357, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 357, + 595 + ], + "score": 1.0, + "content": "should be invariant to discretization schemes (e.g. different", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 357, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 357, + 605 + ], + "score": 1.0, + "content": "types of ODE solvers) as long as the discretization is suf-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 357, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 357, + 617 + ], + "score": 1.0, + "content": "ficiently accurate. We test the Neural ODE using different", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 616, + 357, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 357, + 628 + ], + "score": 1.0, + "content": "solvers without re-training; since ResNet is often viewed as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "a one-step Euler discretization of an ODE (Haber & Ruthotto, 2017), we perform similar exper-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 441, + 650 + ], + "score": 1.0, + "content": "iments. As shown in Table 2, Neural ODE consistently achieves high accuracy", + "type": "text" + }, + { + "bbox": [ + 442, + 638, + 474, + 649 + ], + "score": 0.87, + "content": "( \\sim 7 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 639, + 505, + 650 + ], + "score": 1.0, + "content": ", while", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 245, + 661 + ], + "score": 1.0, + "content": "ResNet drops to random guessing", + "type": "text" + }, + { + "bbox": [ + 246, + 649, + 281, + 660 + ], + "score": 0.89, + "content": "( \\sim 0 . 1 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "because ResNet as a one-step Euler discretization fails", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 353, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 353, + 673 + ], + "score": 1.0, + "content": "to be a meaningful dynamical system (Queiruga et al., 2020).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 572, + 357, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 672 + ], + "lines": [], + "index": 43.5, + "bbox_fs": [ + 104, + 626, + 505, + 673 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "Robustness to adversarial attack Hanshu et al. (2019) demonstrated that Neural ODE is more", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "robust to adversarial attack than ResNet on small-scale datasets such as Cifar10. We validate this", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "result on the large-scale ImageNet dataset. The top-1 accuracy of Neural ODE and ResNet under", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "score": 1.0, + "content": "FGSM attack (Goodfellow et al., 2014) are summarized in Table 3. For Neural ODE, due to its", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "invariance to discretization scheme, we derive the gradient for attack using a certain solver (row in", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 112, + 377, + 160 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 82, + 374, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 374, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 181, + 93 + ], + "score": 1.0, + "content": "Table 4: Test MSE", + "type": "text" + }, + { + "bbox": [ + 181, + 82, + 212, + 93 + ], + "score": 0.87, + "content": "( \\times 0 . 0 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 81, + 374, + 93 + ], + "score": 1.0, + "content": "on Mujoco dataset (lower is better). Results", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 371, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 371, + 104 + ], + "score": 1.0, + "content": "marked with superscript numbers correspond to literature in the footnote.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 112, + 377, + 160 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 112, + 377, + 160 + ], + "spans": [ + { + "bbox": [ + 108, + 112, + 377, + 160 + ], + "score": 0.91, + "html": "
Percentage of training dataRNN1RNN-GRU1Latent-ODE
Adjoint1Naive2ACA²MALI
10%2.4511.9720.4710.3620.3120.35
20%1.7111.4210.4410.30²0.2720.27
50%0.7910.7510.4010.2920.2620.26
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MethodAccuracy (%)
Adjoint392.8± 0.4
SemiNorm392.9 ±0.4
Naive ACA93.2±0.2
MALI93.2±0.2
93.7 ±0.3
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For different combinations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 452, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 452, + 201 + ], + "score": 1.0, + "content": "of solvers and perturbation amplitudes, Neural ODE consistently outperforms ResNet.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 205, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "Summary In image recognition tasks, we demonstrate Neural ODE is accurate, invariant to dis-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "cretization scheme, and more robust to adversarial attack than ResNet. Note that detailed explana-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 241 + ], + "score": 1.0, + "content": "tion on the robustness of Neural ODE is out of the scope for this paper, but to our knowledge, MALI", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 238, + 502, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 502, + 251 + ], + "score": 1.0, + "content": "is the first method to enable training of Neural ODE on large datasets due to constant memory cost.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 237, + 273 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 237, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 237, + 274 + ], + "score": 1.0, + "content": "4.3 TIME-SERIES MODELING", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "We apply MALI to latent-ODE (Rubanova et al., 2019) and Neural Controlled Differential Equation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "(Neural CDE) (Kidger et al., 2020a;b). Our experiment is based on the official implementation from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "the literature. We report the mean squared error (MSE) on the Mujoco test set in Table 4, which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "is generated from the “Hopper” model using DeepMind control suite (Tassa et al., 2018); for all", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "experiments with different ratios of training data, MALI achieves similar MSE to ACA, and both", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "outperform the adjoint and naive method. We report the test accuracy on the Speech Command", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 488, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 488, + 359 + ], + "score": 1.0, + "content": "dataset for Neural CDE in Table 5; MALI achieves a higher accuracy than competing methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 284, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 286, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 286, + 383 + ], + "score": 1.0, + "content": "4.4 CONTINUOUS GENERATIVE MODELS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 388, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "We apply MALI on FFJORD (Grathwohl et al., 2018), a free-from continuous generative model,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "and compare with several variants in the literature (Finlay et al., 2020; Kidger et al., 2020a). Our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "experiment is based on the official implementaion of Finlay et al. (2020); for a fair comparison, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "train with MALI, and test with the same solver as in the literature (Grathwohl et al., 2018; Finlay", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 431, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 104, + 431, + 253, + 445 + ], + "score": 1.0, + "content": "et al., 2020), the Dopri5 solver with", + "type": "text" + }, + { + "bbox": [ + 254, + 432, + 333, + 443 + ], + "score": 0.89, + "content": "\\mathrm { r t o l } = \\mathrm { a t o l } = 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 431, + 506, + 445 + ], + "score": 1.0, + "content": "from the torchdiffeq package (Chen et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "2018). Bits per dim (BPD, lower is better) on validation set for various datasets are reported in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "Table 6. For continuous models, MALI consistently generates the lowest BPD, and outperforms the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "Vanilla FFJORD (trained with adjoint), RNODE (regularized FFJORD) and the SemiNorm Adjoint", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "score": 1.0, + "content": "(Kidger et al., 2020a). Furthermore, FFJORD trained with MALI achieves comparable BPD to state-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 492, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 492, + 499 + ], + "score": 1.0, + "content": "of-the-art discrete-layer flow models in the literature. Please see Sec. B.3 for generated samples.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 505, + 215, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 216, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 216, + 519 + ], + "score": 1.0, + "content": "5 RELATED WORKS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "Besides ALF, the symplectic integrator (Verlet, 1967; Yoshida, 1990) is also able to reconstruct tra-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "jectory accurately, yet it’s typically restricted to second order Hamiltonian systems (De Almeida,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "1990), and are unsuitable for general ODEs. Besides aforementioned methods, there are other meth-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "ods for gradient estimation such as interpolated adjoint (Daulbaev et al., 2020) and spectral method", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "(Quaglino et al., 2019), yet the implementations are involved and not publicly available. Other works", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "focus on the theoretical properties of Neural ODEs (Dupont et al., 2019; Tabuada & Gharesifard,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "2020; Massaroli et al., 2020). Neural ODE is recently applied to stochastic differential equation (Li", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "et al., 2020), jump differential equation (Jia & Benson, 2019) and auto-regressive models (Wehenkel", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 180, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 180, + 626 + ], + "score": 1.0, + "content": "& Louppe, 2019).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 196, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 198, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 198, + 652 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 108, + 657, + 503, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "Based on the asynchronous leapfrog integrator, we propose MALI to estimate the gradient for Neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "ODEs. To our knowledge, our method is the first to achieve accuracy, fast speed and a constant", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "memory cost. We provide comprehensive theoretical analysis on its properties. We validate MALI", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 701, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 117, + 698, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 117, + 698, + 506, + 714 + ], + "score": 1.0, + "content": "01. Rubanova et al. (2019); 2. Zhuang et al. (2020); 3. Kidger et al. (2020a); 4. Chen et al. (2018); 5. Finlay", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "et al. (2020); 6. Dinh et al. (2016); 7. Behrmann et al. (2019); 8. Kingma & Dhariwal (2018); 9. Ho et al.", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 722, + 216, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 722, + 216, + 732 + ], + "score": 1.0, + "content": "(2019); 10. Chen et al. (2019)", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 112, + 377, + 160 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 82, + 374, + 103 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 374, + 93 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 181, + 93 + ], + "score": 1.0, + "content": "Table 4: Test MSE", + "type": "text" + }, + { + "bbox": [ + 181, + 82, + 212, + 93 + ], + "score": 0.87, + "content": "( \\times 0 . 0 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 81, + 374, + 93 + ], + "score": 1.0, + "content": "on Mujoco dataset (lower is better). Results", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 371, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 371, + 104 + ], + "score": 1.0, + "content": "marked with superscript numbers correspond to literature in the footnote.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.0 + }, + { + "type": "table_body", + "bbox": [ + 108, + 112, + 377, + 160 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 112, + 377, + 160 + ], + "spans": [ + { + "bbox": [ + 108, + 112, + 377, + 160 + ], + "score": 0.91, + "html": "
Percentage of training dataRNN1RNN-GRU1Latent-ODE
Adjoint1Naive2ACA²MALI
10%2.4511.9720.4710.3620.3120.35
20%1.7111.4210.4410.30²0.2720.27
50%0.7910.7510.4010.2920.2620.26
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MethodAccuracy (%)
Adjoint392.8± 0.4
SemiNorm392.9 ±0.4
Naive ACA93.2±0.2
MALI93.2±0.2
93.7 ±0.3
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For different combinations", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 452, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 452, + 201 + ], + "score": 1.0, + "content": "of solvers and perturbation amplitudes, Neural ODE consistently outperforms ResNet.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 177, + 505, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 205, + 505, + 249 + ], + "lines": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 217 + ], + "score": 1.0, + "content": "Summary In image recognition tasks, we demonstrate Neural ODE is accurate, invariant to dis-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 228 + ], + "score": 1.0, + "content": "cretization scheme, and more robust to adversarial attack than ResNet. Note that detailed explana-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 241 + ], + "score": 1.0, + "content": "tion on the robustness of Neural ODE is out of the scope for this paper, but to our knowledge, MALI", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 238, + 502, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 502, + 251 + ], + "score": 1.0, + "content": "is the first method to enable training of Neural ODE on large datasets due to constant memory cost.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 205, + 506, + 251 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 262, + 237, + 273 + ], + "lines": [ + { + "bbox": [ + 106, + 262, + 237, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 237, + 274 + ], + "score": 1.0, + "content": "4.3 TIME-SERIES MODELING", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "We apply MALI to latent-ODE (Rubanova et al., 2019) and Neural Controlled Differential Equation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "(Neural CDE) (Kidger et al., 2020a;b). Our experiment is based on the official implementation from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "the literature. We report the mean squared error (MSE) on the Mujoco test set in Table 4, which", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "is generated from the “Hopper” model using DeepMind control suite (Tassa et al., 2018); for all", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "experiments with different ratios of training data, MALI achieves similar MSE to ACA, and both", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "outperform the adjoint and naive method. We report the test accuracy on the Speech Command", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 488, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 488, + 359 + ], + "score": 1.0, + "content": "dataset for Neural CDE in Table 5; MALI achieves a higher accuracy than competing methods.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 281, + 506, + 359 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 284, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 370, + 286, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 286, + 383 + ], + "score": 1.0, + "content": "4.4 CONTINUOUS GENERATIVE MODELS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 388, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "We apply MALI on FFJORD (Grathwohl et al., 2018), a free-from continuous generative model,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "and compare with several variants in the literature (Finlay et al., 2020; Kidger et al., 2020a). Our", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 506, + 423 + ], + "score": 1.0, + "content": "experiment is based on the official implementaion of Finlay et al. (2020); for a fair comparison, we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 435 + ], + "score": 1.0, + "content": "train with MALI, and test with the same solver as in the literature (Grathwohl et al., 2018; Finlay", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 431, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 104, + 431, + 253, + 445 + ], + "score": 1.0, + "content": "et al., 2020), the Dopri5 solver with", + "type": "text" + }, + { + "bbox": [ + 254, + 432, + 333, + 443 + ], + "score": 0.89, + "content": "\\mathrm { r t o l } = \\mathrm { a t o l } = 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 431, + 506, + 445 + ], + "score": 1.0, + "content": "from the torchdiffeq package (Chen et al.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "2018). Bits per dim (BPD, lower is better) on validation set for various datasets are reported in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "Table 6. For continuous models, MALI consistently generates the lowest BPD, and outperforms the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 478 + ], + "score": 1.0, + "content": "Vanilla FFJORD (trained with adjoint), RNODE (regularized FFJORD) and the SemiNorm Adjoint", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 504, + 488 + ], + "score": 1.0, + "content": "(Kidger et al., 2020a). Furthermore, FFJORD trained with MALI achieves comparable BPD to state-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 488, + 492, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 492, + 499 + ], + "score": 1.0, + "content": "of-the-art discrete-layer flow models in the literature. Please see Sec. B.3 for generated samples.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 388, + 506, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 505, + 215, + 517 + ], + "lines": [ + { + "bbox": [ + 105, + 504, + 216, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 216, + 519 + ], + "score": 1.0, + "content": "5 RELATED WORKS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "Besides ALF, the symplectic integrator (Verlet, 1967; Yoshida, 1990) is also able to reconstruct tra-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "jectory accurately, yet it’s typically restricted to second order Hamiltonian systems (De Almeida,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "1990), and are unsuitable for general ODEs. Besides aforementioned methods, there are other meth-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "ods for gradient estimation such as interpolated adjoint (Daulbaev et al., 2020) and spectral method", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "(Quaglino et al., 2019), yet the implementations are involved and not publicly available. Other works", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "focus on the theoretical properties of Neural ODEs (Dupont et al., 2019; Tabuada & Gharesifard,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "2020; Massaroli et al., 2020). Neural ODE is recently applied to stochastic differential equation (Li", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "et al., 2020), jump differential equation (Jia & Benson, 2019) and auto-regressive models (Wehenkel", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 180, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 180, + 626 + ], + "score": 1.0, + "content": "& Louppe, 2019).", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 525, + 505, + 626 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 636, + 196, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 198, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 198, + 652 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 108, + 657, + 503, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 670 + ], + "score": 1.0, + "content": "Based on the asynchronous leapfrog integrator, we propose MALI to estimate the gradient for Neural", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "ODEs. To our knowledge, our method is the first to achieve accuracy, fast speed and a constant", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "memory cost. We provide comprehensive theoretical analysis on its properties. We validate MALI", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46, + "bbox_fs": [ + 106, + 656, + 505, + 691 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 108, + 514, + 154 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 105, + 81, + 505, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 6: Bits per dim (BPD) of generative models, lower is better. Results marked with superscript numbers", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 249, + 102 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 249, + 102 + ], + "score": 1.0, + "content": "correspond to literature in the footnote.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 108, + 514, + 154 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 108, + 514, + 154 + ], + "spans": [ + { + "bbox": [ + 108, + 108, + 514, + 154 + ], + "score": 0.977, + "html": "
DatasetContinuous Flow (FFJORD)Discrete Flow
Vanilla4RNODE5SemiNorm3MALIRealNVP6i-ResNetGlow8Flow++9Residual Flow10
MNIST0.9940.9750.9630.871.0661.0571.058-0.9710
CIFAR103.4043.3853.3533.273.4963.4573.3583.2893.2810
ImageNet64-3.835-3.713.986-3.818=3.7610
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Ffjord:", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 116, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 116, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "Free-form continuous dynamics for scalable reversible generative models. arXiv preprint", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 618, + 219, + 629 + ], + "spans": [ + { + "bbox": [ + 115, + 618, + 219, + 629 + ], + "score": 1.0, + "content": "arXiv:1810.01367, 2018.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 105, + 637, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "Eldad Haber and Lars Ruthotto. Stable architectures for deep neural networks. 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DatasetContinuous Flow (FFJORD)Discrete Flow
Vanilla4RNODE5SemiNorm3MALIRealNVP6i-ResNetGlow8Flow++9Residual Flow10
MNIST0.9940.9750.9630.871.0661.0571.058-0.9710
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Then the local truncation error is", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 106, + 691, + 505, + 716 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 719, + 398, + 733 + ], + "lines": [ + { + "bbox": [ + 213, + 719, + 398, + 733 + ], + "spans": [ + { + "bbox": [ + 213, + 719, + 398, + 733 + ], + "score": 0.92, + "content": "L _ { z } = \\widetilde { z } ( s _ { 0 } + h ) - \\widehat { z } _ { 2 } , L _ { v } = \\widetilde { v } ( s _ { 0 } + h ) - \\widehat { v _ { 2 } }", + "type": "interline_equation", + "image_path": "72b52fe4cf1e9af000a34b253222df4be318c9e16849722e5671385b08afc5e9.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 213, + 719, + 398, + 733 + ], + "spans": [], + "index": 33 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 316, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 317, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 158, + 95 + ], + "score": 1.0, + "content": "We estimate", + "type": "text" + }, + { + "bbox": [ + 158, + 83, + 170, + 93 + ], + "score": 0.89, + "content": "L _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 82, + 188, + 95 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 189, + 83, + 201, + 93 + ], + "score": 0.89, + "content": "L _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 82, + 307, + 95 + ], + "score": 1.0, + "content": "in terms of polynomial of", + "type": "text" + }, + { + "bbox": [ + 307, + 83, + 313, + 92 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 82, + 317, + 95 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 99, + 505, + 133 + ], + "lines": [ + { + "bbox": [ + 105, + 99, + 505, + 112 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 224, + 112 + ], + "score": 1.0, + "content": "Under mild assumptions that", + "type": "text" + }, + { + "bbox": [ + 224, + 100, + 231, + 111 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 99, + 505, + 112 + ], + "score": 1.0, + "content": "is smooth up to 2nd order almost everywhere (this is typically satis-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 108, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 104, + 108, + 478, + 125 + ], + "score": 1.0, + "content": "fied with neural networks with bounded weights), hence Taylor expansion is meaningful for", + "type": "text" + }, + { + "bbox": [ + 479, + 111, + 486, + 122 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 108, + 505, + 125 + ], + "score": 1.0, + "content": ". 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For simplicity, we directly analyze Eq. 7", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 210, + 263, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 252, + 223 + ], + "score": 1.0, + "content": "by performing Taylor Expansion on", + "type": "text" + }, + { + "bbox": [ + 252, + 210, + 258, + 222 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 210, + 263, + 223 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 229, + 458, + 254 + ], + "lines": [ + { + "bbox": [ + 152, + 229, + 458, + 254 + ], + "spans": [ + { + "bbox": [ + 152, + 229, + 458, + 254 + ], + "score": 0.78, + "content": "f ( \\widehat { z _ { 0 } } + \\frac { h } { 2 } \\widehat { v _ { 0 } } , s _ { 0 } + \\frac { h } { 2 } ) = f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\frac { h } { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\frac { h \\widehat { v _ { 0 } } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } )", + "type": "interline_equation", + "image_path": "28958d5e6239f9fab88eec4c0fbea5cadbd57b1ac83861623d584f94b0609b38.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 152, + 229, + 458, + 254 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 264, + 375, + 289 + ], + "lines": [ + { + "bbox": [ + 236, + 264, + 375, + 289 + ], + "spans": [ + { + "bbox": [ + 236, + 264, + 375, + 289 + ], + "score": 0.91, + "content": "\\widehat { z _ { 2 } } = \\widehat { z _ { 0 } } + h f ( \\widehat { z _ { 0 } } + \\frac { h } { 2 } \\widehat { v _ { 0 } } , s _ { 0 } + \\frac { h } { 2 } )", + "type": "interline_equation", + "image_path": "181b075a54a5221db6a9b62f44d80ac60875e1fb7542f24cd8fdfb40bae93119.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 236, + 264, + 375, + 289 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 364, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 365, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 319, + 308 + ], + "score": 1.0, + "content": "Plug Eq. 14, Eq. 15 and E.q. 16 into the definition of", + "type": "text" + }, + { + "bbox": [ + 319, + 294, + 331, + 305 + ], + "score": 0.89, + "content": "L _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 292, + 365, + 308 + ], + "score": 1.0, + "content": ", we get", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 312, + 444, + 405 + ], + "lines": [ + { + "bbox": [ + 165, + 312, + 444, + 405 + ], + "spans": [ + { + "bbox": [ + 165, + 312, + 444, + 405 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { L _ { z } = \\widetilde { z } ( s _ { 0 } + h ) - \\widehat { z _ { 2 } } } \\\\ & { \\quad = \\Big [ \\widehat { z _ { 0 } } + h f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h ^ { 2 } } { 2 } \\Big ( f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) \\Big ] } \\\\ & { \\quad - \\Big [ \\widehat { z _ { 0 } } + h \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h } { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h \\widehat { v _ { 0 } } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) \\Big ] + O ( h ^ { 3 } ) } \\\\ & { \\quad = \\displaystyle \\frac { h ^ { 2 } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } \\Big ) + O ( h ^ { 3 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "abd2ef4f727847efe0bb67ff0d8c150e972d53cff0b8483693e5105d4948750c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 165, + 312, + 444, + 343.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 165, + 343.0, + 444, + 374.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 165, + 374.0, + 444, + 405.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 506, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 160, + 429 + ], + "score": 1.0, + "content": "Therefore, if", + "type": "text" + }, + { + "bbox": [ + 160, + 411, + 225, + 432 + ], + "score": 0.92, + "content": "\\left| f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - 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\\widehat { z _ { 2 } } } \\\\ & { \\quad = \\Big [ \\widehat { z _ { 0 } } + h f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h ^ { 2 } } { 2 } \\Big ( f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) \\Big ] } \\\\ & { \\quad - \\Big [ \\widehat { z _ { 0 } } + h \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h } { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h \\widehat { v _ { 0 } } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) \\Big ] + O ( h ^ { 3 } ) } \\\\ & { \\quad = \\displaystyle \\frac { h ^ { 2 } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } \\Big ) + O ( h ^ { 3 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "abd2ef4f727847efe0bb67ff0d8c150e972d53cff0b8483693e5105d4948750c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 165, + 312, + 444, + 343.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 165, + 343.0, + 444, + 374.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 165, + 374.0, + 444, + 405.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 506, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 160, + 429 + ], + "score": 1.0, + "content": "Therefore, if", + "type": "text" + }, + { + "bbox": [ + 160, + 411, + 225, + 432 + ], + "score": 0.92, + "content": "\\left| f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - 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Denote the ground truth as", + "type": "text" + }, + { + "bbox": [ + 464, + 467, + 501, + 479 + ], + "score": 0.91, + "content": "\\widetilde { v } ( t _ { 0 } + h )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 465, + 505, + 480 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 477, + 142, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 142, + 490 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 465, + 505, + 490 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 150, + 495, + 460, + 529 + ], + "lines": [ + { + "bbox": [ + 150, + 495, + 460, + 529 + ], + "spans": [ + { + "bbox": [ + 150, + 495, + 460, + 529 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\widetilde { v } ( s _ { 0 } + h ) = f \\bigl ( \\widetilde { z } ( s _ { 0 } + h ) , s _ { 0 } + h \\bigr ) } \\\\ & { \\qquad = f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + h f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\bigl ( \\widetilde { z } ( s _ { 0 } + h ) - \\widehat { z _ { 0 } } \\bigr ) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "ee0755d57f7e5bf371beed44f97a77a36d4069aa1faf70e9b9a68a26909456c9.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 150, + 495, + 460, + 506.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 150, + 506.3333333333333, + 460, + 517.6666666666666 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 150, + 517.6666666666666, + 460, + 529.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 436, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 437, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 437, + 549 + ], + "score": 1.0, + "content": "Next we analyze the error in the numerical approximation. Plug Eq. 15 into Eq. 7,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 533, + 437, + 549 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 153, + 553, + 457, + 596 + ], + "lines": [ + { + "bbox": [ + 153, + 553, + 457, + 596 + ], + "spans": [ + { + "bbox": [ + 153, + 553, + 457, + 596 + ], + "score": 0.92, + "content": "\\begin{array} { l } { { \\displaystyle \\widehat { v _ { 2 } } = 2 f \\big ( \\widehat { z _ { 0 } } + \\frac { h } { 2 } \\widehat { v _ { 0 } } , s _ { 0 } + \\frac { h } { 2 } \\big ) - \\widehat { v _ { 0 } } } } \\\\ { { \\displaystyle \\quad = f \\big ( \\widehat { z _ { 0 } } , s _ { 0 } \\big ) + \\big ( f \\big ( \\widehat { z _ { 0 } } , s _ { 0 } \\big ) - \\widehat { v _ { 0 } } \\big ) + h f _ { t } \\big ( \\widehat { z _ { 0 } } , s _ { 0 } \\big ) + h \\widehat { v _ { 0 } } f _ { z } \\big ( \\widehat { z _ { 0 } } , s _ { 0 } \\big ) + O \\big ( h ^ { 2 } \\big ) } } \\end{array}", + "type": "interline_equation", + "image_path": "2f04555dd2bd88b7dda8150bb665f5ec14784347f738f394ee3d64d1a35286cd.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 153, + 553, + 457, + 567.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 153, + 567.3333333333334, + 457, + 581.6666666666667 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 153, + 581.6666666666667, + 457, + 596.0000000000001 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 601, + 272, + 613 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 272, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 272, + 614 + ], + "score": 1.0, + "content": "From Eq. 14, Eq. 21 and Eq. 23, we have", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 600, + 272, + 614 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 619, + 454, + 680 + ], + "lines": [ + { + "bbox": [ + 156, + 619, + 454, + 680 + ], + "spans": [ + { + "bbox": [ + 156, + 619, + 454, + 680 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { L _ { v } = \\widetilde { v } ( s _ { 0 } + h ) - \\widehat { v _ { 2 } } } \\\\ & { \\qquad = \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } \\Big ) + \\Big ( \\widetilde { z } ( s _ { 0 } + h ) - \\big ( \\widehat { z _ { 0 } } + h \\widehat { v _ { 0 } } \\big ) \\Big ) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\\\ & { \\qquad = \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } \\Big ) + h \\Big ( f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } \\Big ) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "e3c4196825177341e711aec19462062508d313a2f2b986ff00f7680c719e37ef.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 156, + 619, + 454, + 639.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 156, + 639.3333333333334, + 454, + 659.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 156, + 659.6666666666667, + 454, + 680.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 685, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 506, + 700 + ], + "score": 1.0, + "content": "The last equation is derived by plugging in Eq. 14. Note that Eq. 26 holds for every single step", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 696, + 506, + 710 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 366, + 709 + ], + "score": 1.0, + "content": "forward in time, and at the start time of integration, we have", + "type": "text" + }, + { + "bbox": [ + 366, + 696, + 457, + 710 + ], + "score": 0.92, + "content": "\\left| \\hat { f } ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } \\right| = \\mathbf { \\bar { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 696, + 506, + 709 + ], + "score": 1.0, + "content": "due to our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 708, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 401, + 722 + ], + "score": 1.0, + "content": "binitialization as in Sec. 3.1 of the main paper. Therefore, by induction,", + "type": "text" + }, + { + "bbox": [ + 401, + 711, + 414, + 721 + ], + "score": 0.86, + "content": "L _ { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 708, + 462, + 722 + ], + "score": 1.0, + "content": "bis of order", + "type": "text" + }, + { + "bbox": [ + 462, + 709, + 489, + 722 + ], + "score": 0.91, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 708, + 506, + 722 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 181, + 734 + ], + "score": 1.0, + "content": "consecutive steps.", + "type": "text" + }, + { + "bbox": [ + 493, + 721, + 505, + 731 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 684, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 225, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 226, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 226, + 95 + ], + "score": 1.0, + "content": "A.4 STABILITY ANALYSIS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 507, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 152, + 154 + ], + "score": 1.0, + "content": "Lemma A.shape, and", + "type": "text" + }, + { + "bbox": [ + 199, + 97, + 275, + 154 + ], + "score": 1.0, + "content": "matrix of the form , then we have det", + "type": "text" + }, + { + "bbox": [ + 277, + 100, + 309, + 127 + ], + "score": 0.8, + "content": "\\left[ \\begin{array} { l l } { A } & { B } \\\\ { C } & { D } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 107, + 369, + 119 + ], + "score": 0.6, + "content": "{ \\mathrm { : } } A , B , C , D", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 97, + 507, + 154 + ], + "score": 1.0, + "content": "square matrices of the same", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 152, + 127, + 388, + 152 + ], + "spans": [ + { + "bbox": [ + 152, + 133, + 199, + 143 + ], + "score": 0.88, + "content": "C D = D C", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 127, + 388, + 152 + ], + "score": 0.83, + "content": "{ \\left[ \\begin{array} { l l } { A } & { B } \\\\ { C } & { D } \\end{array} \\right] } = \\operatorname* { d e t } ( A D - B C )", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 302, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 303, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 303, + 176 + ], + "score": 1.0, + "content": "Proof. See (Silvester, 2000) for a detailed proof.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 303, + 194 + ], + "score": 1.0, + "content": "Theorem A.2. For ALF integrator with stepsize", + "type": "text" + }, + { + "bbox": [ + 303, + 181, + 310, + 191 + ], + "score": 0.6, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 180, + 323, + 194 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 323, + 181, + 338, + 191 + ], + "score": 0.7, + "content": "h \\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 180, + 349, + 194 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 349, + 182, + 356, + 191 + ], + "score": 0.44, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "or is imaginary with norm no larger", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 127, + 207 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 127, + 194, + 132, + 203 + ], + "score": 0.4, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 191, + 162, + 207 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 162, + 194, + 172, + 204 + ], + "score": 0.85, + "content": "\\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 191, + 321, + 207 + ], + "score": 1.0, + "content": "is the i-th eigenvalue of the Jacobian", + "type": "text" + }, + { + "bbox": [ + 322, + 192, + 334, + 206 + ], + "score": 0.89, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 191, + 506, + 207 + ], + "score": 1.0, + "content": ", then the solver is on the critical boundary", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 204, + 311, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 116, + 216 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 205, + 124, + 214 + ], + "score": 0.27, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 204, + 311, + 216 + ], + "score": 1.0, + "content": "-stability; otherwise, the solver is not A-stable.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "score": 1.0, + "content": "Proof. A solver is A-stable is equivalent to the eigenvalue of the numerical forward has a norm", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 306, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 267, + 251 + ], + "score": 1.0, + "content": "below 1. We calculate the eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 268, + 239, + 276, + 250 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 237, + 306, + 251 + ], + "score": 1.0, + "content": "below.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 255, + 305, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 255, + 304, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 304, + 268 + ], + "score": 1.0, + "content": "For the function defined by Eq. 7, the Jacobian is", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 272, + 396, + 315 + ], + "lines": [ + { + "bbox": [ + 214, + 272, + 396, + 315 + ], + "spans": [ + { + "bbox": [ + 214, + 272, + 396, + 315 + ], + "score": 0.94, + "content": "J = \\left[ \\begin{array} { c c } { \\frac { \\partial \\widehat { z } _ { 2 } } { \\partial z _ { 0 } } } & { \\frac { \\partial \\widehat { z } _ { 2 } } { \\partial \\widehat { v _ { 0 } } } } \\\\ { \\frac { \\partial \\widehat { v _ { 2 } } } { \\partial z _ { 0 } } } & { \\frac { \\partial \\widehat { v _ { 2 } } } { \\partial \\widehat { v _ { 0 } } } } \\end{array} \\right] = \\left[ \\begin{array} { c c } { I + h \\frac { \\partial f } { \\partial z } } & { \\frac { h ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } } \\\\ { 2 \\times \\frac { \\partial f } { \\partial z } } & { h \\frac { \\partial f } { \\partial z } - I } \\end{array} \\right]", + "type": "interline_equation", + "image_path": "4f02f625bda4636164067f941f775442cc9b6e6f992a594618471f70f1d06e0a.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 214, + 272, + 396, + 286.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 214, + 286.3333333333333, + 396, + 300.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 214, + 300.66666666666663, + 396, + 314.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 340, + 331 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 339, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 235, + 333 + ], + "score": 1.0, + "content": "We determine the eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 235, + 319, + 243, + 329 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 317, + 339, + 333 + ], + "score": 1.0, + "content": "by solving the equation", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 335, + 426, + 377 + ], + "lines": [ + { + "bbox": [ + 186, + 335, + 426, + 377 + ], + "spans": [ + { + "bbox": [ + 186, + 335, + 426, + 377 + ], + "score": 0.93, + "content": "\\operatorname* { d e t } ( J - \\lambda I ) = \\left[ \\begin{array} { c c } { h \\frac { \\partial f } { \\partial z } + ( 1 - \\lambda ) I } & { \\frac { h ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } } \\\\ { 2 \\times \\frac { \\partial f } { \\partial z } } & { h \\frac { \\partial f } { \\partial z } - ( 1 + \\lambda ) I } \\end{array} \\right] = 0", + "type": "interline_equation", + "image_path": "1f1acec2b3acc46091a666cfed235fa3b79114e1bfa06831ab79f8055ce86b74.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 186, + 335, + 426, + 349.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 186, + 349.0, + 426, + 363.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 186, + 363.0, + 426, + 377.0 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 423, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 422, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 183, + 394 + ], + "score": 1.0, + "content": "It’s trivial to check", + "type": "text" + }, + { + "bbox": [ + 184, + 382, + 191, + 391 + ], + "score": 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\\frac { \\partial f } { \\partial z } + ( \\lambda ^ { 2 } - 1 ) I \\Bigr ] } \\end{array}", + "type": "interline_equation", + "image_path": "a92835e051cb29e8add773e73ff7caadf9a8ac6f82191bcbafcb513c05cc0065.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 125, + 397, + 469, + 414.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 125, + 414.0, + 469, + 431.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 125, + 431.0, + 469, + 448.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 335, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 336, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 252, + 469 + ], + "score": 1.0, + "content": "Suppose the eigen-decompostion of", + "type": "text" + }, + { + "bbox": [ + 252, + 452, + 264, + 467 + ], + "score": 0.91, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 452, + 336, + 469 + ], + "score": 1.0, + "content": "can be written as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 229, + 472, + 383, + 519 + ], + "lines": [ + { + "bbox": [ + 229, + 472, + 383, + 519 + ], + "spans": [ + { + "bbox": [ + 229, + 472, + 383, + 519 + ], + "score": 0.93, + "content": "\\frac { \\partial f } { \\partial z } = \\Lambda \\left[ \\begin{array} { c c c c c } { { \\sigma _ { 1 } } } & { { } } & { { } } & { { } } & { { } } \\\\ { { } } & { { \\sigma _ { 2 } } } & { { } } & { { } } & { { } } \\\\ { { } } & { { } } & { { \\hdots } } & { { } } & { { } } \\\\ { { } } & { { } } & { { } } & { { } } & { { \\sigma _ { N } } } \\end{array} \\right] \\Lambda ^ { - 1 }", + "type": "interline_equation", + "image_path": "12a69c85a889f51aa62417c079669e64c1ad8262928b48c6d96f99d813ed4f1e.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 229, + 472, + 383, + 487.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 229, + 487.6666666666667, + 383, + 503.33333333333337 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 229, + 503.33333333333337, + 383, + 519.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 524, + 258, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 259, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 146, + 538 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 524, + 194, + 535 + ], + "score": 0.94, + "content": "I = \\Lambda I \\lambda ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 523, + 259, + 538 + ], + "score": 1.0, + "content": ", hence we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 541, + 458, + 624 + ], + "lines": [ + { + "bbox": [ + 152, + 541, + 458, + 624 + ], + "spans": [ + { + "bbox": [ + 152, + 541, + 458, + 624 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle \\operatorname * { d e t } ( J - \\lambda I ) = \\operatorname * { d e t } \\ \\Lambda \\Bigg \\{ - 2 \\lambda h \\left[ \\begin{array} { l l l l } { { } } & { { } } & { { } } & { { } } \\\\ { { } } & { { \\sigma _ { 2 } } } & { { } } & { { } } \\\\ { { } } & { { } } & { { \\cdots } } & { { } } \\\\ { { } } & { { } } & { { } } & { { \\sigma _ { N } } } \\end{array} \\right] + ( \\lambda ^ { 2 } - 1 ) I \\Bigg \\} \\Lambda ^ { - 1 } } } \\\\ { { \\displaystyle \\quad = \\prod _ { i = 1 } ^ { N } ( \\lambda ^ { 2 } - 2 h \\sigma _ { i } \\lambda - 1 ) } } \\end{array}", + "type": "interline_equation", + "image_path": "1d79a41742001a2706cb847874383a7b72924ad6e3b9bb407f5aff0aeb2975b8.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 152, + 541, + 458, + 568.6666666666666 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 152, + 568.6666666666666, + 458, + 596.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 152, + 596.3333333333333, + 458, + 623.9999999999999 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 212, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 213, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 213, + 641 + ], + "score": 1.0, + "content": "Hence the eigenvalues are", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 252, + 638, + 360, + 659 + ], + "lines": [ + { + "bbox": [ + 252, + 638, + 360, + 659 + ], + "spans": [ + { + "bbox": [ + 252, + 638, + 360, + 659 + ], + "score": 0.94, + "content": "\\lambda _ { i \\pm } = h \\sigma _ { i } \\pm \\sqrt { h ^ { 2 } \\sigma _ { i } ^ { 2 } + 1 }", + "type": "interline_equation", + "image_path": "b1e7bb129cf25bdd64bd2ce1b9fb48f7226a7814c18d5c1077d917c474a8f537.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 252, + 638, + 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"content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "15", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 163, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 495, + 164, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 495, + 164, + 505, + 174 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 225, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 226, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 226, + 95 + ], + "score": 1.0, + "content": "A.4 STABILITY ANALYSIS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 505, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 97, + 507, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 152, + 154 + ], + "score": 1.0, + "content": "Lemma A.shape, and", + "type": "text" + }, + { + "bbox": [ + 199, + 97, + 275, + 154 + ], + "score": 1.0, + "content": "matrix of the form , then we have det", + "type": "text" + }, + { + "bbox": [ + 277, + 100, + 309, + 127 + ], + "score": 0.8, + "content": "\\left[ \\begin{array} { l l } { A } & { B } \\\\ { C } & { D } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 107, + 369, + 119 + ], + "score": 0.6, + "content": "{ \\mathrm { : } } A , B , C , D", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 97, + 507, + 154 + ], + "score": 1.0, + "content": "square matrices of the same", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 152, + 127, + 388, + 152 + ], + "spans": [ + { + "bbox": [ + 152, + 133, + 199, + 143 + ], + "score": 0.88, + "content": "C D = D C", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 127, + 388, + 152 + ], + "score": 0.83, + "content": "{ \\left[ \\begin{array} { l l } { A } & { B } \\\\ { C } & { D } \\end{array} \\right] } = \\operatorname* { d e t } ( A D - B C )", + "type": "inline_equation" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 97, + 507, + 154 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 302, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 162, + 303, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 303, + 176 + ], + "score": 1.0, + "content": "Proof. See (Silvester, 2000) for a detailed proof.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 162, + 303, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 506, + 216 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 303, + 194 + ], + "score": 1.0, + "content": "Theorem A.2. For ALF integrator with stepsize", + "type": "text" + }, + { + "bbox": [ + 303, + 181, + 310, + 191 + ], + "score": 0.6, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 180, + 323, + 194 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 323, + 181, + 338, + 191 + ], + "score": 0.7, + "content": "h \\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 180, + 349, + 194 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 349, + 182, + 356, + 191 + ], + "score": 0.44, + "content": "O", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "or is imaginary with norm no larger", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 127, + 207 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 127, + 194, + 132, + 203 + ], + "score": 0.4, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 191, + 162, + 207 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 162, + 194, + 172, + 204 + ], + "score": 0.85, + "content": "\\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 191, + 321, + 207 + ], + "score": 1.0, + "content": "is the i-th eigenvalue of the Jacobian", + "type": "text" + }, + { + "bbox": [ + 322, + 192, + 334, + 206 + ], + "score": 0.89, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 191, + 506, + 207 + ], + "score": 1.0, + "content": ", then the solver is on the critical boundary", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 204, + 311, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 116, + 216 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 205, + 124, + 214 + ], + "score": 0.27, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 204, + 311, + 216 + ], + "score": 1.0, + "content": "-stability; otherwise, the solver is not A-stable.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 180, + 506, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 227, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "score": 1.0, + "content": "Proof. A solver is A-stable is equivalent to the eigenvalue of the numerical forward has a norm", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 306, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 267, + 251 + ], + "score": 1.0, + "content": "below 1. 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When", + "type": "text" + }, + { + "bbox": [ + 419, + 513, + 443, + 524 + ], + "score": 0.91, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 512, + 506, + 525 + ], + "score": 1.0, + "content": ", Damped ALF", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 523, + 172, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 172, + 534 + ], + "score": 1.0, + "content": "reduces to ALF.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 489, + 506, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 504, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 553 + ], + "score": 1.0, + "content": "Similar to Sec. 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The proof is similar to Thm. A.3. By similar calculations using the Taylor Expansion in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 218, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 218, + 107 + ], + "score": 1.0, + "content": "Eq. 15 and Eq. 14, we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 110, + 465, + 210 + ], + "lines": [ + { + "bbox": [ + 144, + 110, + 465, + 210 + ], + "spans": [ + { + "bbox": [ + 144, + 110, + 465, + 210 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\widehat { z _ { 2 } } - \\widetilde { z } ( s _ { 0 } + h ) = ( 1 - \\eta ) h \\widehat { v _ { 0 } } + h \\eta \\Big [ f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h } { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h \\widehat { v _ { 0 } } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ] } \\\\ & { \\qquad - h \\Big [ f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h } { 2 } f _ { t } \\widehat { z _ { 0 } } , s _ { 0 } + \\displaystyle \\frac { h } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ] + O ( h ^ { 2 } ) } \\\\ & { \\qquad = ( 1 - \\eta ) h \\Big ( \\widehat { v _ { 0 } } - f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) + \\displaystyle \\frac { \\eta - 1 } { 2 } h ^ { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) } \\\\ & { \\qquad + \\displaystyle \\frac { h ^ { 2 } } { 2 } \\Big ( \\eta \\widehat { v _ { 0 } } - f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "f3d7a08114952338ffc55428195ac40d8c5319086fcea1a0df17656c63d0e0c9.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 144, + 110, + 465, + 143.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 144, + 143.33333333333334, + 465, + 176.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 144, + 176.66666666666669, + 465, + 210.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 275, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 275, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 275, + 226 + ], + "score": 1.0, + "content": "Using Eq. 21, Eq. 15 and Eq. 14, we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 228, + 436, + 306 + ], + "lines": [ + { + "bbox": [ + 173, + 228, + 436, + 306 + ], + "spans": [ + { + "bbox": [ + 173, + 228, + 436, + 306 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\widetilde { v _ { 2 } } - \\widehat { v _ { 2 } } = ( 1 - 2 \\eta ) \\widehat { v _ { 0 } } + ( 2 \\eta - 1 ) f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + ( 1 - \\eta ) h f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) } \\\\ & { \\qquad + \\left( \\widetilde { z } ( s _ { 0 } + h ) - \\widehat { z _ { 0 } } - \\eta h \\widehat { v _ { 0 } } \\right) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\\\ & { \\qquad = ( 2 \\eta - 1 ) \\big [ f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { z _ { 0 } } \\big ] + ( 1 - \\eta ) h f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) } \\\\ & { \\qquad + \\eta \\Big [ h f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - h \\widehat { v _ { 0 } } \\Big ] f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "c3603e3fa15591320cac24097437c6a07cb3176448c5515bc3d0e8240a1cc0a0.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 173, + 228, + 436, + 254.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 173, + 254.0, + 436, + 280.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 173, + 280.0, + 436, + 306.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 171, + 322 + ], + "score": 1.0, + "content": "Note that when", + "type": "text" + }, + { + "bbox": [ + 172, + 309, + 198, + 320 + ], + "score": 0.88, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 307, + 505, + 322 + ], + "score": 1.0, + "content": ", Eq. 51 reduces to Eq. 19, and Eq. 53 reduces to Eq. 26. By initialization,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 142, + 333 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 143, + 320, + 227, + 332 + ], + "score": 0.91, + "content": "| f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } | = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 320, + 488, + 333 + ], + "score": 1.0, + "content": "at initial time, hence by induction, the local truncation error for", + "type": "text" + }, + { + "bbox": [ + 488, + 322, + 495, + 330 + ], + "score": 0.8, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 107, + 330, + 133, + 343 + ], + "score": 0.92, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 330, + 253, + 343 + ], + "score": 1.0, + "content": "b b; the local truncation error for", + "type": "text" + }, + { + "bbox": [ + 253, + 333, + 259, + 341 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 330, + 270, + 343 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 270, + 331, + 292, + 343 + ], + "score": 0.92, + "content": "O ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 330, + 317, + 343 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 318, + 331, + 342, + 343 + ], + "score": 0.9, + "content": "\\eta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 330, + 372, + 343 + ], + "score": 1.0, + "content": ", and is", + "type": "text" + }, + { + "bbox": [ + 372, + 330, + 399, + 343 + ], + "score": 0.92, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 330, + 424, + 343 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 425, + 331, + 449, + 343 + ], + "score": 0.9, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 330, + 453, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 332, + 505, + 342 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 105, + 348, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 371, + 362 + ], + "score": 1.0, + "content": "Theorem A.4 (Theorem 3.2 in the main paper). For Dampled", + "type": "text" + }, + { + "bbox": [ + 372, + 349, + 392, + 359 + ], + "score": 0.25, + "content": "A L F", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 348, + 494, + 362 + ], + "score": 1.0, + "content": "integrator with stepsize", + "type": "text" + }, + { + "bbox": [ + 494, + 349, + 501, + 359 + ], + "score": 0.67, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 348, + 505, + 362 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 356, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 134, + 383 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 365, + 145, + 375 + ], + "score": 0.82, + "content": "\\sigma _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 356, + 174, + 383 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 174, + 365, + 178, + 374 + ], + "score": 0.33, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 356, + 307, + 383 + ], + "score": 1.0, + "content": "-th eigenvalue of the Jacobian", + "type": "text" + }, + { + "bbox": [ + 307, + 361, + 319, + 377 + ], + "score": 0.89, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 356, + 400, + 383 + ], + "score": 1.0, + "content": ", then the solver is", + "type": "text" + }, + { + "bbox": [ + 401, + 365, + 408, + 374 + ], + "score": 0.47, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 356, + 438, + 383 + ], + "score": 1.0, + "content": "-stable", + "type": "text" + }, + { + "bbox": [ + 439, + 360, + 505, + 380 + ], + "score": 0.88, + "content": "i f | 1 + \\eta ( h \\sigma -", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 378, + 274, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 274, + 400 + ], + "score": 0.89, + "content": "1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\Big \\vert < 1 , \\forall i .", + "type": "inline_equation", + "image_path": "ab456ca40601de0d893650b801ced61cf2f0719851cbe8cde610e397015b4d5f.jpg" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 398, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 409, + 399, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 399, + 424 + ], + "score": 1.0, + "content": "Proof. The Jacobian of the forward-pass of a single step damped ALF is", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 427, + 388, + 469 + ], + "lines": [ + { + "bbox": [ + 221, + 427, + 388, + 469 + ], + "spans": [ + { + "bbox": [ + 221, + 427, + 388, + 469 + ], + "score": 0.93, + "content": "J = \\left[ \\begin{array} { c c } { I + \\eta h \\frac { \\partial f } { \\partial z } } & { ( 1 - \\eta ) h I + \\eta \\frac { h ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } } \\\\ { 2 \\eta \\frac { \\partial f } { \\partial z } } & { \\eta h \\frac { \\partial f } { \\partial z } + ( 1 - 2 \\eta ) I } \\end{array} \\right]", + "type": "interline_equation", + "image_path": "0bdba530a638268404b217328de9b3538e5294c5ea6da417ebe5965c1d084114.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 221, + 427, + 388, + 441.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 221, + 441.0, + 388, + 455.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 221, + 455.0, + 388, + 469.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 131, + 486 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 131, + 473, + 156, + 484 + ], + "score": 0.86, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 472, + 161, + 486 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 161, + 474, + 169, + 483 + ], + "score": 0.73, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 472, + 393, + 486 + ], + "score": 1.0, + "content": "reduces to Eq. 27. We can determine the eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 393, + 473, + 401, + 483 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "using similar techniques.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 481, + 327, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 219, + 500 + ], + "score": 1.0, + "content": "Assume the eigenvalues for", + "type": "text" + }, + { + "bbox": [ + 219, + 484, + 231, + 498 + ], + "score": 0.91, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 481, + 248, + 500 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 248, + 484, + 268, + 497 + ], + "score": 0.92, + "content": "\\{ \\sigma _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 481, + 327, + 500 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 503, + 445, + 630 + ], + "lines": [ + { + "bbox": [ + 165, + 503, + 445, + 630 + ], + "spans": [ + { + "bbox": [ + 165, + 503, + 445, + 630 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { d e t } ( J - \\lambda I ) = \\operatorname* { d e t } \\left[ \\begin{array} { l l } { ( 1 - \\lambda ) I + \\eta h \\frac { \\partial f } { \\partial z } } & { ( 1 - \\eta ) h I + \\eta \\frac { h ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } } \\\\ { \\displaystyle \\qquad } & { \\eta h \\frac { \\partial f } { \\partial z } + ( 1 - 2 \\eta - \\lambda ) I } \\end{array} \\right] } \\\\ { \\displaystyle = \\operatorname* { d e t } \\left[ \\Big ( ( 1 - \\lambda ) I + \\eta h \\frac { \\partial f } { \\partial z } \\Big ) \\Big ( \\eta h \\frac { \\partial f } { \\partial z } + ( 1 - 2 \\eta - \\lambda ) I \\Big ) \\right. } \\\\ { \\displaystyle - \\left( ( 1 - \\eta ) h I + \\eta h \\frac { \\partial ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } \\Big ) \\Big ( 2 \\eta \\frac { \\partial f } { \\partial z } \\Big ) \\right] } \\\\ { \\displaystyle = \\prod _ { i = 1 } ^ { N } \\left[ 1 + \\eta ( h \\sigma _ { i } - 1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\right] } \\end{array}", + "type": "interline_equation", + "image_path": "4f5a63a58a85afd4f1df6307ea958525539e2a27f9d6ae4ba1ee08abd927ba5d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 165, + 503, + 445, + 545.3333333333334 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 165, + 545.3333333333334, + 445, + 587.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 165, + 587.6666666666667, + 445, + 630.0000000000001 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 635, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 103, + 633, + 507, + 655 + ], + "spans": [ + { + "bbox": [ + 103, + 633, + 130, + 655 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 640, + 155, + 651 + ], + "score": 0.91, + "content": "\\eta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 633, + 245, + 655 + ], + "score": 1.0, + "content": ", it’s easy to check that", + "type": "text" + }, + { + "bbox": [ + 246, + 635, + 443, + 655 + ], + "score": 0.88, + "content": "\\left| 1 + \\eta ( h \\sigma _ { i } - 1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\right| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 633, + 507, + 655 + ], + "score": 1.0, + "content": "has non-empty", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 652, + 177, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 159, + 666 + ], + "score": 1.0, + "content": "solutions for", + "type": "text" + }, + { + "bbox": [ + 159, + 654, + 172, + 663 + ], + "score": 0.81, + "content": "h \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 652, + 177, + 666 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "For a quick validation, we plot the region of A-stability on the imaginary plane for a single eigen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 183, + 701 + ], + "score": 1.0, + "content": "value in Fig. 1. As", + "type": "text" + }, + { + "bbox": [ + 183, + 690, + 190, + 700 + ], + "score": 0.82, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 689, + 382, + 701 + ], + "score": 1.0, + "content": "increases, the area of stability decreases. When", + "type": "text" + }, + { + "bbox": [ + 382, + 689, + 407, + 699 + ], + "score": 0.9, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 689, + 505, + 701 + ], + "score": 1.0, + "content": ", the system is no-where", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 377, + 713 + ], + "score": 1.0, + "content": "A-stable, and the boundary for A-stability is on the imaginary axis", + "type": "text" + }, + { + "bbox": [ + 378, + 700, + 403, + 712 + ], + "score": 0.92, + "content": "[ - i , i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 699, + 431, + 713 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 432, + 700, + 437, + 709 + ], + "score": 0.76, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "is the imaginary", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 711, + 128, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 128, + 722 + ], + "score": 1.0, + "content": "unit.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Proof. The proof is similar to Thm. A.3. By similar calculations using the Taylor Expansion in", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 218, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 218, + 107 + ], + "score": 1.0, + "content": "Eq. 15 and Eq. 14, we have", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 81, + 505, + 107 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 110, + 465, + 210 + ], + "lines": [ + { + "bbox": [ + 144, + 110, + 465, + 210 + ], + "spans": [ + { + "bbox": [ + 144, + 110, + 465, + 210 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\widehat { z _ { 2 } } - \\widetilde { z } ( s _ { 0 } + h ) = ( 1 - \\eta ) h \\widehat { v _ { 0 } } + h \\eta \\Big [ f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h } { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h \\widehat { v _ { 0 } } } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ] } \\\\ & { \\qquad - h \\Big [ f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + \\displaystyle \\frac { h } { 2 } f _ { t } \\widehat { z _ { 0 } } , s _ { 0 } + \\displaystyle \\frac { h } { 2 } f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ] + O ( h ^ { 2 } ) } \\\\ & { \\qquad = ( 1 - \\eta ) h \\Big ( \\widehat { v _ { 0 } } - f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) + \\displaystyle \\frac { \\eta - 1 } { 2 } h ^ { 2 } f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) } \\\\ & { \\qquad + \\displaystyle \\frac { h ^ { 2 } } { 2 } \\Big ( \\eta \\widehat { v _ { 0 } } - f ( \\widehat { z _ { 0 } } , s _ { 0 } ) \\Big ) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "f3d7a08114952338ffc55428195ac40d8c5319086fcea1a0df17656c63d0e0c9.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 144, + 110, + 465, + 143.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 144, + 143.33333333333334, + 465, + 176.66666666666669 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 144, + 176.66666666666669, + 465, + 210.00000000000003 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 212, + 275, + 225 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 275, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 275, + 226 + ], + "score": 1.0, + "content": "Using Eq. 21, Eq. 15 and Eq. 14, we have", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 106, + 213, + 275, + 226 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 228, + 436, + 306 + ], + "lines": [ + { + "bbox": [ + 173, + 228, + 436, + 306 + ], + "spans": [ + { + "bbox": [ + 173, + 228, + 436, + 306 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\widetilde { v _ { 2 } } - \\widehat { v _ { 2 } } = ( 1 - 2 \\eta ) \\widehat { v _ { 0 } } + ( 2 \\eta - 1 ) f ( \\widehat { z _ { 0 } } , s _ { 0 } ) + ( 1 - \\eta ) h f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) } \\\\ & { \\qquad + \\left( \\widetilde { z } ( s _ { 0 } + h ) - \\widehat { z _ { 0 } } - \\eta h \\widehat { v _ { 0 } } \\right) f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\\\ & { \\qquad = ( 2 \\eta - 1 ) \\big [ f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { z _ { 0 } } \\big ] + ( 1 - \\eta ) h f _ { t } ( \\widehat { z _ { 0 } } , s _ { 0 } ) } \\\\ & { \\qquad + \\eta \\Big [ h f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - h \\widehat { v _ { 0 } } \\Big ] f _ { z } ( \\widehat { z _ { 0 } } , s _ { 0 } ) + O ( h ^ { 2 } ) } \\end{array}", + "type": "interline_equation", + "image_path": "c3603e3fa15591320cac24097437c6a07cb3176448c5515bc3d0e8240a1cc0a0.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 173, + 228, + 436, + 254.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 173, + 254.0, + 436, + 280.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 173, + 280.0, + 436, + 306.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 506, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 171, + 322 + ], + "score": 1.0, + "content": "Note that when", + "type": "text" + }, + { + "bbox": [ + 172, + 309, + 198, + 320 + ], + "score": 0.88, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 307, + 505, + 322 + ], + "score": 1.0, + "content": ", Eq. 51 reduces to Eq. 19, and Eq. 53 reduces to Eq. 26. By initialization,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 142, + 333 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 143, + 320, + 227, + 332 + ], + "score": 0.91, + "content": "| f ( \\widehat { z _ { 0 } } , s _ { 0 } ) - \\widehat { v _ { 0 } } | = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 320, + 488, + 333 + ], + "score": 1.0, + "content": "at initial time, hence by induction, the local truncation error for", + "type": "text" + }, + { + "bbox": [ + 488, + 322, + 495, + 330 + ], + "score": 0.8, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 107, + 330, + 133, + 343 + ], + "score": 0.92, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 330, + 253, + 343 + ], + "score": 1.0, + "content": "b b; the local truncation error for", + "type": "text" + }, + { + "bbox": [ + 253, + 333, + 259, + 341 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 330, + 270, + 343 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 270, + 331, + 292, + 343 + ], + "score": 0.92, + "content": "O ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 330, + 317, + 343 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 318, + 331, + 342, + 343 + ], + "score": 0.9, + "content": "\\eta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 330, + 372, + 343 + ], + "score": 1.0, + "content": ", and is", + "type": "text" + }, + { + "bbox": [ + 372, + 330, + 399, + 343 + ], + "score": 0.92, + "content": "O ( h ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 330, + 424, + 343 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 425, + 331, + 449, + 343 + ], + "score": 0.9, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 330, + 453, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 332, + 505, + 342 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 307, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 348, + 505, + 400 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 371, + 362 + ], + "score": 1.0, + "content": "Theorem A.4 (Theorem 3.2 in the main paper). 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The Jacobian of the forward-pass of a single step damped ALF is", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 409, + 399, + 424 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 427, + 388, + 469 + ], + "lines": [ + { + "bbox": [ + 221, + 427, + 388, + 469 + ], + "spans": [ + { + "bbox": [ + 221, + 427, + 388, + 469 + ], + "score": 0.93, + "content": "J = \\left[ \\begin{array} { c c } { I + \\eta h \\frac { \\partial f } { \\partial z } } & { ( 1 - \\eta ) h I + \\eta \\frac { h ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } } \\\\ { 2 \\eta \\frac { \\partial f } { \\partial z } } & { \\eta h \\frac { \\partial f } { \\partial z } + ( 1 - 2 \\eta ) I } \\end{array} \\right]", + "type": "interline_equation", + "image_path": "0bdba530a638268404b217328de9b3538e5294c5ea6da417ebe5965c1d084114.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 221, + 427, + 388, + 441.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 221, + 441.0, + 388, + 455.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 221, + 455.0, + 388, + 469.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 472, + 505, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 131, + 486 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 131, + 473, + 156, + 484 + ], + "score": 0.86, + "content": "\\eta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 472, + 161, + 486 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 161, + 474, + 169, + 483 + ], + "score": 0.73, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 472, + 393, + 486 + ], + "score": 1.0, + "content": "reduces to Eq. 27. We can determine the eigenvalue of", + "type": "text" + }, + { + "bbox": [ + 393, + 473, + 401, + 483 + ], + "score": 0.84, + "content": "J", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "using similar techniques.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 481, + 327, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 219, + 500 + ], + "score": 1.0, + "content": "Assume the eigenvalues for", + "type": "text" + }, + { + "bbox": [ + 219, + 484, + 231, + 498 + ], + "score": 0.91, + "content": "\\frac { \\partial f } { \\partial z }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 481, + 248, + 500 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 248, + 484, + 268, + 497 + ], + "score": 0.92, + "content": "\\{ \\sigma _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 481, + 327, + 500 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5, + "bbox_fs": [ + 106, + 472, + 505, + 500 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 503, + 445, + 630 + ], + "lines": [ + { + "bbox": [ + 165, + 503, + 445, + 630 + ], + "spans": [ + { + "bbox": [ + 165, + 503, + 445, + 630 + ], + "score": 0.95, + "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { d e t } ( J - \\lambda I ) = \\operatorname* { d e t } \\left[ \\begin{array} { l l } { ( 1 - \\lambda ) I + \\eta h \\frac { \\partial f } { \\partial z } } & { ( 1 - \\eta ) h I + \\eta \\frac { h ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } } \\\\ { \\displaystyle \\qquad } & { \\eta h \\frac { \\partial f } { \\partial z } + ( 1 - 2 \\eta - \\lambda ) I } \\end{array} \\right] } \\\\ { \\displaystyle = \\operatorname* { d e t } \\left[ \\Big ( ( 1 - \\lambda ) I + \\eta h \\frac { \\partial f } { \\partial z } \\Big ) \\Big ( \\eta h \\frac { \\partial f } { \\partial z } + ( 1 - 2 \\eta - \\lambda ) I \\Big ) \\right. } \\\\ { \\displaystyle - \\left( ( 1 - \\eta ) h I + \\eta h \\frac { \\partial ^ { 2 } } { 2 } \\frac { \\partial f } { \\partial z } \\Big ) \\Big ( 2 \\eta \\frac { \\partial f } { \\partial z } \\Big ) \\right] } \\\\ { \\displaystyle = \\prod _ { i = 1 } ^ { N } \\left[ 1 + \\eta ( h \\sigma _ { i } - 1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\right] } \\end{array}", + "type": "interline_equation", + "image_path": "4f5a63a58a85afd4f1df6307ea958525539e2a27f9d6ae4ba1ee08abd927ba5d.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 165, + 503, + 445, + 545.3333333333334 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 165, + 545.3333333333334, + 445, + 587.6666666666667 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 165, + 587.6666666666667, + 445, + 630.0000000000001 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 635, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 103, + 633, + 507, + 655 + ], + "spans": [ + { + "bbox": [ + 103, + 633, + 130, + 655 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 640, + 155, + 651 + ], + "score": 0.91, + "content": "\\eta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 633, + 245, + 655 + ], + "score": 1.0, + "content": ", it’s easy to check that", + "type": "text" + }, + { + "bbox": [ + 246, + 635, + 443, + 655 + ], + "score": 0.88, + "content": "\\left| 1 + \\eta ( h \\sigma _ { i } - 1 ) \\pm \\sqrt { \\eta \\big [ 2 h \\sigma _ { i } + \\eta ( h \\sigma _ { i } - 1 ) ^ { 2 } \\big ] } \\right| < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 633, + 507, + 655 + ], + "score": 1.0, + "content": "has non-empty", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 652, + 177, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 159, + 666 + ], + "score": 1.0, + "content": "solutions for", + "type": "text" + }, + { + "bbox": [ + 159, + 654, + 172, + 663 + ], + "score": 0.81, + "content": "h \\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 652, + 177, + 666 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 103, + 633, + 507, + 666 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "For a quick validation, we plot the region of A-stability on the imaginary plane for a single eigen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 183, + 701 + ], + "score": 1.0, + "content": "value in Fig. 1. 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From left", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 244, + 238 + ], + "score": 1.0, + "content": "to right, the region of stability for", + "type": "text" + }, + { + "bbox": [ + 244, + 226, + 282, + 237 + ], + "score": 0.89, + "content": "\\eta = 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 226, + 286, + 238 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 286, + 226, + 354, + 237 + ], + "score": 0.9, + "content": "\\eta = 0 . 7 , \\eta = 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 226, + 422, + 238 + ], + "score": 1.0, + "content": "respectively. As", + "type": "text" + }, + { + "bbox": [ + 423, + 227, + 429, + 237 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "increases to 1, the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 237, + 242, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 242, + 249 + ], + "score": 1.0, + "content": "area of stability region decreases.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 267, + 257, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 258, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 258, + 282 + ], + "score": 1.0, + "content": "B EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 291, + 226, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 227, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 227, + 304 + ], + "score": 1.0, + "content": "B.1 IMAGE RECOGNITION", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 312, + 254, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 255, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 255, + 325 + ], + "score": 1.0, + "content": "B.1.1 EXPERIMENT ON CIFAR10", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 483, + 345 + ], + "score": 1.0, + "content": "We directly modify a ResNet18 into a Neural ODE, where the forward of a residual block", + "type": "text" + }, + { + "bbox": [ + 483, + 333, + 505, + 344 + ], + "score": 0.8, + "content": "( y =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 342, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 148, + 357 + ], + "score": 0.92, + "content": "x + f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 344, + 289, + 359 + ], + "score": 1.0, + "content": "and the forward of an ODE block", + "type": "text" + }, + { + "bbox": [ + 289, + 342, + 381, + 358 + ], + "score": 0.93, + "content": "\\begin{array} { r } { ( y = x + \\int _ { 0 } ^ { T } f ( z , t ) d t } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 344, + 410, + 359 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 410, + 345, + 439, + 356 + ], + "score": 0.87, + "content": "T = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 344, + 506, + 359 + ], + "score": 1.0, + "content": ") share the same", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 176, + 368 + ], + "score": 1.0, + "content": "parameterization", + "type": "text" + }, + { + "bbox": [ + 177, + 357, + 183, + 367 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 356, + 505, + 368 + ], + "score": 1.0, + "content": ", hence they have the same number of parameters. Our experiment is based on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 367, + 487, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 487, + 380 + ], + "score": 1.0, + "content": "the official implementation by Zhuang et al. (2020) and an open-source repository (Liu, 2017).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "All models are trained with SGD optimizer for 90 epochs, with an initial learning rate of 0.01, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "decayed by a factor of 10 at 30th epoch and 60th epoch respectively. Training scheme is the same", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "for all models (ResNet, Neural ODE trained with adjoint, naive, ACA and MALI). For ACA, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "follow the settings in (Zhuang et al., 2020) and use the official implementation torch ACA 1, and use", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 208, + 440 + ], + "score": 1.0, + "content": "a Heun-Euler solver with", + "type": "text" + }, + { + "bbox": [ + 209, + 428, + 260, + 438 + ], + "score": 0.85, + "content": "r t o \\bar { l } = 1 0 ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 427, + 263, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 263, + 427, + 315, + 438 + ], + "score": 0.87, + "content": "a t o l = 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "during training. 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For the naive and adjoint method, we use the default", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 449, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 311, + 462 + ], + "score": 1.0, + "content": "Dopri5 solver from the torchdiffeq2 package with", + "type": "text" + }, + { + "bbox": [ + 311, + 450, + 393, + 460 + ], + "score": 0.88, + "content": "\\mathrm { r t o l } = \\mathrm { a t o l } = \\mathrm { \\bar { 1 0 } } ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 449, + 507, + 462 + ], + "score": 1.0, + "content": ". We train all models for 5", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 460, + 404, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 404, + 473 + ], + "score": 1.0, + "content": "independent runs, and report the mean and standard deviation across runs.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 484, + 268, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 270, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 270, + 497 + ], + "score": 1.0, + "content": "B.1.2 EXPERIMENTS ON IMAGENET", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "Training scheme We conduct experiments on ImageNet with ResNet18 and Neural-ODE18. All", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "models are trained on 4 GTX-1080Ti GPUs with a batchsize of 256. All models are trained for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 444, + 538 + ], + "score": 1.0, + "content": "80 epochs, with an initial learning rate of 0.1, and decayed by a factor of 10 at", + "type": "text" + }, + { + "bbox": [ + 445, + 525, + 464, + 536 + ], + "score": 0.28, + "content": "3 0 \\mathrm { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "and 60th", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 282, + 550 + ], + "score": 1.0, + "content": "epoch. Note that due to the large size input", + "type": "text" + }, + { + "bbox": [ + 282, + 536, + 326, + 547 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 535, + 506, + 550 + ], + "score": 1.0, + "content": ", the naive method and ACA requires a huge", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "memory, and is infeasible to train. MALI and the adjoint method requires a constant memory hence", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "is suitable for large-scale experiments. For both MALI and the adjoint menthod, we use a fixed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 271, + 582 + ], + "score": 1.0, + "content": "stepsize of 0.25, and integrates from 0 to", + "type": "text" + }, + { + "bbox": [ + 272, + 569, + 299, + 579 + ], + "score": 0.89, + "content": "T = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 568, + 505, + 582 + ], + "score": 1.0, + "content": ". As shown in Table. 2 in the main paper, a stepsize", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "of 0.25 is sufficiently small to train a meaningful continuous model that is robust to discretization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 591, + 142, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 142, + 604 + ], + "score": 1.0, + "content": "scheme.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "Invariance to discretization scheme To test the influence of discretization scheme, we test our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "score": 1.0, + "content": "Neural ODE with different solvers without re-training. For fixed-stepsize solvers, we tested various", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 190, + 648 + ], + "score": 1.0, + "content": "step sizes including", + "type": "text" + }, + { + "bbox": [ + 190, + 636, + 292, + 649 + ], + "score": 0.89, + "content": "\\{ 0 . 1 , 0 . 1 5 , 0 . 2 5 , 0 . 5 , 1 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 636, + 426, + 648 + ], + "score": 1.0, + "content": "; for adaptive solvers, we set rto", + "type": "text" + }, + { + "bbox": [ + 426, + 637, + 446, + 647 + ], + "score": 0.65, + "content": "\\scriptstyle \\mathbf { - 0 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 636, + 465, + 648 + ], + "score": 1.0, + "content": ", atol", + "type": "text" + }, + { + "bbox": [ + 465, + 637, + 488, + 647 + ], + "score": 0.52, + "content": "= 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 281, + 660 + ], + "score": 1.0, + "content": "MALI and Heun-Euler method, and set rto", + "type": "text" + }, + { + "bbox": [ + 282, + 647, + 319, + 658 + ], + "score": 0.76, + "content": "1 \\stackrel { { \\textstyle \\sum } } { = } 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 646, + 322, + 660 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 322, + 647, + 374, + 658 + ], + "score": 0.76, + "content": "\\mathrm { a t o l } = 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 646, + 493, + 660 + ], + "score": 1.0, + "content": "for RK23 solver, and set rtol", + "type": "text" + }, + { + "bbox": [ + 493, + 648, + 505, + 658 + ], + "score": 0.64, + "content": "=", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 128, + 669 + ], + "score": 0.85, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 657, + 132, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 132, + 658, + 185, + 669 + ], + "score": 0.54, + "content": "\\mathrm { a t o l } = 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "for Dopri5 solver. As shown in Table. 2, Neural ODE trained with MALI is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "robust to discretization scheme, and MALI significantly outperforms the adjoint method in terms", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 159, + 693 + ], + "score": 1.0, + "content": "of accuracy (", + "type": "text" + }, + { + "bbox": [ + 159, + 680, + 178, + 691 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 680, + 196, + 693 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 197, + 680, + 217, + 691 + ], + "score": 0.86, + "content": "63 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "top-1 accuracy on the validation dataset). An interesting finding is that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 689, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 705 + ], + "score": 1.0, + "content": "when trained with MALI which is a second-order solver, and tested with higher-order solver (e.g.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 711, + 356, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 708, + 357, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 708, + 357, + 723 + ], + "score": 1.0, + "content": "1https://github.com/juntang-zhuang/torch_ACA", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 719, + 336, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 336, + 734 + ], + "score": 1.0, + "content": "2https://github.com/rtqichen/torchdiffeq", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 79, + 495, + 207 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 79, + 495, + 207 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 495, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 495, + 207 + ], + "score": 0.97, + "type": "image", + "image_path": "101ac7d51d52296b4d11b9f4136ff79e830a99b276655dff89546fd5738863bb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 79, + 495, + 121.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 121.66666666666666, + 495, + 164.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 164.33333333333331, + 495, + 206.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 214, + 505, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "Figure 1: Region of A-stability for eigenvalue on the imaginary plane for damped ALF. From left", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 244, + 238 + ], + "score": 1.0, + "content": "to right, the region of stability for", + "type": "text" + }, + { + "bbox": [ + 244, + 226, + 282, + 237 + ], + "score": 0.89, + "content": "\\eta = 0 . 2 5", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 226, + 286, + 238 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 286, + 226, + 354, + 237 + ], + "score": 0.9, + "content": "\\eta = 0 . 7 , \\eta = 0 . 8", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 226, + 422, + 238 + ], + "score": 1.0, + "content": "respectively. As", + "type": "text" + }, + { + "bbox": [ + 423, + 227, + 429, + 237 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "increases to 1, the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 237, + 242, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 242, + 249 + ], + "score": 1.0, + "content": "area of stability region decreases.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 108, + 267, + 257, + 280 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 258, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 258, + 282 + ], + "score": 1.0, + "content": "B EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 291, + 226, + 303 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 227, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 227, + 304 + ], + "score": 1.0, + "content": "B.1 IMAGE RECOGNITION", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 312, + 254, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 312, + 255, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 255, + 325 + ], + "score": 1.0, + "content": "B.1.1 EXPERIMENT ON CIFAR10", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 483, + 345 + ], + "score": 1.0, + "content": "We directly modify a ResNet18 into a Neural ODE, where the forward of a residual block", + "type": "text" + }, + { + "bbox": [ + 483, + 333, + 505, + 344 + ], + "score": 0.8, + "content": "( y =", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 342, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 148, + 357 + ], + "score": 0.92, + "content": "x + f ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 344, + 289, + 359 + ], + "score": 1.0, + "content": "and the forward of an ODE block", + "type": "text" + }, + { + "bbox": [ + 289, + 342, + 381, + 358 + ], + "score": 0.93, + "content": "\\begin{array} { r } { ( y = x + \\int _ { 0 } ^ { T } f ( z , t ) d t } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 344, + 410, + 359 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 410, + 345, + 439, + 356 + ], + "score": 0.87, + "content": "T = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 344, + 506, + 359 + ], + "score": 1.0, + "content": ") share the same", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 176, + 368 + ], + "score": 1.0, + "content": "parameterization", + "type": "text" + }, + { + "bbox": [ + 177, + 357, + 183, + 367 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 356, + 505, + 368 + ], + "score": 1.0, + "content": ", hence they have the same number of parameters. Our experiment is based on", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 367, + 487, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 487, + 380 + ], + "score": 1.0, + "content": "the official implementation by Zhuang et al. (2020) and an open-source repository (Liu, 2017).", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 330, + 506, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 384, + 505, + 472 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 396 + ], + "score": 1.0, + "content": "All models are trained with SGD optimizer for 90 epochs, with an initial learning rate of 0.01, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 506, + 407 + ], + "score": 1.0, + "content": "decayed by a factor of 10 at 30th epoch and 60th epoch respectively. Training scheme is the same", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "for all models (ResNet, Neural ODE trained with adjoint, naive, ACA and MALI). For ACA, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 429 + ], + "score": 1.0, + "content": "follow the settings in (Zhuang et al., 2020) and use the official implementation torch ACA 1, and use", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 427, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 208, + 440 + ], + "score": 1.0, + "content": "a Heun-Euler solver with", + "type": "text" + }, + { + "bbox": [ + 209, + 428, + 260, + 438 + ], + "score": 0.85, + "content": "r t o \\bar { l } = 1 0 ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 427, + 263, + 440 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 263, + 427, + 315, + 438 + ], + "score": 0.87, + "content": "a t o l = 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 427, + 505, + 440 + ], + "score": 1.0, + "content": "during training. For MALI, we use an adaptive", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 171, + 451 + ], + "score": 1.0, + "content": "version and set", + "type": "text" + }, + { + "bbox": [ + 171, + 438, + 224, + 450 + ], + "score": 0.62, + "content": "r t o l = 1 0 ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 438, + 227, + 451 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 228, + 439, + 281, + 449 + ], + "score": 0.71, + "content": "a t o l = 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 438, + 506, + 451 + ], + "score": 1.0, + "content": ". For the naive and adjoint method, we use the default", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 449, + 507, + 462 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 311, + 462 + ], + "score": 1.0, + "content": "Dopri5 solver from the torchdiffeq2 package with", + "type": "text" + }, + { + "bbox": [ + 311, + 450, + 393, + 460 + ], + "score": 0.88, + "content": "\\mathrm { r t o l } = \\mathrm { a t o l } = \\mathrm { \\bar { 1 0 } } ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 449, + 507, + 462 + ], + "score": 1.0, + "content": ". We train all models for 5", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 460, + 404, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 404, + 473 + ], + "score": 1.0, + "content": "independent runs, and report the mean and standard deviation across runs.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 384, + 507, + 473 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 484, + 268, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 270, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 270, + 497 + ], + "score": 1.0, + "content": "B.1.2 EXPERIMENTS ON IMAGENET", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "Training scheme We conduct experiments on ImageNet with ResNet18 and Neural-ODE18. All", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "models are trained on 4 GTX-1080Ti GPUs with a batchsize of 256. All models are trained for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 444, + 538 + ], + "score": 1.0, + "content": "80 epochs, with an initial learning rate of 0.1, and decayed by a factor of 10 at", + "type": "text" + }, + { + "bbox": [ + 445, + 525, + 464, + 536 + ], + "score": 0.28, + "content": "3 0 \\mathrm { t h }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "and 60th", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 282, + 550 + ], + "score": 1.0, + "content": "epoch. Note that due to the large size input", + "type": "text" + }, + { + "bbox": [ + 282, + 536, + 326, + 547 + ], + "score": 0.9, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 535, + 506, + 550 + ], + "score": 1.0, + "content": ", the naive method and ACA requires a huge", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "memory, and is infeasible to train. MALI and the adjoint method requires a constant memory hence", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "is suitable for large-scale experiments. For both MALI and the adjoint menthod, we use a fixed", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 271, + 582 + ], + "score": 1.0, + "content": "stepsize of 0.25, and integrates from 0 to", + "type": "text" + }, + { + "bbox": [ + 272, + 569, + 299, + 579 + ], + "score": 0.89, + "content": "T = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 568, + 505, + 582 + ], + "score": 1.0, + "content": ". As shown in Table. 2 in the main paper, a stepsize", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "of 0.25 is sufficiently small to train a meaningful continuous model that is robust to discretization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 591, + 142, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 142, + 604 + ], + "score": 1.0, + "content": "scheme.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 503, + 506, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 627 + ], + "score": 1.0, + "content": "Invariance to discretization scheme To test the influence of discretization scheme, we test our", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "score": 1.0, + "content": "Neural ODE with different solvers without re-training. For fixed-stepsize solvers, we tested various", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 190, + 648 + ], + "score": 1.0, + "content": "step sizes including", + "type": "text" + }, + { + "bbox": [ + 190, + 636, + 292, + 649 + ], + "score": 0.89, + "content": "\\{ 0 . 1 , 0 . 1 5 , 0 . 2 5 , 0 . 5 , 1 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 636, + 426, + 648 + ], + "score": 1.0, + "content": "; for adaptive solvers, we set rto", + "type": "text" + }, + { + "bbox": [ + 426, + 637, + 446, + 647 + ], + "score": 0.65, + "content": "\\scriptstyle \\mathbf { - 0 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 636, + 465, + 648 + ], + "score": 1.0, + "content": ", atol", + "type": "text" + }, + { + "bbox": [ + 465, + 637, + 488, + 647 + ], + "score": 0.52, + "content": "= 0 . 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 646, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 281, + 660 + ], + "score": 1.0, + "content": "MALI and Heun-Euler method, and set rto", + "type": "text" + }, + { + "bbox": [ + 282, + 647, + 319, + 658 + ], + "score": 0.76, + "content": "1 \\stackrel { { \\textstyle \\sum } } { = } 1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 646, + 322, + 660 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 322, + 647, + 374, + 658 + ], + "score": 0.76, + "content": "\\mathrm { a t o l } = 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 646, + 493, + 660 + ], + "score": 1.0, + "content": "for RK23 solver, and set rtol", + "type": "text" + }, + { + "bbox": [ + 493, + 648, + 505, + 658 + ], + "score": 0.64, + "content": "=", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 128, + 669 + ], + "score": 0.85, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 657, + 132, + 671 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 132, + 658, + 185, + 669 + ], + "score": 0.54, + "content": "\\mathrm { a t o l } = 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "for Dopri5 solver. As shown in Table. 2, Neural ODE trained with MALI is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "robust to discretization scheme, and MALI significantly outperforms the adjoint method in terms", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 159, + 693 + ], + "score": 1.0, + "content": "of accuracy (", + "type": "text" + }, + { + "bbox": [ + 159, + 680, + 178, + 691 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 680, + 196, + 693 + ], + "score": 1.0, + "content": "v.s.", + "type": "text" + }, + { + "bbox": [ + 197, + 680, + 217, + 691 + ], + "score": 0.86, + "content": "63 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "top-1 accuracy on the validation dataset). An interesting finding is that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 689, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 705 + ], + "score": 1.0, + "content": "when trained with MALI which is a second-order solver, and tested with higher-order solver (e.g.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 613, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 81, + 504, + 244 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 81, + 504, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 81, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 109, + 81, + 504, + 244 + ], + "score": 0.965, + "type": "image", + "image_path": "a2eb3a813f5f7dc301a07eb82c50a2ff52e673b7a547da9299eb769bdf3b5a0e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 81, + 504, + 135.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 135.33333333333334, + 504, + 189.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 189.66666666666669, + 504, + 244.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 241, + 253, + 369, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 241, + 252, + 369, + 266 + ], + "spans": [ + { + "bbox": [ + 241, + 252, + 369, + 266 + ], + "score": 1.0, + "content": "Figure 2: Results on ImageNet.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 503, + 316 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 239, + 306 + ], + "score": 1.0, + "content": "RK4), our Neural ODE achieves", + "type": "text" + }, + { + "bbox": [ + 239, + 294, + 271, + 305 + ], + "score": 0.88, + "content": "7 0 . 2 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "top-1 accuracy, which is higher than both the same solver", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 304, + 430, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 202, + 318 + ], + "score": 1.0, + "content": "during training (MALI,", + "type": "text" + }, + { + "bbox": [ + 202, + 305, + 234, + 316 + ], + "score": 0.85, + "content": "6 9 . 5 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 304, + 353, + 318 + ], + "score": 1.0, + "content": "accuracy) and the ResNet18 (", + "type": "text" + }, + { + "bbox": [ + 353, + 305, + 385, + 316 + ], + "score": 0.82, + "content": "7 0 . 0 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 304, + 430, + 318 + ], + "score": 1.0, + "content": "accuracy).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 321, + 505, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 334 + ], + "score": 1.0, + "content": "Furthermore, many papers claim ResNet to be an approximation for an ODE (Lu et al., 2018).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 345 + ], + "score": 1.0, + "content": "However, Queiruga et al. (2020) argues that many numerical discretizations fail to be meaningful", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "dynamical systems, while our experiments demonstrate that our model is continuous hence invariant", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 355, + 211, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 211, + 366 + ], + "score": 1.0, + "content": "to discretization schemes.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "Adversarial robustness Besides the high accuracy and robustness to discretization scheme, an-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 410 + ], + "score": 1.0, + "content": "other advantage of Neural ODE is the robustness to adversarial attack. The adversary robustness of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "Neural ODE is extensively studied in (Hanshu et al., 2019), but not only validated on small-scale", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 434 + ], + "score": 1.0, + "content": "datasets such as Cifar10. To our knowledge, our method is the first to enable effectuve training", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "of Neural ODE on large-scale datasets such as ImageNet and achieve a high accuracy, and we are", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 455 + ], + "score": 1.0, + "content": "the first to validate the robustness of Neural ODE on ImageNet. We use the advertorch 3 toolbox", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "to perform adversarial attack. We test the performance of ResNet and Neural ODE under FGSM", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "attack. To be more convincing, we conduct experiment on the pretrained ResNet18 provided by the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "official PyTorch website 4. Since Neural ODE is invariant to discretization scheme, it’s possible to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 485, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 500 + ], + "score": 1.0, + "content": "derive the gradient for attack using one ODE solver, and inference on the perturbed image using", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 496, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 511 + ], + "score": 1.0, + "content": "another solver. As summarized in Table. 3, Neural ODE consistently achieves a higher accuracy", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 509, + 249, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 249, + 519 + ], + "score": 1.0, + "content": "than ResNet under the same attack.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 542, + 237, + 553 + ], + "lines": [ + { + "bbox": [ + 105, + 542, + 238, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 238, + 554 + ], + "score": 1.0, + "content": "B.2 TIME SERIES MODELING", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 566, + 505, + 687 + ], + "lines": [ + { + "bbox": [ + 106, + 567, + 504, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 504, + 578 + ], + "score": 1.0, + "content": "We conduct experiments on Latent-ODE models (Rubanova et al., 2019) and Neural CDE (con-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 578, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 504, + 590 + ], + "score": 1.0, + "content": "trolled differential equation) (Kidger et al., 2020a). For all experiments, we use the official imple-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 588, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 504, + 601 + ], + "score": 1.0, + "content": "mentation, and only replace the solver with MALI. The latent-ODE model is trained on the Mujoco", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "dataset processed with code provided by the official implementation, and we experiment with dif-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 160, + 623 + ], + "score": 1.0, + "content": "ferent ratios", + "type": "text" + }, + { + "bbox": [ + 160, + 610, + 225, + 622 + ], + "score": 0.89, + "content": "( 1 0 \\% , 2 0 \\% , 5 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 610, + 506, + 623 + ], + "score": 1.0, + "content": "of training data as described in (Rubanova et al., 2019). All models", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "are trained for 300 epochs with Adamax optimizer, with an initial learning rate of 0.01 and scaled", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 645 + ], + "score": 1.0, + "content": "by 0.999 for each epoch. For the Neural CDE model, for the naive method, ACA and MALI, we", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "perform 5 independent runs and report the mean value and standard deviation; results for the adjoint", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "and seminorm adjoint are from (Kidger et al., 2020a). 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m1.00.950.90.85
Test Accuracy on Speech Commands (Higher is better)93.7 ± 0.393.7 ± 0.193.5± 0.293.7 ± 0.3
Test MSEof latent ODE on Mujoco (Lower is better)10% training data0.350.360.330.33
20% training data0.270.250.260.27
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Test Accuracy on Speech Commands (Higher is better)93.7 ± 0.393.7 ± 0.193.5± 0.293.7 ± 0.3
Test MSEof latent ODE on Mujoco (Lower is better)10% training data0.350.360.330.33
20% training data0.270.250.260.27
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