Daily Papers

Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76\% over the strongest competing baselines while reducing the number of parameters by 81.98\%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable training. These challenges arise particularly from the ill-conditioning of the optimization problem caused by the differential terms in the loss function. To address these issues, we propose learning a solver, i.e., solving PDEs using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance, significantly accelerating and stabilizing the optimization process and enabling faster convergence of physics-aware models. Furthermore, while traditional physics-informed methods solve for a single PDE instance, our approach extends to parametric PDEs. Specifically, we integrate the physical loss gradient with PDE parameters, allowing our method to solve over a distribution of PDE parameters, including coefficients, initial conditions, and boundary conditions. We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets, comparing both training and test-time optimization performance. The code is available at https://github.com/2ailesB/neural-parametric-solver.

Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine learning with a specific focus on the effectiveness of physics-informed neural networks in predicting outcomes of physical systems and discovering hidden physics from noisy data. We will also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training. To address this limitation we present a learning rate annealing algorithm that utilizes gradient statistics during model training to balance the interplay between different terms in composite loss functions. We also propose a novel neural network architecture that is more resilient to such gradient pathologies. Taken together, our developments provide new insights into the training of constrained neural networks and consistently improve the predictive accuracy of physics-informed neural networks by a factor of 50-100x across a range of problems in computational physics. All code and data accompanying this manuscript are publicly available at https://github.com/PredictiveIntelligenceLab/GradientPathologiesPINNs.

The loss functions of many learning problems contain multiple additive terms that can disagree and yield conflicting update directions. For Physics-Informed Neural Networks (PINNs), loss terms on initial/boundary conditions and physics equations are particularly interesting as they are well-established as highly difficult tasks. To improve learning the challenging multi-objective task posed by PINNs, we propose the ConFIG method, which provides conflict-free updates by ensuring a positive dot product between the final update and each loss-specific gradient. It also maintains consistent optimization rates for all loss terms and dynamically adjusts gradient magnitudes based on conflict levels. We additionally leverage momentum to accelerate optimizations by alternating the back-propagation of different loss terms. We provide a mathematical proof showing the convergence of the ConFIG method, and it is evaluated across a range of challenging PINN scenarios. ConFIG consistently shows superior performance and runtime compared to baseline methods. We also test the proposed method in a classic multi-task benchmark, where the ConFIG method likewise exhibits a highly promising performance. Source code is available at https://tum-pbs.github.io/ConFIG

Imposing known physical constraints, such as conservation laws, during neural network training introduces an inductive bias that can improve accuracy, reliability, convergence, and data efficiency for modeling physical dynamics. While such constraints can be softly imposed via loss function penalties, recent advancements in differentiable physics and optimization improve performance by incorporating PDE-constrained optimization as individual layers in neural networks. This enables a stricter adherence to physical constraints. However, imposing hard constraints significantly increases computational and memory costs, especially for complex dynamical systems. This is because it requires solving an optimization problem over a large number of points in a mesh, representing spatial and temporal discretizations, which greatly increases the complexity of the constraint. To address this challenge, we develop a scalable approach to enforce hard physical constraints using Mixture-of-Experts (MoE), which can be used with any neural network architecture. Our approach imposes the constraint over smaller decomposed domains, each of which is solved by an "expert" through differentiable optimization. During training, each expert independently performs a localized backpropagation step by leveraging the implicit function theorem; the independence of each expert allows for parallelization across multiple GPUs. Compared to standard differentiable optimization, our scalable approach achieves greater accuracy in the neural PDE solver setting for predicting the dynamics of challenging non-linear systems. We also improve training stability and require significantly less computation time during both training and inference stages.

Physics-informed Neural Networks (PINNs) have recently achieved remarkable progress in solving Partial Differential Equations (PDEs) in various fields by minimizing a weighted sum of PDE loss and boundary loss. However, there are several critical challenges in the training of PINNs, including the lack of theoretical frameworks and the imbalance between PDE loss and boundary loss. In this paper, we present an analysis of second-order non-homogeneous PDEs, which are classified into three categories and applicable to various common problems. We also characterize the connections between the training loss and actual error, guaranteeing convergence under mild conditions. The theoretical analysis inspires us to further propose MultiAdam, a scale-invariant optimizer that leverages gradient momentum to parameter-wisely balance the loss terms. Extensive experiment results on multiple problems from different physical domains demonstrate that our MultiAdam solver can improve the predictive accuracy by 1-2 orders of magnitude compared with strong baselines.

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.

High-fidelity simulations and complex inverse problems, such as detector modeling and unfolding, are computationally intensive bottlenecks across subatomic physics, yet essential for accurate physical interpretation. While Conditional Flow Matching (CFM) offers a robust acceleration approach, we demonstrate its standard training loss is fundamentally misleading. Specifically, utilizing a Jefferson Lab Nuclear Physics (NP) kinematic dataset (γp to ρ^0 p to π^+π^- p), we expose that CFM loss plateaus prematurely, obscuring ongoing physical refinement. To verify this disconnect is a dataset-agnostic pathology, we introduce ScatterPrism, an efficient generative surrogate evaluated against both the NP data and synthetic stress tests modeling challenging 1D distribution topologies. Coupling these benchmarks, we establish that physics-informed metrics continue improving long after standard loss converges. Consequently, we propose a multi-metric diagnostic protocol to ensure true kinematic fidelity without data memorization. Driven by NP challenges relevant to the forthcoming Electron-Ion Collider (EIC), this unified machinery has strong potential to extend to High-Energy Physics (HEP) applications, such as jet modeling. Furthermore, the framework holds promise for broader domains requiring rigorous generative reliability, including medical imaging, astrophysics, and quantitative finance.

Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic L^2(Ω)-error bounds for FK-PINNs with tanh activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of tanh neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schrödinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40times faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

We propose a hybrid neural network (NN) and PDE approach for learning generalizable PDE dynamics from motion observations. Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and constitutive models (or material models). Without explicit PDE knowledge, these approaches cannot guarantee physical correctness and have limited generalizability. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. Instead, constitutive models are particularly suitable for learning due to their data-fitting nature. To this end, we introduce a new framework termed "Neural Constitutive Laws" (NCLaw), which utilizes a network architecture that strictly guarantees standard constitutive priors, including rotation equivariance and undeformed state equilibrium. We embed this network inside a differentiable simulation and train the model by minimizing a loss function based on the difference between the simulation and the motion observation. We validate NCLaw on various large-deformation dynamical systems, ranging from solids to fluids. After training on a single motion trajectory, our method generalizes to new geometries, initial/boundary conditions, temporal ranges, and even multi-physics systems. On these extremely out-of-distribution generalization tasks, NCLaw is orders-of-magnitude more accurate than previous NN approaches. Real-world experiments demonstrate our method's ability to learn constitutive laws from videos.

Generative machine learning methods, such as diffusion models and flow matching, have shown great potential in modeling complex system behaviors and building efficient surrogate models. However, these methods typically learn the underlying physics implicitly from data. We propose Physics-Based Flow Matching (PBFM), a novel generative framework that explicitly embeds physical constraints, both PDE residuals and algebraic relations, into the flow matching objective. We also introduce temporal unrolling at training time that improves the accuracy of the final, noise-free sample prediction. Our method jointly minimizes the flow matching loss and the physics-based residual loss without requiring hyperparameter tuning of their relative weights. Additionally, we analyze the role of the minimum noise level, sigma_{min}, in the context of physical constraints and evaluate a stochastic sampling strategy that helps to reduce physical residuals. Through extensive benchmarks on three representative PDE problems, we show that our approach yields up to an 8times more accurate physical residuals compared to FM, while clearly outperforming existing algorithms in terms of distributional accuracy. PBFM thus provides a principled and efficient framework for surrogate modeling, uncertainty quantification, and accelerated simulation in physics and engineering applications.

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called Walk-on-Spheres Neural Operator (WoS-NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75times improvement in L_2-error compared to standard physics-informed training schemes, up to 6.31times improvement in training speed, and reductions of up to 2.97times in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a spectral loss. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a 10times increase in performance speed.

Inferring the physical properties of 3D scenes from visual information is a critical yet challenging task for creating interactive and realistic virtual worlds. While humans intuitively grasp material characteristics such as elasticity or stiffness, existing methods often rely on slow, per-scene optimization, limiting their generalizability and application. To address this problem, we introduce PIXIE, a novel method that trains a generalizable neural network to predict physical properties across multiple scenes from 3D visual features purely using supervised losses. Once trained, our feed-forward network can perform fast inference of plausible material fields, which coupled with a learned static scene representation like Gaussian Splatting enables realistic physics simulation under external forces. To facilitate this research, we also collected PIXIEVERSE, one of the largest known datasets of paired 3D assets and physic material annotations. Extensive evaluations demonstrate that PIXIE is about 1.46-4.39x better and orders of magnitude faster than test-time optimization methods. By leveraging pretrained visual features like CLIP, our method can also zero-shot generalize to real-world scenes despite only ever been trained on synthetic data. https://pixie-3d.github.io/

Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to O(d^{k}) scaling of the derivative tensor size and the O(2^{k-1}L) scaling in the computation graph, where d is the dimension of the domain, L is the number of ops in the forward computation graph, and k is the derivative order. In previous works, the polynomial scaling in d was addressed by amortizing the computation over the optimization process via randomization. Separately, the exponential scaling in k for univariate functions (d=1) was addressed with high-order auto-differentiation (AD). In this work, we show how to efficiently perform arbitrary contraction of the derivative tensor of arbitrary order for multivariate functions, by properly constructing the input tangents to univariate high-order AD, which can be used to efficiently randomize any differential operator. When applied to Physics-Informed Neural Networks (PINNs), our method provides >1000times speed-up and >30times memory reduction over randomization with first-order AD, and we can now solve 1-million-dimensional PDEs in 8 minutes on a single NVIDIA A100 GPU. This work opens the possibility of using high-order differential operators in large-scale problems.

This document is a hands-on, comprehensive guide to deep learning in the realm of physical simulations. Rather than just theory, we emphasize practical application: every concept is paired with interactive Jupyter notebooks to get you up and running quickly. Beyond traditional supervised learning, we dive into physical loss-constraints, differentiable simulations, diffusion-based approaches for probabilistic generative AI, as well as reinforcement learning and advanced neural network architectures. These foundations are paving the way for the next generation of scientific foundation models. We are living in an era of rapid transformation. These methods have the potential to redefine what's possible in computational science.

Reduced-order modeling (ROM) of time-dependent and parameterized differential equations aims to accelerate the simulation of complex high-dimensional systems by learning a compact latent manifold representation that captures the characteristics of the solution fields and their time-dependent dynamics. Although high-fidelity numerical solvers generate the training datasets, they have thus far been excluded from the training process, causing the learned latent dynamics to drift away from the discretized governing physics. This mismatch often limits generalization and forecasting capabilities. In this work, we propose Physics-informed ROM (Φ-ROM) by incorporating differentiable PDE solvers into the training procedure. Specifically, the latent space dynamics and its dependence on PDE parameters are shaped directly by the governing physics encoded in the solver, ensuring a strong correspondence between the full and reduced systems. Our model outperforms state-of-the-art data-driven ROMs and other physics-informed strategies by accurately generalizing to new dynamics arising from unseen parameters, enabling long-term forecasting beyond the training horizon, maintaining continuity in both time and space, and reducing the data cost. Furthermore, Φ-ROM learns to recover and forecast the solution fields even when trained or evaluated with sparse and irregular observations of the fields, providing a flexible framework for field reconstruction and data assimilation. We demonstrate the framework's robustness across various PDE solvers and highlight its broad applicability by providing an open-source JAX implementation that is readily extensible to other PDE systems and differentiable solvers, available at https://phi-rom.github.io.

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative L_2 and maximum absolute errors among all compared methods, reducing relative L_2 error by up to 97.8\% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high-fidelity PDEs remains computationally prohibitive, particularly for parametric systems requiring multiple evaluations across varying parameter configurations. This paper presents MF-BPINN, a novel multi-fidelity framework that synergistically combines physics-informed neural networks with Bayesian uncertainty quantification and adaptive residual learning. Our approach leverages abundant low-fidelity simulations alongside sparse high-fidelity data through a hierarchical neural architecture that learns nonlinear correlations across fidelity levels. We introduce an adaptive residual network with learnable gating mechanisms that dynamically balances linear and nonlinear fidelity discrepancies. Furthermore, we develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.

Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a neural-operator framework for surrogate modeling of power system components, using Deep Operator Networks (DeepONets) to learn mappings from system states and time-varying inputs to full trajectories without step-by-step integration. To enhance generalization and data efficiency, we introduce Physics-Informed DeepONets (PI-DeepONets), which embed the residuals of governing equations into the training loss. Our results show that DeepONets, and especially PI-DeepONets, achieve accurate predictions under diverse scenarios, providing over 30 times speedup compared to high-order ODE solvers. Benchmarking against Physics-Informed Neural Networks (PINNs) highlights superior stability and scalability. Our results demonstrate neural operators as a promising path toward real-time, physics-aware simulation of power system dynamics.

Physics-Informed Neural Networks (PINNs) have emerged as a promising deep learning framework for approximating numerical solutions to partial differential equations (PDEs). However, conventional PINNs, relying on multilayer perceptrons (MLP), neglect the crucial temporal dependencies inherent in practical physics systems and thus fail to propagate the initial condition constraints globally and accurately capture the true solutions under various scenarios. In this paper, we introduce a novel Transformer-based framework, termed PINNsFormer, designed to address this limitation. PINNsFormer can accurately approximate PDE solutions by utilizing multi-head attention mechanisms to capture temporal dependencies. PINNsFormer transforms point-wise inputs into pseudo sequences and replaces point-wise PINNs loss with a sequential loss. Additionally, it incorporates a novel activation function, Wavelet, which anticipates Fourier decomposition through deep neural networks. Empirical results demonstrate that PINNsFormer achieves superior generalization ability and accuracy across various scenarios, including PINNs failure modes and high-dimensional PDEs. Moreover, PINNsFormer offers flexibility in integrating existing learning schemes for PINNs, further enhancing its performance.

Neural operators have shown promise in learning solution maps of partial differential equations (PDEs), but they often struggle to generalize when test inputs lie outside the training distribution, such as novel initial conditions, unseen PDE coefficients or unseen physics. Prior works address this limitation with large-scale multiple physics pretraining followed by fine-tuning, but this still requires examples from the new dynamics, falling short of true zero-shot generalization. In this work, we propose a method to enhance generalization at test time, i.e., without modifying pretrained weights. Building on DISCO, which provides a dictionary of neural operators trained across different dynamics, we introduce a neural operator splitting strategy that, at test time, searches over compositions of training operators to approximate unseen dynamics. On challenging out-of-distribution tasks including parameter extrapolation and novel combinations of physics phenomena, our approach achieves state-of-the-art zero-shot generalization results, while being able to recover the underlying PDE parameters. These results underscore test-time computation as a key avenue for building flexible, compositional, and generalizable neural operators.

Recent advances in text-to-video (T2V) generation have achieved good visual quality, yet synthesizing videos that faithfully follow physical laws remains an open challenge. Existing methods mainly based on graphics or prompt extension struggle to generalize beyond simple simulated environments or learn implicit physical reasoning. The scarcity of training data with rich physics interactions and phenomena is also a problem. In this paper, we first introduce a Physics-Augmented video data construction Pipeline, PhyAugPipe, that leverages a vision-language model (VLM) with chain-of-thought reasoning to collect a large-scale training dataset, PhyVidGen-135K. Then we formulate a principled Physics-aware Groupwise Direct Preference Optimization, PhyGDPO, framework that builds upon the groupwise Plackett-Luce probabilistic model to capture holistic preferences beyond pairwise comparisons. In PhyGDPO, we design a Physics-Guided Rewarding (PGR) scheme that embeds VLM-based physics rewards to steer optimization toward physical consistency. We also propose a LoRA-Switch Reference (LoRA-SR) scheme that eliminates memory-heavy reference duplication for efficient training. Experiments show that our method significantly outperforms state-of-the-art open-source methods on PhyGenBench and VideoPhy2. Please check our project page at https://caiyuanhao1998.github.io/project/PhyGDPO for more video results. Our code, models, and data will be released at https://github.com/caiyuanhao1998/Open-PhyGDPO

Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework that unifies generative modeling and partial differential equation fulfillment by introducing a first-principle-based loss term that enforces generated samples to fulfill the underlying physical constraints. Our approach reduces the residual error by up to two orders of magnitude compared to previous work in a fluid flow case study and outperforms task-specific frameworks in relevant metrics for structural topology optimization. We also present numerical evidence that our extended training objective acts as a natural regularization mechanism against overfitting. Our framework is simple to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32\% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/ZhouChaunge/PhysGuard.

Differentiable physics enables efficient gradient-based optimizations of neural network (NN) controllers. However, existing work typically only delivers NN controllers with limited capability and generalizability. We present a practical learning framework that outputs unified NN controllers capable of tasks with significantly improved complexity and diversity. To systematically improve training robustness and efficiency, we investigated a suite of improvements over the baseline approach, including periodic activation functions, and tailored loss functions. In addition, we find our adoption of batching and an Adam optimizer effective in training complex locomotion tasks. We evaluate our framework on differentiable mass-spring and material point method (MPM) simulations, with challenging locomotion tasks and multiple robot designs. Experiments show that our learning framework, based on differentiable physics, delivers better results than reinforcement learning and converges much faster. We demonstrate that users can interactively control soft robot locomotion and switch among multiple goals with specified velocity, height, and direction instructions using a unified NN controller trained in our system. Code is available at https://github.com/erizmr/Complex-locomotion-skill-learning-via-differentiable-physics.

Deep generative models such as flow matching and diffusion models have shown great potential in learning complex distributions and dynamical systems, but often act as black-boxes, neglecting underlying physics. In contrast, physics-based simulation models described by ODEs/PDEs remain interpretable, but may have missing or unknown terms, unable to fully describe real-world observations. We bridge this gap with a novel grey-box method that integrates incomplete physics models directly into generative models. Our approach learns dynamics from observational trajectories alone, without ground-truth physics parameters, in a simulation-free manner that avoids scalability and stability issues of Neural ODEs. The core of our method lies in modelling a structured variational distribution within the flow matching framework, by using two latent encodings: one to model the missing stochasticity and multi-modal velocity, and a second to encode physics parameters as a latent variable with a physics-informed prior. Furthermore, we present an adaptation of the framework to handle second-order dynamics. Our experiments on representative ODE/PDE problems show that our method performs on par with or superior to fully data-driven approaches and previous grey-box baselines, while preserving the interpretability of the physics model. Our code is available at https://github.com/DMML-Geneva/VGB-DM.

Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a black box by directly predicting the next state. While this is a natural and easy framework for applying neural surrogates, it can be an over-simplified and rigid framework for predicting physics. In this work, we propose an alternative framework in which neural solvers predict the temporal derivative and an ODE integrator forwards the solution in time, which has little overhead and is broadly applicable across model architectures and PDEs. We find that by simply changing the training target and introducing numerical integration during inference, neural surrogates can gain accuracy and stability. Predicting temporal derivatives also allows models to not be constrained to a specific temporal discretization, allowing for flexible time-stepping during inference or training on higher-resolution PDE data. Lastly, we investigate why this new framework can be beneficial and in what situations does it work well.

Recent advances of data-driven machine learning have revolutionized fields like computer vision, reinforcement learning, and many scientific and engineering domains. In many real-world and scientific problems, systems that generate data are governed by physical laws. Recent work shows that it provides potential benefits for machine learning models by incorporating the physical prior and collected data, which makes the intersection of machine learning and physics become a prevailing paradigm. By integrating the data and mathematical physics models seamlessly, it can guide the machine learning model towards solutions that are physically plausible, improving accuracy and efficiency even in uncertain and high-dimensional contexts. In this survey, we present this learning paradigm called Physics-Informed Machine Learning (PIML) which is to build a model that leverages empirical data and available physical prior knowledge to improve performance on a set of tasks that involve a physical mechanism. We systematically review the recent development of physics-informed machine learning from three perspectives of machine learning tasks, representation of physical prior, and methods for incorporating physical prior. We also propose several important open research problems based on the current trends in the field. We argue that encoding different forms of physical prior into model architectures, optimizers, inference algorithms, and significant domain-specific applications like inverse engineering design and robotic control is far from being fully explored in the field of physics-informed machine learning. We believe that the interdisciplinary research of physics-informed machine learning will significantly propel research progress, foster the creation of more effective machine learning models, and also offer invaluable assistance in addressing long-standing problems in related disciplines.

We present a new approach for evaluating Feynman integrals numerically. We apply the recently-proposed framework of physics-informed deep learning to train neural networks to approximate the solution to the differential equations satisfied by the Feynman integrals. This approach relies neither on a canonical form of the differential equations, which is often a bottleneck for the analytical techniques, nor on the availability of a large dataset, and after training yields essentially instantaneous evaluation times. We provide a proof-of-concept implementation within the PyTorch framework, and apply it to a number of one- and two-loop examples, achieving a mean magnitude of relative difference of around 1% at two loops in the physical phase space with network training times on the order of an hour on a laptop GPU.

Efficiently exploring complex loss landscapes is key to the performance of deep neural networks. While momentum-based optimizers are widely used in state-of-the-art setups, classical momentum can still struggle with large, misaligned gradients, leading to oscillations. To address this, we propose Torque-Aware Momentum (TAM), which introduces a damping factor based on the angle between the new gradients and previous momentum, stabilizing the update direction during training. Empirical results show that TAM, which can be combined with both SGD and Adam, enhances exploration, handles distribution shifts more effectively, and improves generalization performance across various tasks, including image classification and large language model fine-tuning, when compared to classical momentum-based optimizers.

Recent advances in deep learning for physics have focused on discovering shared representations of target systems by incorporating physics priors or inductive biases into neural networks. While effective, these methods are limited to the system domain, where the type of system remains consistent and thus cannot ensure the adaptation to new, or unseen physical systems governed by different laws. For instance, a neural network trained on a mass-spring system cannot guarantee accurate predictions for the behavior of a two-body system or any other system with different physical laws. In this work, we take a significant leap forward by targeting cross domain generalization within the field of Hamiltonian dynamics. We model our system with a graph neural network and employ a meta learning algorithm to enable the model to gain experience over a distribution of tasks and make it adapt to new physics. Our approach aims to learn a unified Hamiltonian representation that is generalizable across multiple system domains, thereby overcoming the limitations of system-specific models. Our results demonstrate that the meta-trained model not only adapts effectively to new systems but also captures a generalized Hamiltonian representation that is consistent across different physical domains. Overall, through the use of meta learning, we offer a framework that achieves cross domain generalization, providing a step towards a unified model for understanding a wide array of dynamical systems via deep learning.

In this paper, we propose a probabilistic physics-guided framework, termed Physics-guided Deep Markov Model (PgDMM). The framework targets the inference of the characteristics and latent structure of nonlinear dynamical systems from measurement data, where exact inference of latent variables is typically intractable. A recently surfaced option pertains to leveraging variational inference to perform approximate inference. In such a scheme, transition and emission functions of the system are parameterized via feed-forward neural networks (deep generative models). However, due to the generalized and highly versatile formulation of neural network functions, the learned latent space often lacks physical interpretation and structured representation. To address this, we bridge physics-based state space models with Deep Markov Models, thus delivering a hybrid modeling framework for unsupervised learning and identification of nonlinear dynamical systems. The proposed framework takes advantage of the expressive power of deep learning, while retaining the driving physics of the dynamical system by imposing physics-driven restrictions on the side of the latent space. We demonstrate the benefits of such a fusion in terms of achieving improved performance on illustrative simulation examples and experimental case studies of nonlinear systems. Our results indicate that the physics-based models involved in the employed transition and emission functions essentially enforce a more structured and physically interpretable latent space, which is essential for enhancing and generalizing the predictive capabilities of deep learning-based models.

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the use of costly data and improve the generalization ability. However, these physics constraints, based on certain finite dimensional approximations over the function space, must resolve the smallest scaled physics to ensure the accuracy and stability of the simulation, resulting in high computational costs from large input, output, and neural networks. This paper proposes a general acceleration methodology called NeuralStagger by spatially and temporally decomposing the original learning tasks into several coarser-resolution subtasks. We define a coarse-resolution neural solver for each subtask, which requires fewer computational resources, and jointly train them with the vanilla physics-constrained loss by simply arranging their outputs to reconstruct the original solution. Due to the perfect parallelism between them, the solution is achieved as fast as a coarse-resolution neural solver. In addition, the trained solvers bring the flexibility of simulating with multiple levels of resolution. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations, which leads to an additional 10sim100times speed-up. Moreover, the experiment also shows that the learned model could be well used for optimal control.

Video generators are increasingly evaluated as potential world models, which requires them to encode and understand physical laws. We investigate their representation of a fundamental law: gravity. Out-of-the-box video generators consistently generate objects falling at an effectively slower acceleration. However, these physical tests are often confounded by ambiguous metric scale. We first investigate if observed physical errors are artifacts of these ambiguities (e.g., incorrect frame rate assumptions). We find that even temporal rescaling cannot correct the high-variance gravity artifacts. To rigorously isolate the underlying physical representation from these confounds, we introduce a unit-free, two-object protocol that tests the timing ratio t_1^2/t_2^2 = h_1/h_2, a relationship independent of g, focal length, and scale. This relative test reveals violations of Galileo's equivalence principle. We then demonstrate that this physical gap can be partially mitigated with targeted specialization. A lightweight low-rank adaptor fine-tuned on only 100 single-ball clips raises g_{eff} from 1.81,m/s^2 to 6.43,m/s^2 (reaching 65% of terrestrial gravity). This specialist adaptor also generalizes zero-shot to two-ball drops and inclined planes, offering initial evidence that specific physical laws can be corrected with minimal data.

Shape optimization is essential in aerospace vehicle design, including reentry systems, and propulsion system components, as it directly influences aerodynamic efficiency, structural integrity, and overall mission success. Rapid and accurate prediction of external and internal flows accelerates design iterations. To this end, we develop a new variant of DeepONet, called Fusion-DeepONet as a fast surrogate model for geometry-dependent hypersonic and supersonic flow fields. We evaluated Fusion-DeepONet in learning two external hypersonic flows and a supersonic shape-dependent internal flow problem. First, we compare the performance of Fusion-DeepONet with state-of-the-art neural operators to learn inviscid hypersonic flow around semi-elliptic blunt bodies for two grid types: uniform Cartesian and irregular grids. Fusion-DeepONet provides comparable accuracy to parameter-conditioned U-Net on uniform grids while outperforming MeshGraphNet and Vanilla-DeepONet on irregular grids. Fusion-DeepONet requires significantly fewer trainable parameters than U-Net, MeshGraphNet, and FNO. For the second hypersonic problem, we set up Fusion-DeepONet to map from geometry and free stream Mach number to the temperature field around a reentry capsule traveling at hypersonic speed. This fast surrogate is then improved to predict the spatial derivative of the temperature, resulting in an accurate prediction of heat flux at the surfaces of the capsule. To enhance the accuracy of spatial derivative prediction, we introduce a derivative-enhanced loss term with the least computation overhead. For the third problem, we show that Fusion-DeepONet outperforms MeshGraphNet in learning geometry-dependent supersonic flow in a converging-diverging nozzle configuration. For all the problems, we used high-fidelity simulations with a high-order entropy-stable DGSEM solver to generate training datasets with limited samples.

Generative video models achieve high visual fidelity but often violate basic physical principles, limiting reliability in real-world settings. Prior attempts to inject physics rely on conditioning: frame-level signals are domain-specific and short-horizon, while global text prompts are coarse and noisy, missing fine-grained dynamics. We present PhysVid, a physics-aware local conditioning scheme that operates over temporally contiguous chunks of frames. Each chunk is annotated with physics-grounded descriptions of states, interactions, and constraints, which are fused with the global prompt via chunk-aware cross-attention during training. At inference, we introduce negative physics prompts (descriptions of locally relevant law violations) to steer generation away from implausible trajectories. On VideoPhy, PhysVid improves physical commonsense scores by approx 33% over baseline video generators, and by up to approx 8% on VideoPhy2. These results show that local, physics-aware guidance substantially increases physical plausibility in generative video and marks a step toward physics-grounded video models.

Transfer learning is widely used for training deep neural networks (DNN) for building a powerful representation. Even after the pre-trained model is adapted for the target task, the representation performance of the feature extractor is retained to some extent. As the performance of the pre-trained model can be considered the private property of the owner, it is natural to seek the exclusive right of the generalized performance of the pre-trained weight. To address this issue, we suggest a new paradigm of transfer learning called disposable transfer learning (DTL), which disposes of only the source task without degrading the performance of the target task. To achieve knowledge disposal, we propose a novel loss named Gradient Collision loss (GC loss). GC loss selectively unlearns the source knowledge by leading the gradient vectors of mini-batches in different directions. Whether the model successfully unlearns the source task is measured by piggyback learning accuracy (PL accuracy). PL accuracy estimates the vulnerability of knowledge leakage by retraining the scrubbed model on a subset of source data or new downstream data. We demonstrate that GC loss is an effective approach to the DTL problem by showing that the model trained with GC loss retains the performance on the target task with a significantly reduced PL accuracy.

Existing imitation learning (IL) methods such as inverse reinforcement learning (IRL) usually have a double-loop training process, alternating between learning a reward function and a policy and tend to suffer long training time and high variance. In this work, we identify the benefits of differentiable physics simulators and propose a new IL method, i.e., Imitation Learning via Differentiable Physics (ILD), which gets rid of the double-loop design and achieves significant improvements in final performance, convergence speed, and stability. The proposed ILD incorporates the differentiable physics simulator as a physics prior into its computational graph for policy learning. It unrolls the dynamics by sampling actions from a parameterized policy, simply minimizing the distance between the expert trajectory and the agent trajectory, and back-propagating the gradient into the policy via temporal physics operators. With the physics prior, ILD policies can not only be transferable to unseen environment specifications but also yield higher final performance on a variety of tasks. In addition, ILD naturally forms a single-loop structure, which significantly improves the stability and training speed. To simplify the complex optimization landscape induced by temporal physics operations, ILD dynamically selects the learning objectives for each state during optimization. In our experiments, we show that ILD outperforms state-of-the-art methods in a variety of continuous control tasks with Brax, requiring only one expert demonstration. In addition, ILD can be applied to challenging deformable object manipulation tasks and can be generalized to unseen configurations.

Integrating physics models within machine learning models holds considerable promise toward learning robust models with improved interpretability and abilities to extrapolate. In this work, we focus on the integration of incomplete physics models into deep generative models. In particular, we introduce an architecture of variational autoencoders (VAEs) in which a part of the latent space is grounded by physics. A key technical challenge is to strike a balance between the incomplete physics and trainable components such as neural networks for ensuring that the physics part is used in a meaningful manner. To this end, we propose a regularized learning method that controls the effect of the trainable components and preserves the semantics of the physics-based latent variables as intended. We not only demonstrate generative performance improvements over a set of synthetic and real-world datasets, but we also show that we learn robust models that can consistently extrapolate beyond the training distribution in a meaningful manner. Moreover, we show that we can control the generative process in an interpretable manner.

We present ConDiff, a novel dataset for scientific machine learning. ConDiff focuses on the parametric diffusion equation with space dependent coefficients, a fundamental problem in many applications of partial differential equations (PDEs). The main novelty of the proposed dataset is that we consider discontinuous coefficients with high contrast. These coefficient functions are sampled from a selected set of distributions. This class of problems is not only of great academic interest, but is also the basis for describing various environmental and industrial problems. In this way, ConDiff shortens the gap with real-world problems while remaining fully synthetic and easy to use. ConDiff consists of a diverse set of diffusion equations with coefficients covering a wide range of contrast levels and heterogeneity with a measurable complexity metric for clearer comparison between different coefficient functions. We baseline ConDiff on standard deep learning models in the field of scientific machine learning. By providing a large number of problem instances, each with its own coefficient function and right-hand side, we hope to encourage the development of novel physics-based deep learning approaches, such as neural operators, ultimately driving progress towards more accurate and efficient solutions of complex PDE problems.

Diffusion models can generate realistic videos, but existing methods rely on implicitly learning physical reasoning from large-scale text-video datasets, which is costly, difficult to scale, and still prone to producing implausible motions that violate fundamental physical laws. We introduce a training-free framework that improves physical plausibility at inference time by explicitly reasoning about implausibility and guiding the generation away from it. Specifically, we employ a lightweight physics-aware reasoning pipeline to construct counterfactual prompts that deliberately encode physics-violating behaviors. Then, we propose a novel Synchronized Decoupled Guidance (SDG) strategy, which leverages these prompts through synchronized directional normalization to counteract lagged suppression and trajectory-decoupled denoising to mitigate cumulative trajectory bias, ensuring that implausible content is suppressed immediately and consistently throughout denoising. Experiments across different physical domains show that our approach substantially enhances physical fidelity while maintaining photorealism, despite requiring no additional training. Ablation studies confirm the complementary effectiveness of both the physics-aware reasoning component and SDG. In particular, the aforementioned two designs of SDG are also individually validated to contribute critically to the suppression of implausible content and the overall gains in physical plausibility. This establishes a new and plug-and-play physics-aware paradigm for video generation.

Foundation models have revolutionized natural language processing through a ``train once, deploy anywhere'' paradigm, where a single pre-trained model adapts to countless downstream tasks without retraining. Access to a Physics Foundation Model (PFM) would be transformative -- democratizing access to high-fidelity simulations, accelerating scientific discovery, and eliminating the need for specialized solver development. Yet current physics-aware machine learning approaches remain fundamentally limited to single, narrow domains and require retraining for each new system. We present the General Physics Transformer (GPhyT), trained on 1.8 TB of diverse simulation data, that demonstrates foundation model capabilities are achievable for physics. Our key insight is that transformers can learn to infer governing dynamics from context, enabling a single model to simulate fluid-solid interactions, shock waves, thermal convection, and multi-phase dynamics without being told the underlying equations. GPhyT achieves three critical breakthroughs: (1) superior performance across multiple physics domains, outperforming specialized architectures by up to 29x, (2) zero-shot generalization to entirely unseen physical systems through in-context learning, and (3) stable long-term predictions through 50-timestep rollouts. By establishing that a single model can learn generalizable physical principles from data alone, this work opens the path toward a universal PFM that could transform computational science and engineering.

Neural surrogates promise large speedups over classical solvers for physical dynamics but fail silently at sharp dynamical events such as shocks, fronts, and contact. We present hybrid neural world models for physical dynamics: a recipe for training and deploying multi-horizon surrogates in physical state space, where a single network with continuous horizon conditioning is trained with direct supervision against textbook reference solvers to predict any future state at horizon T in one forward pass. Although no part of the training data, loss function, or architecture supervises discontinuity location, the trained surrogate encodes it implicitly, recoverable from its forward passes alone as a per-trajectory error map that concentrates on shocks, fronts, and contacts, and stays small elsewhere. The map is competitive with or better than standard label-free baselines including deep ensembles, learned error heads, gradient-magnitude indicators, and locally-adaptive conformal prediction, while using only a single trained network and requiring no calibration set or governing-equation knowledge. The recipe supports two operating points. Mode 1 runs the surrogate alone for maximum throughput, with same-hardware CPU speedups of 26x to 72x against textbook solvers on the PDE environments. Mode 2 uses the error map to gate a reference-solver fallback, deferring uncertain trajectories and roughly halving the surrogate's residual error at the default operating point. The recipe applies without modification across reaction-diffusion, compressible Euler, and rigid-body collision dynamics.

We propose a neural physics system for real-time, interactive fluid simulations. Traditional physics-based methods, while accurate, are computationally intensive and suffer from latency issues. Recent machine-learning methods reduce computational costs while preserving fidelity; yet most still fail to satisfy the latency constraints for real-time use and lack support for interactive applications. To bridge this gap, we introduce a novel hybrid method that integrates numerical simulation, neural physics, and generative control. Our neural physics jointly pursues low-latency simulation and high physical fidelity by employing a fallback safeguard to classical numerical solvers. Furthermore, we develop a diffusion-based controller that is trained using a reverse modeling strategy to generate external dynamic force fields for fluid manipulation. Our system demonstrates robust performance across diverse 2D/3D scenarios, material types, and obstacle interactions, achieving real-time simulations at high frame rates (11~29% latency) while enabling fluid control guided by user-friendly freehand sketches. We present a significant step towards practical, controllable, and physically plausible fluid simulations for real-time interactive applications. We promise to release both models and data upon acceptance.

With diffusion and flow matching models achieving state-of-the-art generating performance, the interest of the community now turned to reducing the inference time without sacrificing sample quality. Consistency Models (CMs), which are trained to be consistent on diffusion or probability flow ordinary differential equation (PF-ODE) trajectories, enable one or two-step flow or diffusion sampling. However, CMs typically require prolonged training with large batch sizes to obtain competitive sample quality. In this paper, we examine the training dynamics of CMs near convergence and discover that CM tangents -- CM output update directions -- are quite oscillatory, in the sense that they move parallel to the data manifold, not towards the manifold. To mitigate oscillatory tangents, we propose a new loss function, called the manifold feature distance (MFD), which provides manifold-aligned tangents that point toward the data manifold. Consequently, our method -- dubbed Align Your Tangent (AYT) -- can accelerate CM training by orders of magnitude and even out-perform the learned perceptual image patch similarity metric (LPIPS). Furthermore, we find that our loss enables training with extremely small batch sizes without compromising sample quality. Code: https://github.com/1202kbs/AYT

Discovering the underlying physical behavior of complex systems is a crucial, but less well-understood topic in many engineering disciplines. This study proposes a finite-difference inspired convolutional neural network framework to learn hidden partial differential equations from given data and iteratively estimate future dynamical behavior. The methodology designs the filter sizes such that they mimic the finite difference between the neighboring points. By learning the governing equation, the network predicts the future evolution of the solution by using only a few trainable parameters. In this paper, we provide numerical results to compare the efficiency of the second-order Trust-Region Conjugate Gradient (TRCG) method with the first-order ADAM optimizer.

Offline goal-conditioned reinforcement learning (GCRL) learns goal-conditioned policies from static pre-collected datasets. However, accurate value estimation remains a challenge due to the limited coverage of the state-action space. Recent physics-informed approaches have sought to address this by imposing physical and geometric constraints on the value function through regularization defined over first-order partial differential equations (PDEs), such as the Eikonal equation. However, these formulations can often be ill-posed in complex, high-dimensional environments. In this work, we propose a physics-informed regularization derived from the viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. By providing a physics-based inductive bias, our approach grounds the learning process in optimal control theory, explicitly regularizing and bounding updates during value iterations. Furthermore, we leverage the Feynman-Kac theorem to recast the PDE solution as an expectation, enabling a tractable Monte Carlo estimation of the objective that avoids numerical instability in higher-order gradients. Experiments demonstrate that our method improves geometric consistency, making it broadly applicable to navigation and high-dimensional, complex manipulation tasks. Open-source codes are available at https://github.com/HrishikeshVish/phys-fk-value-GCRL.

State-of-the-art video generative models produce promising visual content yet often violate basic physics principles, limiting their utility. While some attribute this deficiency to insufficient physics understanding from pre-training, we find that the shortfall in physics plausibility also stems from suboptimal inference strategies. We therefore introduce WMReward and treat improving physics plausibility of video generation as an inference-time alignment problem. In particular, we leverage the strong physics prior of a latent world model (here, VJEPA-2) as a reward to search and steer multiple candidate denoising trajectories, enabling scaling test-time compute for better generation performance. Empirically, our approach substantially improves physics plausibility across image-conditioned, multiframe-conditioned, and text-conditioned generation settings, with validation from human preference study. Notably, in the ICCV 2025 Perception Test PhysicsIQ Challenge, we achieve a final score of 62.64%, winning first place and outperforming the previous state of the art by 7.42%. Our work demonstrates the viability of using latent world models to improve physics plausibility of video generation, beyond this specific instantiation or parameterization.

Foundation models have achieved remarkable success across video, image, and language domains. By scaling up the number of parameters and training datasets, these models acquire generalizable world knowledge and often surpass task-specific approaches. However, such progress has yet to extend to the domain of physics simulation. A primary bottleneck is data scarcity: while millions of images, videos, and textual resources are readily available on the internet, the largest physics simulation datasets contain only tens of thousands of samples. This data limitation hinders the use of large models, as overfitting becomes a major concern. As a result, physics applications typically rely on small models, which struggle with long-range prediction due to limited context understanding. Additionally, unlike images, videos, or text-which typically exhibit fixed granularity-physics datasets often vary drastically in scale, amplifying the challenges of scaling up multitask training. We introduce PhysiX, the first large-scale foundation model for physics simulation. PhysiX is a 4.5B parameter autoregressive generative model. It uses a discrete tokenizer to encode physical processes at different scales into a sequence of discrete tokens, and employs an autoregressive next-token prediction objective to model such processes in the token space. To mitigate the rounding error in the discretization process, PhysiX incorporates a specialized refinement module. Through extensive experiments, we show that PhysiX effectively addresses the data bottleneck, outperforming task-specific baselines under comparable settings as well as the previous absolute state-of-the-art approaches on The Well benchmark. Our results indicate that knowledge learned from natural videos can be successfully transferred to physics simulation, and that joint training across diverse simulation tasks enables synergistic learning.

Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural operators offer a fast alternative, they typically rely on large-scale labeled datasets or struggle with unstable training dynamics when using physics-informed residual losses. We propose NPSolver, a neural Poisson solver trained without solution labels via iterative physics supervision. Instead of relying on fully converged numerical solutions or raw PDE residuals, NPSolver utilizes a small number of preconditioned conjugate gradient (PCG) steps to refine its own predictions, providing a more stable and well-scaled training signal. Theoretical analysis confirms that this iterative supervision serves as a well-conditioned error proxy and that a stop-gradient design is essential for optimization stability. To better capture boundary-driven features under mixed boundary conditions, we further introduce the Boundary-Aware Transolver (BA-Transolver) architecture that explicitly separates interior and boundary tokenization. Extensive evaluations on 2D and 3D irregular geometries demonstrate that NPSolver outperforms both physics-informed and data-driven baselines. Furthermore, a downstream thermal control task highlights the model's capability for conducting efficient and reliable gradient-based boundary control. We will release our codes and data at https://github.com/intell-sci-comput/NPSolver.

We introduce novel variants of momentum by incorporating the variance of the stochastic loss function. The variance characterizes the confidence or uncertainty of the local features of the averaged loss surface across the i.i.d. subsets of the training data defined by the mini-batches. We show two applications of the gradient of the variance of the loss function. First, as a bias to the conventional momentum update to encourage conformity of the local features of the loss function (e.g. local minima) across mini-batches to improve generalization and the cumulative training progress made per epoch. Second, as an alternative direction for "exploration" in the parameter space, especially, for non-convex objectives, that exploits both the optimistic and pessimistic views of the loss function in the face of uncertainty. We also introduce a novel data-driven stochastic regularization technique through the parameter update rule that is model-agnostic and compatible with arbitrary architectures. We further establish connections to probability distributions over loss functions and the REINFORCE policy gradient update with baseline in RL. Finally, we incorporate the new variants of momentum proposed into Adam, and empirically show that our methods improve the rate of convergence of training based on our experiments on the MNIST and CIFAR-10 datasets.

Humans possess an exceptional ability to imagine 4D scenes, encompassing both motion and 3D geometry, from a single still image. This ability is rooted in our accumulated observations of similar scenes and an intuitive understanding of physics. In this paper, we aim to replicate this capacity in neural networks, specifically focusing on natural fluid imagery. Existing methods for this task typically employ simplistic 2D motion estimators to animate the image, leading to motion predictions that often defy physical principles, resulting in unrealistic animations. Our approach introduces a novel method for generating 4D scenes with physics-consistent animation from a single image. We propose the use of a physics-informed neural network that predicts motion for each surface point, guided by a loss term derived from fundamental physical principles, including the Navier-Stokes equations. To capture appearance, we predict feature-based 3D Gaussians from the input image and its estimated depth, which are then animated using the predicted motions and rendered from any desired camera perspective. Experimental results highlight the effectiveness of our method in producing physically plausible animations, showcasing significant performance improvements over existing methods. Our project page is https://physfluid.github.io/ .

Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime.

Deep Learning using the eponymous deep neural networks (DNNs) has become an attractive approach towards various data-based problems of theoretical physics in the past decade. There has been a clear trend to deeper architectures containing increasingly more powerful and involved layers. Contrarily, Taylor coefficients of DNNs still appear mainly in the light of interpretability studies, where they are computed at most to first order. However, especially in theoretical physics numerous problems benefit from accessing higher orders, as well. This gap motivates a general formulation of neural network (NN) Taylor expansions. Restricting our analysis to multilayer perceptrons (MLPs) and introducing quantities we refer to as propagators and vertices, both depending on the MLP's weights and biases, we establish a graph-theoretical approach. Similarly to Feynman rules in quantum field theories, we can systematically assign diagrams containing propagators and vertices to the corresponding partial derivative. Examining this approach for S-wave scattering lengths of shallow potentials, we observe NNs to adapt their derivatives mainly to the leading order of the target function's Taylor expansion. To circumvent this problem, we propose an iterative NN perturbation theory. During each iteration we eliminate the leading order, such that the next-to-leading order can be faithfully learned during the subsequent iteration. After performing two iterations, we find that the first- and second-order Born terms are correctly adapted during the respective iterations. Finally, we combine both results to find a proxy that acts as a machine-learned second-order Born approximation.

Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.

Physics-Informed Neural Networks (PINNs), which incorporate PDEs as soft constraints, train with a composite loss function that contains multiple training point types: different types of collocation points chosen during training to enforce each PDE and initial/boundary conditions, and experimental points which are usually costly to obtain via experiments or simulations. Training PINNs using this loss function is challenging as it typically requires selecting large numbers of points of different types, each with different training dynamics. Unlike past works that focused on the selection of either collocation or experimental points, this work introduces PINN Adaptive ColLocation and Experimental points selection (PINNACLE), the first algorithm that jointly optimizes the selection of all training point types, while automatically adjusting the proportion of collocation point types as training progresses. PINNACLE uses information on the interaction among training point types, which had not been considered before, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). We theoretically show that the criterion used by PINNACLE is related to the PINN generalization error, and empirically demonstrate that PINNACLE is able to outperform existing point selection methods for forward, inverse, and transfer learning problems.

Recent advances in Neural Fields have enabled powerful, discretization-invariant methods for learning neural operators that approximate solutions of Partial Differential Equations (PDEs) on general geometries. Building on these developments, we introduce enf2enf, an encoder--decoder methodology for predicting steady-state Partial Differential Equations with non-parameterized geometric variability, based on recently proposed Equivariant Neural Field architectures. In enf2enf, input geometries are encoded into latent point cloud embeddings that inherently preserve geometric grounding and capture local phenomena. The resulting representations are then combined with global parameters and directly decoded into continuous output fields, thus efficiently modeling the coupling between geometry and physics. By leveraging the inductive biases of locality and translation invariance, our approach is able to capture fine-scale physical features as well as complex shape variations, thereby enhancing generalization and physical compliance. Extensive experiments on a high-fidelity aerodynamic dataset, a hyper-elastic material benchmark, and multi-element airfoil geometries, demonstrate that the proposed model achieves superior or competitive performance compared to state-of-the-art graph based, operator learning, and neural field methods. Notably, our method supports real time inference and zero-shot super-resolution, enabling efficient training on low-resolution meshes while maintaining high accuracy on full-scale discretizations.

High-fidelity scientific simulations are now producing unprecedented amounts of data, creating a storage and analysis bottleneck. A single simulation can generate tremendous data volumes, often forcing researchers to discard valuable information. A prime example of this is plasma turbulence described by the gyrokinetic equations: nonlinear, multiscale, and 5D in phase space. It constitutes one of the most computationally demanding frontiers of modern science, with runs taking weeks and yielding tens of terabytes of data dumps. The increasing storage demands underscore the importance of compression. However, reconstructed snapshots do not necessarily preserve essential physical quantities. We present a spatiotemporal evaluation pipeline, accounting for structural phenomena and multi-scale transient fluctuations to assess the degree of physical fidelity. Indeed, we find that various compression techniques lack preservation of both spatial mode structure and temporal turbulence characteristics. Therefore, we explore Physics-Informed Neural Compression (PINC), which incorporates physics-informed losses tailored to gyrokinetics and enables extreme compressions ratios of over 70,000x. Entropy coding on top of PINC further pushes it to 120,000x. This direction provides a viable and scalable solution to the prohibitive storage demands of gyrokinetics, enabling post-hoc analyses that were previously infeasible.

Recently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schr\"odinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches.

The approximation of Partial Differential Equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and non-linear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive biases of neural networks. However, they usually require very high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting issues. In addition, the fixed positions of the mesh parameters restrict their flexibility, making it challenging to accurately approximate complex PDEs. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

In-line digital holography (DIH) is a widely used lensless imaging technique, valued for its simplicity and capability to image samples at high throughput. However, capturing only intensity of the interference pattern during the recording process gives rise to some unwanted terms such as cross-term and twin-image. The cross-term can be suppressed by adjusting the intensity of reference wave, but the twin-image problem remains. The twin-image is a spectral artifact that superimposes a defocused conjugate wave onto the reconstructed object, severely degrading image quality. While deep learning has recently emerged as a powerful tool for phase retrieval, traditional Convolutional Neural Networks (CNNs) are limited by their local receptive fields, making them less effective at capturing the global diffraction patterns inherent in holography. In this study, we introduce HoloPASWIN, a physics-aware deep learning framework based on the Swin Transformer architecture. By leveraging hierarchical shifted-window attention, our model efficiently captures both local details and long-range dependencies essential for accurate holographic reconstruction. We propose a comprehensive loss function that integrates frequency-domain constraints with physical consistency via a differentiable angular spectrum propagator, ensuring high spectral fidelity. Validated on a large-scale synthetic dataset of 25,000 samples with diverse noise configurations (speckle, shot, read, and dark noise), HoloPASWIN demonstrates effective twin-image suppression and robust reconstruction quality.

Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing the "failure modes" of PINNs, although a thorough understanding of the connection between PINN failure modes and sampling strategies is missing. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that training PINNs relies on successful "propagation" of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures, characterized by highly imbalanced PDE residual fields. To mitigate propagation failures, we propose a novel Retain-Resample-Release sampling (R3) algorithm that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of R3 sampling to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of R3 sampling and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.

We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of parametric PDEs without requiring task-specific fine-tuning. Our approach combines a Swin Transformer-based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics-informed objectives. The model maps PDE parametrizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions with varying source terms, geometries, and mixed Dirichlet/Neumann boundary conditions, including interior boundaries. HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN literature, outperforming U-Nets, Poseidon, and Physics-Informed Neural Operators (PINO). Further, we introduce an iterative refinement procedure that compares the physics of the generated PINN to the requested PDE and uses the discrepancy to generate a "delta" PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves over 100x gain in average L_2 loss in the best case, while retaining forward-only inference. Additionally, we evaluate the fine-tuning behavior of PINNs initialized by HyPINO and show that they converge faster and to lower final error than both randomly initialized and Reptile-meta-learned PINNs on five benchmarks, performing on par on the remaining two. Our results highlight the potential of this scalable approach as a foundation for extending neural operators toward solving increasingly complex, nonlinear, and high-dimensional PDE problems with significantly improved accuracy and reduced computational cost.

When solving inverse problems, it is increasingly popular to use pre-trained diffusion models as plug-and-play priors. This framework can accommodate different forward models without re-training while preserving the generative capability of diffusion models. Despite their success in many imaging inverse problems, most existing methods rely on privileged information such as derivative, pseudo-inverse, or full knowledge about the forward model. This reliance poses a substantial limitation that restricts their use in a wide range of problems where such information is unavailable, such as in many scientific applications. To address this issue, we propose Ensemble Kalman Diffusion Guidance (EnKG) for diffusion models, a derivative-free approach that can solve inverse problems by only accessing forward model evaluations and a pre-trained diffusion model prior. We study the empirical effectiveness of our method across various inverse problems, including scientific settings such as inferring fluid flows and astronomical objects, which are highly non-linear inverse problems that often only permit black-box access to the forward model.

Physical principles are fundamental to realistic visual simulation, but remain a significant oversight in transformer-based video generation. This gap highlights a critical limitation in rendering rigid body motion, a core tenet of classical mechanics. While computer graphics and physics-based simulators can easily model such collisions using Newton formulas, modern pretrain-finetune paradigms discard the concept of object rigidity during pixel-level global denoising. Even perfectly correct mathematical constraints are treated as suboptimal solutions (i.e., conditions) during model optimization in post-training, fundamentally limiting the physical realism of generated videos. Motivated by these considerations, we introduce, for the first time, a physics-aware reinforcement learning paradigm for video generation models that enforces physical collision rules directly in high-dimensional spaces, ensuring the physics knowledge is strictly applied rather than treated as conditions. Subsequently, we extend this paradigm to a unified framework, termed Mimicry-Discovery Cycle (MDcycle), which allows substantial fine-tuning while fully preserving the model's ability to leverage physics-grounded feedback. To validate our approach, we construct new benchmark PhysRVGBench and perform extensive qualitative and quantitative experiments to thoroughly assess its effectiveness.

Traditional experimental and numerical approaches for fluid dynamics problems often suffer from high computational cost, mesh sensitivity, and limited capability in capturing complex physical behaviors. Moreover, conventional physics-informed neural networks (PINNs) frequently struggle in chaotic and highly unsteady flow regimes. In this work, we propose PhysicsFormer, a fast and efficient transformer-based physics-informed framework that incorporates multi-head encoder-decoder cross-attention. Unlike multilayer perceptron-based PINNs, PhysicsFormer operates on sequential representations constructed from spatio-temporal data, enabling effective learning of long-range temporal dependencies and improved propagation of initial condition information. A data-embedding strategy is employed to convert spatio-temporal points into pseudo-sequences, while a dynamics-weighted loss function replaces the standard PINNs formulation. Owing to its parallel learning structure, PhysicsFormer demonstrates superior computational efficiency compared to existing transformer-based approaches. The framework is validated on Burgers' equation and flow reconstruction governed by the Navier-Stokes equations, achieving mean squared errors on the order of 10^{-6}. In addition, an inverse problem involving parameter identification in the two-dimensional incompressible Navier-Stokes equations is investigated. For clean data, PhysicsFormer achieves zero identification error for both λ_1 and λ_2; under 1% Gaussian noise, the errors are 0.07% for λ_1 and 0% for λ_2. These results demonstrate that PhysicsFormer provides a reliable and computationally efficient surrogate modeling framework for time-dependent fluid flow problems.

Physics-informed neural networks (PINNs) leverage neural-networks to find the solutions of partial differential equation (PDE)-constrained optimization problems with initial conditions and boundary conditions as soft constraints. These soft constraints are often considered to be the sources of the complexity in the training phase of PINNs. Here, we demonstrate that the challenge of training (i) persists even when the boundary conditions are strictly enforced, and (ii) is closely related to the Kolmogorov n-width associated with problems demonstrating transport, convection, traveling waves, or moving fronts. Given this realization, we describe the mechanism underlying the training schemes such as those used in eXtended PINNs (XPINN), curriculum regularization, and sequence-to-sequence learning. For an important category of PDEs, i.e., governed by non-linear convection-diffusion equation, we propose reformulating PINNs on a Lagrangian frame of reference, i.e., LPINNs, as a PDE-informed solution. A parallel architecture with two branches is proposed. One branch solves for the state variables on the characteristics, and the second branch solves for the low-dimensional characteristics curves. The proposed architecture conforms to the causality innate to the convection, and leverages the direction of travel of the information in the domain. Finally, we demonstrate that the loss landscapes of LPINNs are less sensitive to the so-called "complexity" of the problems, compared to those in the traditional PINNs in the Eulerian framework.

Reconstructing physics-based 3D assets -- geometry, materials, and illumination -- from multi-view images is a core problem in computer graphics and vision, and a prerequisite for realistic relighting and editing. Physics-based inverse rendering offers an accurate image-formation model, but is severely underconstrained: without strong priors, illumination is baked into materials, and reconstructions generalize poorly to novel views and lighting. Data-driven diffusion models, in contrast, predict visually plausible materials, yet their predictions rarely satisfy the rendering equation and are not directly usable for physics-based rendering. We bridge these two paradigms rather than replacing either. Our key idea is to treat the predictions of a state-of-the-art diffusion model not as target material values but as a similarity kernel for optimization: we introduce a regularization loss that penalizes deviations in the optimized material over surface regions where the diffusion predictions are near-constant, while leaving the optimization free to match the input images. Built on this regularizer, our end-to-end pipeline jointly reconstructs geometry, materials, and illumination, yielding high-quality assets that drop into standard rendering pipelines and relight faithfully. On the Synthetic4Relight, Stanford-ORB, and DTC-Synthetic datasets, our method significantly outperforms state-of-the-art baselines in both reconstruction accuracy and relighting quality.

Cross-entropy loss has long been the standard choice for training deep neural networks, yet it suffers from interpretability limitations, unbounded weight growth, and inefficiencies that can contribute to costly training dynamics. The harmonic loss is a distance-based alternative grounded in Euclidean geometry that improves interpretability and mitigates phenomena such as grokking, or delayed generalization on the test set. However, the study of harmonic loss remains narrow: only Euclidean distance is explored, and no systematic evaluation of computational efficiency or sustainability was conducted. We extend harmonic loss by systematically investigating a broad spectrum of distance metrics as replacements for the Euclidean distance. We comprehensively evaluate distance-tailored harmonic losses on both vision backbones and large language models. Our analysis is framed around a three-way evaluation of model performance, interpretability, and sustainability. On vision tasks, cosine distances provide the most favorable trade-off, consistently improving accuracy while lowering carbon emissions, whereas Bray-Curtis and Mahalanobis further enhance interpretability at varying efficiency costs. On language models, cosine-based harmonic losses improve gradient and learning stability, strengthen representation structure, and reduce emissions relative to cross-entropy and Euclidean heads. Our code is available at: https://anonymous.4open.science/r/rethinking-harmonic-loss-5BAB/.

Loss landscape is a useful tool to characterize and compare neural network models. The main challenge for analysis of loss landscape for the deep neural networks is that they are generally highly non-convex in very high dimensional space. In this paper, we develop "the roughness"concept for understanding such landscapes in high dimensions and apply this technique to study two neural network models arising from solving differential equations. Our main innovation is the proposal of a well-defined and easy-to-compute roughness index (RI) which is based on the mean and variance of the (normalized) total variation for one-dimensional functions projected on randomly sampled directions. A large RI at the local minimizer hints an oscillatory landscape profile and indicates a severe challenge for the first-order optimization method. Particularly, we observe the increasing-then-decreasing pattern for RI along the gradient descent path in most models. We apply our method to two types of loss functions used to solve partial differential equations (PDEs) when the solution of PDE is parametrized by neural networks. Our empirical results on these PDE problems reveal important and consistent observations that the landscapes from the deep Galerkin method around its local minimizers are less rough than the deep Ritz method.

Cross-entropy loss and focal loss are the most common choices when training deep neural networks for classification problems. Generally speaking, however, a good loss function can take on much more flexible forms, and should be tailored for different tasks and datasets. Motivated by how functions can be approximated via Taylor expansion, we propose a simple framework, named PolyLoss, to view and design loss functions as a linear combination of polynomial functions. Our PolyLoss allows the importance of different polynomial bases to be easily adjusted depending on the targeting tasks and datasets, while naturally subsuming the aforementioned cross-entropy loss and focal loss as special cases. Extensive experimental results show that the optimal choice within the PolyLoss is indeed dependent on the task and dataset. Simply by introducing one extra hyperparameter and adding one line of code, our Poly-1 formulation outperforms the cross-entropy loss and focal loss on 2D image classification, instance segmentation, object detection, and 3D object detection tasks, sometimes by a large margin.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

Graph Neural Networks (GNNs) with equivariant properties have emerged as powerful tools for modeling complex dynamics of multi-object physical systems. However, their generalization ability is limited by the inadequate consideration of physical inductive biases: (1) Existing studies overlook the continuity of transitions among system states, opting to employ several discrete transformation layers to learn the direct mapping between two adjacent states; (2) Most models only account for first-order velocity information, despite the fact that many physical systems are governed by second-order motion laws. To incorporate these inductive biases, we propose the Second-order Equivariant Graph Neural Ordinary Differential Equation (SEGNO). Specifically, we show how the second-order continuity can be incorporated into GNNs while maintaining the equivariant property. Furthermore, we offer theoretical insights into SEGNO, highlighting that it can learn a unique trajectory between adjacent states, which is crucial for model generalization. Additionally, we prove that the discrepancy between this learned trajectory of SEGNO and the true trajectory is bounded. Extensive experiments on complex dynamical systems including molecular dynamics and motion capture demonstrate that our model yields a significant improvement over the state-of-the-art baselines.

Many evaluation metrics can be used to assess the performance of models in binary classification tasks. However, most of them are derived from a confusion matrix in a non-differentiable form, making it very difficult to generate a differentiable loss function that could directly optimize them. The lack of solutions to bridge this challenge not only hinders our ability to solve difficult tasks, such as imbalanced learning, but also requires the deployment of computationally expensive hyperparameter search processes in model selection. In this paper, we propose a general-purpose approach that transforms any confusion matrix-based metric into a loss function, AnyLoss, that is available in optimization processes. To this end, we use an approximation function to make a confusion matrix represented in a differentiable form, and this approach enables any confusion matrix-based metric to be directly used as a loss function. The mechanism of the approximation function is provided to ensure its operability and the differentiability of our loss functions is proved by suggesting their derivatives. We conduct extensive experiments under diverse neural networks with many datasets, and we demonstrate their general availability to target any confusion matrix-based metrics. Our method, especially, shows outstanding achievements in dealing with imbalanced datasets, and its competitive learning speed, compared to multiple baseline models, underscores its efficiency.

The Mechanistic Interpretability (MI) program has mapped the Transformer as a precise computational graph. We extend this graph with a conservation law and time-varying AC dynamics, viewing it as a physical circuit. We introduce Momentum Attention, a symplectic augmentation embedding physical priors via the kinematic difference operator p_t = q_t - q_{t-1}, implementing the symplectic shear q_t = q_t + γp_t on queries and keys. We identify a fundamental Symplectic-Filter Duality: the physical shear is mathematically equivalent to a High-Pass Filter. This duality is our cornerstone contribution -- by injecting kinematic momentum, we sidestep the topological depth constraint (L geq 2) for induction head formation. While standard architectures require two layers for induction from static positions, our extension grants direct access to velocity, enabling Single-Layer Induction and Spectral Forensics via Bode Plots. We formalize an Orthogonality Theorem proving that DC (semantic) and AC (mechanistic) signals segregate into orthogonal frequency bands when Low-Pass RoPE interacts with High-Pass Momentum. Validated through 5,100+ controlled experiments (documented in Supplementary Appendices A--R and 27 Jupyter notebooks), our 125M Momentum model exceeds expectations on induction-heavy tasks while tracking a 350M baseline within sim2.9% validation loss. Dedicated associative recall experiments reveal a scaling law γ^* = 4.17 times N^{-0.74} establishing momentum-depth fungibility. We offer this framework as a complementary analytical toolkit connecting Generative AI, Hamiltonian Physics, and Signal Processing.

We propose Neural Field Thermal Tomography (NeFTY), a differentiable physics framework for the quantitative 3D reconstruction of material properties from transient surface temperature measurements. While traditional thermography relies on pixel-wise 1D approximations that neglect lateral diffusion, and soft-constrained Physics-Informed Neural Networks (PINNs) often fail in transient diffusion scenarios due to gradient stiffness, NeFTY parameterizes the 3D diffusivity field as a continuous neural field optimized through a rigorous numerical solver. By leveraging a differentiable physics solver, our approach enforces thermodynamic laws as hard constraints while maintaining the memory efficiency required for high-resolution 3D tomography. Our discretize-then-optimize paradigm effectively mitigates the spectral bias and ill-posedness inherent in inverse heat conduction, enabling the recovery of subsurface defects at arbitrary scales. Experimental validation on synthetic data demonstrates that NeFTY significantly improves the accuracy of subsurface defect localization over baselines. Additional details at https://cab-lab-princeton.github.io/nefty/

Simulating physical systems governed by Lagrangian dynamics often entails solving partial differential equations (PDEs) over high-resolution spatial domains, leading to significant computational expense. Reduced-order modeling (ROM) mitigates this cost by evolving low-dimensional latent representations of the underlying system. While neural ROMs enable querying solutions from latent states at arbitrary spatial points, their latent states typically represent the global domain and struggle to capture localized, highly dynamic behaviors such as fluids. We propose a sampling-based reduction framework that evolves Lagrangian systems directly in physical space over the particles themselves, reducing the number of active degrees of freedom via data-driven neural PDE operators. To enable querying at arbitrary spatial locations, we introduce a learnable kernel parameterization that uses local spatial information from time-evolved sample particles to infer the underlying solution manifold. Empirically, our approach achieves a 6.6x to 32x reduction in input dimensionality while maintaining high-fidelity evaluations across diverse Lagrangian regimes, including fluid flows, granular media, and elastoplastic dynamics. We refer to this framework as GIOROM (Geometry-Informed Reduced-Order Modeling). All code and data are available at: https://github.com/HrishikeshVish/GIOROM

To accelerate diffusion model inference, numerical solvers perform poorly at extremely small steps, while distillation techniques often introduce complexity and instability. This work presents an intermediate strategy, balancing performance and cost, by learning ODE integration using loss functions derived from the derivative-integral relationship, inspired by Monte Carlo integration and Picard iteration. From a geometric perspective, the losses operate by gradually extending the tangent to the secant, thus are named as secant losses. The target of secant losses is the same as that of diffusion models, or the diffusion model itself, leading to great training stability. By fine-tuning or distillation, the secant version of EDM achieves a 10-step FID of 2.14 on CIFAR-10, while the secant version of SiT-XL/2 attains a 4-step FID of 2.27 and an 8-step FID of 1.96 on ImageNet-256times256. Code is available at https://github.com/poppuppy/secant-expectation.

Recent advances in deep learning have inspired numerous works on data-driven solutions to partial differential equation (PDE) problems. These neural PDE solvers can often be much faster than their numerical counterparts; however, each presents its unique limitations and generally balances training cost, numerical accuracy, and ease of applicability to different problem setups. To address these limitations, we introduce several methods to apply latent diffusion models to physics simulation. Firstly, we introduce a mesh autoencoder to compress arbitrarily discretized PDE data, allowing for efficient diffusion training across various physics. Furthermore, we investigate full spatio-temporal solution generation to mitigate autoregressive error accumulation. Lastly, we investigate conditioning on initial physical quantities, as well as conditioning solely on a text prompt to introduce text2PDE generation. We show that language can be a compact, interpretable, and accurate modality for generating physics simulations, paving the way for more usable and accessible PDE solvers. Through experiments on both uniform and structured grids, we show that the proposed approach is competitive with current neural PDE solvers in both accuracy and efficiency, with promising scaling behavior up to sim3 billion parameters. By introducing a scalable, accurate, and usable physics simulator, we hope to bring neural PDE solvers closer to practical use.

Fast, geometry-generalizing surrogates for unsteady flow remain challenging. We present a time-dependent, geometry-aware Deep Operator Network that predicts velocity fields for moderate-Re flows around parametric and non-parametric shapes. The model encodes geometry via a signed distance field (SDF) trunk and flow history via a CNN branch, trained on 841 high-fidelity simulations. On held-out shapes, it attains sim 5% relative L2 single-step error and up to 1000X speedups over CFD. We provide physics-centric rollout diagnostics, including phase error at probes and divergence norms, to quantify long-horizon fidelity. These reveal accurate near-term transients but error accumulation in fine-scale wakes, most pronounced for sharp-cornered geometries. We analyze failure modes and outline practical mitigations. Code, splits, and scripts are openly released at: https://github.com/baskargroup/TimeDependent-DeepONet to support reproducibility and benchmarking.

Learning complex dynamics driven by partial differential equations directly from data holds great promise for fast and accurate simulations of complex physical systems. In most cases, this problem can be formulated as an operator learning task, where one aims to learn the operator representing the physics of interest, which entails discretization of the continuous system. However, preserving key continuous properties at the discrete level, such as boundary conditions, and addressing physical systems with complex geometries is challenging for most existing approaches. We introduce a family of operator learning architectures, structure-preserving operator networks (SPONs), that allows to preserve key mathematical and physical properties of the continuous system by leveraging finite element (FE) discretizations of the input-output spaces. SPONs are encode-process-decode architectures that are end-to-end differentiable, where the encoder and decoder follows from the discretizations of the input-output spaces. SPONs can operate on complex geometries, enforce certain boundary conditions exactly, and offer theoretical guarantees. Our framework provides a flexible way of devising structure-preserving architectures tailored to specific applications, and offers an explicit trade-off between performance and efficiency, all thanks to the FE discretization of the input-output spaces. Additionally, we introduce a multigrid-inspired SPON architecture that yields improved performance at higher efficiency. Finally, we release a software to automate the design and training of SPON architectures.

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields.Neural networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space transformations. While existing benchmarks of machine learning methods for PDEs primarily focus on the forward problem, there are no similar comprehensive studies and benchmark datasets on PDE inverse problems, i.e., mapping solution fields to underlying physical parameters. We fill this gap by introducing PDEInvBench, a comprehensive benchmark dataset consisting of numerical simulations for both time-dependent and time-independent PDEs across a wide range of physical behaviors and parameters. Our dataset includes evaluation splits that assess performance in both in-distribution and various out-of-distribution settings. Using our benchmark dataset, we comprehensively explore the design space of neural networks for PDE inverse problems along three key dimensions: (1) optimization procedures, analyzing the role of supervised, self-supervised, and test-time training objectives on performance, (2) problem representations, where we study the value of architectural choices with different inductive biases and various conditioning strategies, and (3) scaling, which we perform with respect to both model and data size. Our experiments reveal several practical insights: 1) neural networks perform best with a two-stage training procedure: initial supervision with PDE parameters followed by test-time fine-tuning using the PDE residual, 2) incorporating PDE derivatives as input features consistently improves accuracy, and 3) increasing the diversity of initial conditions in the training data yields greater performance gains than expanding the range of PDE parameters. We make our dataset and codebase publicly available.

Training large neural networks (NNs) requires optimizing high-dimensional data-dependent loss functions. The optimization landscape of these functions is often highly complex and textured, even fractal-like, with many spurious local minima, ill-conditioned valleys, degenerate points, and saddle points. Complicating things further is the fact that these landscape characteristics are a function of the data, meaning that noise in the training data can propagate forward and give rise to unrepresentative small-scale geometry. This poses a difficulty for gradient-based optimization methods, which rely on local geometry to compute updates and are, therefore, vulnerable to being derailed by noisy data. In practice,this translates to a strong dependence of the optimization dynamics on the noise in the data, i.e., poor generalization performance. To remediate this problem, we propose a new optimization procedure: Rolling Ball Optimizer (RBO), that breaks this spatial locality by incorporating information from a larger region of the loss landscape in its updates. We achieve this by simulating the motion of a rigid sphere of finite radius rolling on the loss landscape, a straightforward generalization of Gradient Descent (GD) that simplifies into it in the infinitesimal limit. The radius serves as a hyperparameter that determines the scale at which RBO sees the loss landscape, allowing control over the granularity of its interaction therewith. We are motivated by the intuition that the large-scale geometry of the loss landscape is less data-specific than its fine-grained structure, and that it is easier to optimize. We support this intuition by proving that our algorithm has a smoothing effect on the loss function. Evaluation against SGD, SAM, and Entropy-SGD, on MNIST and CIFAR-10/100 demonstrates promising results in terms of convergence speed, training accuracy, and generalization performance.

Neural scaling laws approximate a language model's loss as a power-law function of parameter count N and token count D. Following Chinchilla-style compute-optimal training, many studies fit scaling laws from runs performed under a fixed tokens-per-parameter (TPP) ratio k and set D = kN. We show that this collinear design, combined with the empirically common near-equality of the exponents governing N and D, induces an inherent ill-conditioning in the Gauss-Newton least-squares problem: the condition number of the design grows as the inverse square of the gap between the N and D-exponents. The scale coefficients become practically unidentifiable, with confidence intervals inflating by an order of magnitude or more, yielding a ``sloppy'' model whose extrapolations degrade sharply off the training ray. We prove this for four scaling-law formalisms and derive a closed-form TPP-diversity threshold that is necessary and sufficient for well-conditioned estimation. Empirically, non-collinear designs outperform collinear ones on held-out splits with a 97.3\% win rate across four laws, five corpora, multiple floating point precision modes. We further show the degeneracy is rooted in Jacobian geometry and is not an artifact of the loss function: any smooth estimation objective whose curvature involves the Jacobian inherits the same ill-conditioning.

We introduce Gradient Descent with Adaptive Momentum Scaling (Grams), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning.

A key aspect of learned partial differential equation (PDE) solvers is that the main cost often comes from generating training data with classical solvers rather than learning the model itself. Another is that there are clear axes of difficulty--e.g., more complex geometries and higher Reynolds numbers--along which problems become (1) harder for classical solvers and thus (2) more likely to benefit from neural speedups. Towards addressing this chicken-and-egg challenge, we study difficulty transfer on 2D incompressible Navier-Stokes, systematically varying task complexity along geometry (number and placement of obstacles), physics (Reynolds number), and their combination. Similar to how it is possible to spend compute to pre-train foundation models and improve their performance on downstream tasks, we find that by classically solving (analogously pre-generating) many low and medium difficulty examples and including them in the training set, it is possible to learn high-difficulty physics from far fewer samples. Furthermore, we show that by combining low and high difficulty data, we can spend 8.9x less compute on pre-generating a dataset to achieve the same error as using only high difficulty examples. Our results highlight that how we allocate classical-solver compute across difficulty levels is as important as how much we allocate overall, and suggest substantial gains from principled curation of pre-generated PDE data for neural solvers. Our code is available at https://github.com/Naman-Choudhary-AI-ML/pregenerating-pde

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under structured shifts in coefficients, boundary conditions, discretization, and rollout horizon. Yet evaluation is still often dominated by in-distribution test error, making robustness difficult to assess. We introduce a standardized stress-testing framework for neural PDE solvers under deployment-relevant shift. We instantiate it on three representative architectures -- Fourier Neural Operators (FNOs), a DeepONet-style model, and convolutional neural operators (CNOs) -- across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Across 750 trained models, we measure robustness using baseline-normalized degradation factors together with spectral and rollout diagnostics. The resulting comparisons reveal that strong in-distribution accuracy does not reliably predict robustness, and that failure patterns depend jointly on architecture and PDE family. Our results provide a clearer basis for evaluating robustness claims in neural PDE solvers and suggest that function-space generalization under structured shift should be treated as a first-class evaluation target.

Empirical risk minimization (ERM) with a computationally feasible surrogate loss is a widely accepted approach for classification. Notably, the convexity and calibration (CC) properties of a loss function ensure consistency of ERM in maximizing accuracy, thereby offering a wide range of options for surrogate losses. In this article, we propose a novel ensemble method, namely EnsLoss, which extends the ensemble learning concept to combine loss functions within the ERM framework. A key feature of our method is the consideration on preserving the "legitimacy" of the combined losses, i.e., ensuring the CC properties. Specifically, we first transform the CC conditions of losses into loss-derivatives, thereby bypassing the need for explicit loss functions and directly generating calibrated loss-derivatives. Therefore, inspired by Dropout, EnsLoss enables loss ensembles through one training process with doubly stochastic gradient descent (i.e., random batch samples and random calibrated loss-derivatives). We theoretically establish the statistical consistency of our approach and provide insights into its benefits. The numerical effectiveness of EnsLoss compared to fixed loss methods is demonstrated through experiments on a broad range of 14 OpenML tabular datasets and 46 image datasets with various deep learning architectures. Python repository and source code are available on GitHub at https://github.com/statmlben/ensloss.

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

Transfer learning has become crucial in computer vision tasks due to the vast availability of pre-trained deep learning models. However, selecting the optimal pre-trained model from a diverse pool for a specific downstream task remains a challenge. Existing methods for measuring the transferability of pre-trained models rely on statistical correlations between encoded static features and task labels, but they overlook the impact of underlying representation dynamics during fine-tuning, leading to unreliable results, especially for self-supervised models. In this paper, we present an insightful physics-inspired approach named PED to address these challenges. We reframe the challenge of model selection through the lens of potential energy and directly model the interaction forces that influence fine-tuning dynamics. By capturing the motion of dynamic representations to decline the potential energy within a force-driven physical model, we can acquire an enhanced and more stable observation for estimating transferability. The experimental results on 10 downstream tasks and 12 self-supervised models demonstrate that our approach can seamlessly integrate into existing ranking techniques and enhance their performances, revealing its effectiveness for the model selection task and its potential for understanding the mechanism in transfer learning. Code will be available at https://github.com/lixiaotong97/PED.

In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions. By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach. Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity. We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets.

Modern artificial intelligence excels at statistical interpolation within seen manifolds but fundamentally fails at the exact reasoning required for theoretical physics and mathematics. We identify the "Float Wall" -- a catastrophic collapse of neural extrapolation at scales beyond 10^{16} -- caused by standard floating-point representation and linguistic tokenization (BPE). To resolve this, we introduce the Active Discoverer Framework, a digit-native neuro-symbolic architecture designed for invariant discovery. At its core is NumberNet, a Siamese Arithmetic Transformer that utilizes least-significant-bit (LSB) sequence encoding to achieve 0% precision loss and cosmic-scale extrapolation up to 10^{50}. To enforce physical grounding, we implement a Hamiltonian-based energy descent and Symmetry Grouping layer, ensuring the model respects Noether's theorem natively. The primary innovation is the Symbolic LaTeX Bottleneck: an active discovery loop where the model is forced to hypothesize unknown physical variables through an autoregressive LaTeX decoder. By reconciling numeric "hallucinations" with structurally valid mathematical expressions, the framework ensures that any discovered physics is parsimonious and human-interpretable. We evaluate this system against a 30-billion scale benchmark and the Universal Physics Pantheon, featuring 50 "Chaos Mode" systemic perturbations. Our results demonstrate that while traditional GBDT and LLM-based architectures collapse at cosmic scales, the Active Discoverer autonomously deduces universal constants such as the Gravitational Constant (G) with high fidelity. This framework establishes a path toward zero-hallucination artificial intelligence and truly autonomous scientific research agents.

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

Video imaging is often affected by complex degradations such as blur, noise, and compression artifacts. Traditional restoration methods follow a "single-task single-model" paradigm, resulting in poor generalization and high computational cost, limiting their applicability in real-world scenarios with diverse degradation types. We propose UniFlowRestore, a general video restoration framework that models restoration as a time-continuous evolution under a prompt-guided and physics-informed vector field. A physics-aware backbone PhysicsUNet encodes degradation priors as potential energy, while PromptGenerator produces task-relevant prompts as momentum. These components define a Hamiltonian system whose vector field integrates inertial dynamics, decaying physical gradients, and prompt-based guidance. The system is optimized via a fixed-step ODE solver to achieve efficient and unified restoration across tasks. Experiments show that UniFlowRestore delivers stateof-the-art performance with strong generalization and efficiency. Quantitative results demonstrate that UniFlowRestore achieves state-of-the-art performance, attaining the highest PSNR (33.89 dB) and SSIM (0.97) on the video denoising task, while maintaining top or second-best scores across all evaluated tasks.

Existing diffusion-based text-to-3D generation methods primarily focus on producing visually realistic shapes and appearances, often neglecting the physical constraints necessary for downstream tasks. Generated models frequently fail to maintain balance when placed in physics-based simulations or 3D printed. This balance is crucial for satisfying user design intentions in interactive gaming, embodied AI, and robotics, where stable models are needed for reliable interaction. Additionally, stable models ensure that 3D-printed objects, such as figurines for home decoration, can stand on their own without requiring additional supports. To fill this gap, we introduce Atlas3D, an automatic and easy-to-implement method that enhances existing Score Distillation Sampling (SDS)-based text-to-3D tools. Atlas3D ensures the generation of self-supporting 3D models that adhere to physical laws of stability under gravity, contact, and friction. Our approach combines a novel differentiable simulation-based loss function with physically inspired regularization, serving as either a refinement or a post-processing module for existing frameworks. We verify Atlas3D's efficacy through extensive generation tasks and validate the resulting 3D models in both simulated and real-world environments.

Machine learning approaches to spatiotemporal physical systems have primarily focused on next-frame prediction, with the goal of learning an accurate emulator for the system's evolution in time. However, these emulators are computationally expensive to train and are subject to performance pitfalls, such as compounding errors during autoregressive rollout. In this work, we take a different perspective and look at scientific tasks further downstream of predicting the next frame, such as estimation of a system's governing physical parameters. Accuracy on these tasks offers a uniquely quantifiable glimpse into the physical relevance of the representations of these models. We evaluate the effectiveness of general-purpose self-supervised methods in learning physics-grounded representations that are useful for downstream scientific tasks. Surprisingly, we find that not all methods designed for physical modeling outperform generic self-supervised learning methods on these tasks, and methods that learn in the latent space (e.g., joint embedding predictive architectures, or JEPAs) outperform those optimizing pixel-level prediction objectives. Code is available at https://github.com/helenqu/physical-representation-learning.

While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date PINNs have not been successful in simulating dynamical systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In this work we attribute this shortcoming to the inability of existing PINNs formulations to respect the spatio-temporal causal structure that is inherent to the evolution of physical systems. We argue that this is a fundamental limitation and a key source of error that can ultimately steer PINN models to converge towards erroneous solutions. We address this pathology by proposing a simple re-formulation of PINNs loss functions that can explicitly account for physical causality during model training. We demonstrate that this simple modification alone is enough to introduce significant accuracy improvements, as well as a practical quantitative mechanism for assessing the convergence of a PINNs model. We provide state-of-the-art numerical results across a series of benchmarks for which existing PINNs formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. To the best of our knowledge, this is the first time that PINNs have been successful in simulating such systems, introducing new opportunities for their applicability to problems of industrial complexity.

To balance quality and cost, various domain areas of science and engineering run simulations at multiple levels of sophistication. Multi-fidelity active learning aims to learn a direct mapping from input parameters to simulation outputs at the highest fidelity by actively acquiring data from multiple fidelity levels. However, existing approaches based on Gaussian processes are hardly scalable to high-dimensional data. Deep learning-based methods often impose a hierarchical structure in hidden representations, which only supports passing information from low-fidelity to high-fidelity. These approaches can lead to the undesirable propagation of errors from low-fidelity representations to high-fidelity ones. We propose a novel framework called Disentangled Multi-fidelity Deep Bayesian Active Learning (D-MFDAL), which learns the surrogate models conditioned on the distribution of functions at multiple fidelities. On benchmark tasks of learning deep surrogates of partial differential equations including heat equation, Poisson's equation and fluid simulations, our approach significantly outperforms state-of-the-art in prediction accuracy and sample efficiency.

Initial value problems -- a system of ordinary differential equations and corresponding initial conditions -- can be used to describe many physical phenomena including those arise in classical mechanics. We have developed a novel approach to solve physics-based initial value problems using unsupervised machine learning. We propose a deep learning framework that models the dynamics of a variety of mechanical systems through neural networks. Our framework is flexible, allowing us to solve non-linear, coupled, and chaotic dynamical systems. We demonstrate the effectiveness of our approach on systems including a free particle, a particle in a gravitational field, a classical pendulum, and the H\'enon--Heiles system (a pair of coupled harmonic oscillators with a non-linear perturbation, used in celestial mechanics). Our results show that deep neural networks can successfully approximate solutions to these problems, producing trajectories which conserve physical properties such as energy and those with stationary action. We note that probabilistic activation functions, as defined in this paper, are required to learn any solutions of initial value problems in their strictest sense, and we introduce coupled neural networks to learn solutions of coupled systems.

We show that Fréchet Distance (FD), long considered impractical as a training objective, can in fact be effectively optimized in the representation space. Our idea is simple: decouple the population size for FD estimation (e.g., 50k) from the batch size for gradient computation (e.g., 1024). We term this approach FD-loss. Optimizing FD-loss reveals several surprising findings. First, post-training a base generator with FD-loss in different representation spaces consistently improves visual quality. Under the Inception feature space, a one-step generator achieves0.72 FID on ImageNet 256x256. Second, the same FD-loss repurposes multi-step generators into strong one-step generators without teacher distillation, adversarial training or per-sample targets. Third, FID can misrank visual quality: modern representations can yield better samples despite worse Inception FID. This motivates FDr^k, a multi-representation metric. We hope this work will encourage further exploration of distributional distances in diverse representation spaces as both training objectives and evaluation metrics for generative models.

We introduce Gravity, another algorithm for gradient-based optimization. In this paper, we explain how our novel idea change parameters to reduce the deep learning model's loss. It has three intuitive hyper-parameters that the best values for them are proposed. Also, we propose an alternative to moving average. To compare the performance of the Gravity optimizer with two common optimizers, Adam and RMSProp, five standard datasets were trained on two VGGNet models with a batch size of 128 for 100 epochs. Gravity hyper-parameters did not need to be tuned for different models. As will be explained more in the paper, to investigate the direct impact of the optimizer itself on loss reduction no overfitting prevention technique was used. The obtained results show that the Gravity optimizer has more stable performance than Adam and RMSProp and gives greater values of validation accuracy for datasets with more output classes like CIFAR-100 (Fine).

Physics-informed neural networks (PINNs) are a newly emerging research frontier in machine learning, which incorporate certain physical laws that govern a given data set, e.g., those described by partial differential equations (PDEs), into the training of the neural network (NN) based on such a data set. In PINNs, the NN acts as the solution approximator for the PDE while the PDE acts as the prior knowledge to guide the NN training, leading to the desired generalization performance of the NN when facing the limited availability of training data. However, training PINNs is a non-trivial task largely due to the complexity of the loss composed of both NN and physical law parts. In this work, we propose a new PINN training framework based on the multi-task optimization (MTO) paradigm. Under this framework, multiple auxiliary tasks are created and solved together with the given (main) task, where the useful knowledge from solving one task is transferred in an adaptive mode to assist in solving some other tasks, aiming to uplift the performance of solving the main task. We implement the proposed framework and apply it to train the PINN for addressing the traffic density prediction problem. Experimental results demonstrate that our proposed training framework leads to significant performance improvement in comparison to the traditional way of training the PINN.

Recent advances in text-to-video generation have achieved impressive perceptual quality, yet generated content often violates fundamental principles of physical plausibility - manifesting as implausible object dynamics, incoherent interactions, and unrealistic motion patterns. Such failures hinder the deployment of video generation models in embodied AI, robotics, and simulation-intensive domains. To bridge this gap, we propose PhysCorr, a unified framework for modeling, evaluating, and optimizing physical consistency in video generation. Specifically, we introduce PhysicsRM, the first dual-dimensional reward model that quantifies both intra-object stability and inter-object interactions. On this foundation, we develop PhyDPO, a novel direct preference optimization pipeline that leverages contrastive feedback and physics-aware reweighting to guide generation toward physically coherent outputs. Our approach is model-agnostic and scalable, enabling seamless integration into a wide range of video diffusion and transformer-based backbones. Extensive experiments across multiple benchmarks demonstrate that PhysCorr achieves significant improvements in physical realism while preserving visual fidelity and semantic alignment. This work takes a critical step toward physically grounded and trustworthy video generation.

Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent manifolds of arbitrary topology, we propose to learn a mixture model of variational autoencoders. Here, every encoder-decoder pair represents one chart of a manifold. We propose a loss function for maximum likelihood estimation of the model weights and choose an architecture that provides us the analytical expression of the charts and of their inverses. Once the manifold is learned, we use it for solving inverse problems by minimizing a data fidelity term restricted to the learned manifold. To solve the arising minimization problem we propose a Riemannian gradient descent algorithm on the learned manifold. We demonstrate the performance of our method for low-dimensional toy examples as well as for deblurring and electrical impedance tomography on certain image manifolds.

Machine learning for differential equations paves the way for computationally efficient alternatives to numerical solvers, with potentially broad impacts in science and engineering. Though current algorithms typically require simulated training data tailored to a given setting, one may instead wish to learn useful information from heterogeneous sources, or from real dynamical systems observations that are messy or incomplete. In this work, we learn general-purpose representations of PDEs from heterogeneous data by implementing joint embedding methods for self-supervised learning (SSL), a framework for unsupervised representation learning that has had notable success in computer vision. Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.

The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce a surrogate model that not only sheds new light on the design of Muon, but more importantly leads to a new optimizer. In the same spirit as the derivation of Newton's method, the surrogate approximates the loss as a quadratic function of the perturbation to a weight matrix W using only three matrices: the gradient G, an output-space curvature matrix H, and the data matrix Z that stacks the layer inputs. By minimizing this surrogate in one step and adopting a certain isotropic assumption on the weights, we obtain the closed-form update rule (up to momentum and weight decay) W leftarrow W - ηcdot msgn(G(ZZ^top)^{-1}), where η is the learning rate and msgn(X)=UV^top if X=USV^top is a compact singular value decomposition. This new optimization method, which we refer to as Newton-Muon, shows that standard Muon can be interpreted as an implicit Newton-type method that neglects the right preconditioning induced by the input second moment. Empirically, on a reproduction of the earliest publicly released Modded-NanoGPT speedrun configuration using Muon for GPT-2 pretraining, Newton-Muon reaches the target validation loss in 6\% fewer iteration steps and reduces wall-clock training time by about 4\%.

Democratization of machine learning requires architectures that automatically adapt to new problems. Neural Differential Equations (NDEs) have emerged as a popular modeling framework by removing the need for ML practitioners to choose the number of layers in a recurrent model. While we can control the computational cost by choosing the number of layers in standard architectures, in NDEs the number of neural network evaluations for a forward pass can depend on the number of steps of the adaptive ODE solver. But, can we force the NDE to learn the version with the least steps while not increasing the training cost? Current strategies to overcome slow prediction require high order automatic differentiation, leading to significantly higher training time. We describe a novel regularization method that uses the internal cost heuristics of adaptive differential equation solvers combined with discrete adjoint sensitivities to guide the training process towards learning NDEs that are easier to solve. This approach opens up the blackbox numerical analysis behind the differential equation solver's algorithm and directly uses its local error estimates and stiffness heuristics as cheap and accurate cost estimates. We incorporate our method without any change in the underlying NDE framework and show that our method extends beyond Ordinary Differential Equations to accommodate Neural Stochastic Differential Equations. We demonstrate how our approach can halve the prediction time and, unlike other methods which can increase the training time by an order of magnitude, we demonstrate similar reduction in training times. Together this showcases how the knowledge embedded within state-of-the-art equation solvers can be used to enhance machine learning.

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization, called multi-grid tensorized neural operator (MG-TFNO). MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena, through a decomposition of both the input domain and the operator's parameter space. Our contributions are threefold: i) we enable parallelization over input samples with a novel multi-grid-based domain decomposition, ii) we represent the parameters of the model in a high-order latent subspace of the Fourier domain, through a global tensor factorization, resulting in an extreme reduction in the number of parameters and improved generalization, and iii) we propose architectural improvements to the backbone FNO. Our approach can be used in any operator learning setting. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression. The tensorization combined with the domain decomposition, yields over 150x reduction in the number of parameters and 7x reduction in the domain size without losses in accuracy, while slightly enabling parallelism.

Combinatorial inverse problems in high energy physics span enormous algorithmic challenges. This work presents a new deep learning driven clustering algorithm that utilizes a space-time non-local trainable graph constructor, a graph neural network, and a set transformer. The model is trained with loss functions at the graph node, edge and object level, including contrastive learning and meta-supervision. The algorithm can be applied to problems such as charged particle tracking, calorimetry, pile-up discrimination, jet physics, and beyond. We showcase the effectiveness of this cutting-edge AI approach through particle tracking simulations. The code is available online.

Motion blur, caused by relative movement between camera and scene during exposure, significantly degrades image quality and impairs downstream computer vision tasks such as object detection, tracking, and recognition in dynamic environments. While deep learning-based motion deblurring methods have achieved remarkable progress, existing approaches face fundamental challenges in capturing the long-range spatial dependencies inherent in motion blur patterns. Traditional convolutional methods rely on limited receptive fields and require extremely deep networks to model global spatial relationships. These limitations motivate the need for alternative approaches that incorporate physical priors to guide feature evolution during restoration. In this paper, we propose a progressive training framework that integrates physics-informed PDE dynamics into state-of-the-art restoration architectures. By leveraging advection-diffusion equations to model feature evolution, our approach naturally captures the directional flow characteristics of motion blur while enabling principled global spatial modeling. Our PDE-enhanced deblurring models achieve superior restoration quality with minimal overhead, adding only approximately 1\% to inference GMACs while providing consistent improvements in perceptual quality across multiple state-of-the-art architectures. Comprehensive experiments on standard motion deblurring benchmarks demonstrate that our physics-informed approach improves PSNR and SSIM significantly across four diverse architectures, including FFTformer, NAFNet, Restormer, and Stripformer. These results validate that incorporating mathematical physics principles through PDE-based global layers can enhance deep learning-based image restoration, establishing a promising direction for physics-informed neural network design in computer vision applications.

Despite the great promise of the physics-informed neural networks (PINNs) in solving forward and inverse problems, several technical challenges are present as roadblocks for more complex and realistic applications. First, most existing PINNs are based on point-wise formulation with fully-connected networks to learn continuous functions, which suffer from poor scalability and hard boundary enforcement. Second, the infinite search space over-complicates the non-convex optimization for network training. Third, although the convolutional neural network (CNN)-based discrete learning can significantly improve training efficiency, CNNs struggle to handle irregular geometries with unstructured meshes. To properly address these challenges, we present a novel discrete PINN framework based on graph convolutional network (GCN) and variational structure of PDE to solve forward and inverse partial differential equations (PDEs) in a unified manner. The use of a piecewise polynomial basis can reduce the dimension of search space and facilitate training and convergence. Without the need of tuning penalty parameters in classic PINNs, the proposed method can strictly impose boundary conditions and assimilate sparse data in both forward and inverse settings. The flexibility of GCNs is leveraged for irregular geometries with unstructured meshes. The effectiveness and merit of the proposed method are demonstrated over a variety of forward and inverse computational mechanics problems governed by both linear and nonlinear PDEs.

Offline optimization has recently emerged as an increasingly popular approach to mitigate the prohibitively expensive cost of online experimentation. The key idea is to learn a surrogate of the black-box function that underlines the target experiment using a static (offline) dataset of its previous input-output queries. Such an approach is, however, fraught with an out-of-distribution issue where the learned surrogate becomes inaccurate outside the offline data regimes. To mitigate this, existing offline optimizers have proposed numerous conditioning techniques to prevent the learned surrogate from being too erratic. Nonetheless, such conditioning strategies are often specific to particular surrogate or search models, which might not generalize to a different model choice. This motivates us to develop a model-agnostic approach instead, which incorporates a notion of model sharpness into the training loss of the surrogate as a regularizer. Our approach is supported by a new theoretical analysis demonstrating that reducing surrogate sharpness on the offline dataset provably reduces its generalized sharpness on unseen data. Our analysis extends existing theories from bounding generalized prediction loss (on unseen data) with loss sharpness to bounding the worst-case generalized surrogate sharpness with its empirical estimate on training data, providing a new perspective on sharpness regularization. Our extensive experimentation on a diverse range of optimization tasks also shows that reducing surrogate sharpness often leads to significant improvement, marking (up to) a noticeable 9.6% performance boost. Our code is publicly available at https://github.com/cuong-dm/IGNITE

Recent physics foundation models claim general spatiotemporal forecasting ability, yet their evaluations often collapse performance into a single average score under a fixed training distribution. This makes it difficult to determine whether a model has learned generalizable physical dynamics or only performs well under particular settings. We construct a benchmark with 8 physical dynamics, 3 training-data mixtures, and 25 test regimes induced by dynamic-scale and initial-condition complexity shifts, covering in-distribution, distribution-shift, and out-of-distribution settings. We evaluate five physics foundation model architectures and four model variants per architecture (scratch and three pretrained sizes), resulting in 60,000 measurements. Our results show that current physics foundation models behave as conditional rather than universal generalists: their generality depends on the physical regime, temporal scale, initial-condition setting, pretraining, model size, and architecture. Improving the training data distribution only partially mitigates this limitation. Pretraining and scaling are also unable to reliably remove their ability biases. We argue that improving physics foundation models requires moving beyond scaling models or expanding data, toward learning mechanisms that better capture transferable physical knowledge across regimes, temporal scales, and distribution shifts.

There is a widely-spread claim that GANs are difficult to train, and GAN architectures in the literature are littered with empirical tricks. We provide evidence against this claim and build a modern GAN baseline in a more principled manner. First, we derive a well-behaved regularized relativistic GAN loss that addresses issues of mode dropping and non-convergence that were previously tackled via a bag of ad-hoc tricks. We analyze our loss mathematically and prove that it admits local convergence guarantees, unlike most existing relativistic losses. Second, our new loss allows us to discard all ad-hoc tricks and replace outdated backbones used in common GANs with modern architectures. Using StyleGAN2 as an example, we present a roadmap of simplification and modernization that results in a new minimalist baseline -- R3GAN. Despite being simple, our approach surpasses StyleGAN2 on FFHQ, ImageNet, CIFAR, and Stacked MNIST datasets, and compares favorably against state-of-the-art GANs and diffusion models.

Accurate and efficient physical simulations are essential in science and engineering, yet traditional numerical solvers face significant challenges in computational cost when handling simulations across dynamic scenarios involving complex geometries, varying boundary/initial conditions, and diverse physical parameters. While deep learning offers promising alternatives, existing methods often struggle with flexibility and generalization, particularly on unstructured meshes, which significantly limits their practical applicability. To address these challenges, we propose PhysGTO, an efficient Graph-Transformer Operator for learning physical dynamics through explicit manifold embeddings in both physical and latent spaces. In the physical space, the proposed Unified Graph Embedding module aligns node-level conditions and constructs sparse yet structure-preserving graph connectivity to process heterogeneous inputs. In the latent space, PhysGTO integrates a lightweight flux-oriented message-passing scheme with projection-inspired attention to capture local and global dependencies, facilitating multilevel interactions among complex physical correlations. This design ensures linear complexity relative to the number of mesh points, reducing both the number of trainable parameters and computational costs in terms of floating-point operations (FLOPs), and thereby allowing efficient inference in real-time applications. We introduce a comprehensive benchmark spanning eleven datasets, covering problems with unstructured meshes, transient flow dynamics, and large-scale 3D geometries. PhysGTO consistently achieves state-of-the-art accuracy while significantly reducing computational costs, demonstrating superior flexibility, scalability, and generalization in a wide range of simulation tasks.