PDE Solving

PDE: Partial Differential Equation

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19 papers in the last four weeks, up 111% on the four weeks before. 0.2% of all new papers.

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Latest papers 170

Jul 6, 2026cs.LG

PDEFlow: Autonomous Agentic PDE Pipelines for Neural Operator Learning and Solver-Free Inference

We present PDEFlow, an autonomous agentic framework that turns user-level ODE and PDE descriptions into solver-backed neural-operator pipelines. The workflow links problem specification, data generation, operator training, and checkpoint-based inference. A stateful input graph converts multi-turn natural-language input and user edits into validated problem specifications. The data-generation module then samples parameters, solves the configured governing-equation with FEniCSx finite-element backend, and stores the solutions as operator-ready tensors. The training and inference stages use a registry-based interface, allowing different neural operators to be trained and deployed without changing the surrounding pipeline. In the current implementation, we instantiate this interface with a multi-branch Bayesian DeepONet. Experiments on benchmark ODE and PDE tasks show that PDEFlow can construct valid specifications, generate solver-backed datasets, train neural operators across steady and transient problem classes, and provide solver-free predictions from saved checkpoints. The framework is designed for repeatable scientific and engineering workflows where many related physics configurations must be specified, simulated, learned, and queried with minimal manual intervention.

Akshat Jani, Prathamesh Gadekar, Sakhinana Sagar Srinivas +1

Tata Research Development and Design Center, Pune, India 411057

PDE Solving PDE Surrogate Modeling PDE Operator Learning

Jul 4, 2026math.AP

LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies

d/\|w\|

as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than

30\%

of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

Zihao Guo, Xin Li, Zhihong Xia

Institute for Advanced Research, Great Bay University, Dongguan, China · College of Computer Science and Technology, Dongguan University of Technology, Dongguan, China · Department of Mathematics, Northwestern University, Evanston, USA

Neural PDE Solvers PDE Solving Neural Network Activation Functions Neural Network Approximation Theory Physics-Informed ML

Jul 2, 2026cs.LG

An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present

\textbf{DSGNAR}

: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed.

\textbf{DSGNAR}

couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative

\ell_2

errors as low as

3\times10^{-16}

in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to

\ell_2^{\text{rel}} = 4.75 \times 10^{-7}

in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

Joseph Webb, Sadok Jerad, Coralia Cartis

Mathematical Institute, University of Oxford, Oxford, United Kingdom

Gauss-Newton Optimization PDE Solving Second-Order Optimization Physics-Informed ML

Jul 2, 2026cs.LG

Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations

Solving partial differential equations (PDEs) with high-frequency solutions remains a central challenge in physics-informed machine learning due to spectral bias -- the tendency of neural networks to learn low-frequency components preferentially. This paper proposes a Frequency Shift Physics-Informed Extreme Learning Machine (FS-PIELM) framework that addresses this limitation through an additive mechanism for weight initialization. Rather than multiplying random weights by a scaling factor, the method translates the mean of the Gaussian weight distribution while keeping the variance fixed at unity, thereby avoiding the variance amplification inherent in scaling-based methods. Two variants are developed: FS-PIELM-L assigns independent frequency magnitudes to individual neurons, while FS-PIELM-G groups neurons for improved robustness. Theoretical analysis shows that the frequency variance under the proposed framework remains bounded and approaches unity regardless of target frequency, in contrast to the quadratic growth of conventional approaches. The method preserves the computational efficiency of extreme learning machines, requiring only a single linear solve. Experiments on seven benchmark problems spanning six equation types -- Helmholtz, wave, Poisson, Klein-Gordon, heat, and advection-diffusion -- on both regular and complex geometries show that the linear variant achieves the best accuracy in six of seven cases, with improvements of one to nearly five orders of magnitude over existing PIELM variants. The code and data accompanying this manuscript will be made publicly available at https://github.com/xgxgnpu/Physics-informed-vibe-coding/tree/main/FS-PIELM.

Xiong Xiong, Ruonan Zhai, Zheng Zeng +3

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China · MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China · College of Science, Shijiazhuang University, Shijiazhuang, 050035, China +2

PDE Solving Frequency-Domain Feature Learning Spectral Bias Physics-Informed ML

Jul 1, 2026cs.LG

GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative

L^2

error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

Meenakshi Krishnan, Pranav Pulijala, Ke Chen +2

University of Maryland, College Park · University of Delaware, Newark

PDE Solving Image Reconstruction PDE Operator Learning PDE Inverse Problems Neural Operators

Jun 30, 2026math.NA

Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov--Galerkin or collocation formulations. We develop a Petrov--Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

Haixin Wang, Haoning Dang, Fei Wang +1

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Neural PDE Solvers PDE Solving Neural Network Approximation Theory

Jun 29, 2026math.NA

McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Jiwei Jia, Xinliang Liu, Juntao Wang +1

School of Mathematics, Jilin University, Changchun 130012, China · Shenzhen Loop Area Institute, Shenzhen 518038, China · Ocean University of China, Qingdao 266100, China +1

Neural PDE Solvers PDE Solving PDE Operator Learning

Jun 28, 2026math.NA

Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

Chenhui Zhu, Fei Wang

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P. R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Fourier Neural Operator PDE Solving PDE Operator Learning

Jun 26, 2026physics.comp-ph

Mosaic: A Benchmark Suite for Differentiable Physics Solvers

Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, computational cost, gradient accuracy, and numerical conditioning vary widely across solvers and are discoverable only by trial and error. We introduce Mosaic, an extensible benchmarking framework for differentiable PDE solvers that standardizes access to solver gradients. Each solver is packaged as a containerized component (Tesseract) exposing a uniform gradient API regardless of language or automatic differentiation (AD) strategy, enabling researchers to evaluate, compare, and build on non-trivial physical solvers. Our evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer demonstrates that the benchmark surfaces practically relevant differences: order-of-magnitude variation in computational cost and Jacobian conditioning, alongside structural incompatibilities that eliminate solvers from realistic tasks entirely. Despite this variation, all solvers that produce gradients converge to similar optima, indicating that the practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy alone. Mosaic is open-source and available at https://github.com/pasteurlabs/mosaic.

Andrin Rehmann, Heiko Zimmermann, Dion Häfner

Pasteur Labs, Brooklyn, NY, USA

Automatic Differentiation PDE Solving Benchmark Design Differentiable Programming Differentiable Physics

Jun 25, 2026cs.LG

Operator Learning for Cubic Nonlinear Schrödinger Equation on Periodic Domains

We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter

ω^2

. The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger

H^2

-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including

ω^2

improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.

Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu

Fourier Neural Operator PDE Solving PDE Operator Learning

Jun 25, 2026cs.LG

fTNN: a tensor neural network for fractional PDEs

We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field. Then the singular radial integrals are treated by Gauss-Jacobi quadrature, the regular radial integrals by Gauss quadrature, and the angular variables by deterministic angular quadrature, yielding a fully deterministic integration framework of the fractional Laplacian operator. To accurately resolve low-regularity solutions and the associated loss functional, we construct boundary-singularity-aware trial functions enriched with explicit boundary features, and propose two strategies for automatically selecting the leading exponent and evaluating the loss function from the singularity structure induced by the fractional operator, or jointly by the fractional operator and the source term. For time-dependent fractional PDEs, we design a spatiotemporally separable neural network that factorizes the time-space residual into a sum of low-dimensional temporal and spatial integrals, and we integrate this representation with an alternating neural network subspace optimization strategy for efficient training. Numerical experiments show that the proposed framework attains high accuracy on the tested benchmarks and improves substantially over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.

Qingkui Ma, Hehu Xie, Xiaobo Yin

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China · SKLMS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No.55, Zhongguancun Donglu, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China +1

Tensor Networks Neural PDE Solvers PDE Solving

Jun 23, 2026cs.LG

A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

Yanwei Jia, Du Ouyang, Huyên Pham +1

Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, New Territories, Hong Kong SAR · Department of Mathematical Sciences Tsinghua University Beijing, Beijing 100084, China · CMAP Ecole Polytechnique Route de Saclay, Palaiseau 91128, France +1

Neural PDE Solvers PDE Solving

Jun 22, 2026math.OC

Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs

A data-driven method is developed for approximating value functions in deterministic optimal control problems with nonlinear control-affine dynamics. The Pontryagin Maximum Principle optimality system is solved from multiple initial conditions to generate training data consisting of values, gradients, and Hessians of the value function, where Hessian information is obtained from a matrix Riccati equation along optimal trajectories. These quantities augment a weighted least-squares regression over sparse polynomial bases on hyperbolic cross index sets, with gradients and Hessians contributing additional linear equations per sample and substantially reducing sample complexity compared to value-only regression. Feedback laws are recovered analytically from the learned value function. In high dimensions, a partial Hessian strategy controls the cost of data generation. The approach is validated on problems of increasing state dimension, where second-order data augmentation is shown to improve approximation accuracy and closed-loop performance, with up to an order-of-magnitude reduction in the number of training samples required relative to lower-order methods.

Matías Gómez-Aedo, Behzad Azmi, Yuyang Huang +2

Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK · Department of Mathematics and Statistics, University of Konstanz, Germany · RICAM and Institute of Mathematics and Scientific Computing, University of Graz, Austria

Value Function Estimation PDE Solving Data-Driven Control Optimal Control

Jun 22, 2026cs.LG

Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solving

Physics-informed neural networks (PINNs) provide an effective way to solve partial differential equations (PDEs) by embedding physical principles into the learning process. However, the conventional PINN formulation, in which all constraints are imposed as soft penalty terms within a composite loss, often exhibits slow convergence, sensitivity to loss weight scaling, and inaccurate boundary enforcement due to poor conditioning of the optimization landscape. To address these limitations, this study proposes a unified hard--soft physics--informed neural network (HSPINN) with adaptive loss weighting. In this framework, Dirichlet and periodic boundary conditions are enforced exactly by construction through analytical or polynomial lifting, masking functions, and periodic feature mappings, while the governing PDE residuals, Neumann fluxes, and initial conditions are treated as soft constraints. An inverse-share softmax strategy dynamically balances the relative importance of individual loss components during training, eliminating manual penalty tuning and improving gradient stability. This formulation ensures boundary admissibility throughout optimization and enhances convergence efficiency and numerical robustness. Applications to representative elliptic (Poisson), parabolic (Burgers), and hyperbolic (convection with periodic boundaries) problems demonstrate that HSPINN consistently achieves faster convergence, higher accuracy, and greater stability than conventional PINNs, establishing a general and scalable foundation for physics-constrained deep learning across science and technology.

Duc Tien Nguyen, Trinh Minh Tuan, Nguyen Duc Manh +2

aCollege of Engineering and Computer Science, VinUniversity, Hanoi, Vietnam · bCenter for AI Research, VinUniversity, Hanoi, Vietnam · dDepartment of Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam +1

PDE Solving Adaptive Loss Weighting Physics-Informed ML

Jun 20, 2026cs.LG

Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers

Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.

Zhangyong Liang, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, PR China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130025, PR China

Neural PDE Solvers PDE Solving

Jun 20, 2026math.NA

Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers

Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative

L^2

error

O(10^{-3})

can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses

\mathrm{det} F

well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4

\times

wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom.

Jaemin Oh, Youngkyu Lee, Jerome Darbon +1

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Spectral Regularization PDE Solving Hyperelasticity Neural Operators

Jun 19, 2026cs.LG

TF-SNO: Time-Frequency Gated Spectral Neural Operators for Learning Non-Stationary Partial Differential Equations

Non-stationary partial differential equations (PDEs) arise throughout scientific computing, where the dominant frequency content and energy distribution can drift over time. While efficient in PDE solving, many spectral neural operators apply a shared spectral response across rollout stages, leading to mismatch with time-varying spectra in non-stationary systems. To address this issue, we propose Time-Frequency Gated Spectral Neural Operator (TF-SNO), a state-adaptive framework with learnable time-frequency gating inside spectral blocks. TF-SNO extracts compact frequency-domain and physical-space statistics from the current state to generate modulation coefficients, enabling the spectral response to evolve with the dynamics. TF-SNO learns temporal variation implicitly from the evolving state without introducing an explicit time dimension or time embedding, keeping the modeling complexity low. We further embed the adaptive operator blocks to accurately capture the multi-scale features, thereby improving long-horizon stability. Experiments on six non-stationary PDE benchmarks in 1D and 2D demonstrate that TF-SNO significantly reduces prediction errors and improves robustness compared to strong baselines, with particularly clear gains in long rollout, suggesting the effectiveness of state-dependent spectral adaptation in modeling non-stationary physical systems.

Yitian Zhou, Chaoning Zhang, Zhenzhen Huang +8

University of Electronic Science and Technology of China Chengdu, China · Institute of Software, Chinese Academy of Sciences Beijing, China · Harbin Institute of Technology Shenzhen, China

PDE Solving PDE Operator Learning Spectral Methods Neural Operators

Jun 18, 2026cs.LG

Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

Zongmin Yu, Liu Yang

National University of Singapore

AI Agents for Scientific Discovery PDE Solving PDE Discovery Symbolic Regression

Jun 18, 2026cs.LG

Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs

We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.

Xiang Rao, Yuxuan Shen

1. School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · 2. School of Computer Science, Yangtze University, Jingzhou 434023, China · 3. State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China +2

PDE Solving Hybrid Quantum-Classical ML Kolmogorov-Arnold Networks Physics-Informed ML

Jun 18, 2026math.NA

A fast direct solver based neural network for solving PDEs

The matrices arising from large scale

N

-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions. HODLR (Hierarchical Off-Diagonal Low-Rank) matrices are a subclass of hierarchical matrices in which all off-diagonal submatrices at every level of a recursive binary partition are low-rank. In this article, we present a neural network that learns the inverse operation of HODLR matrices based on the fast direct solver for HODLR matrices developed by Ambikasaran and Darve (2013). We further extend the architecture to learn nonlinear solution operators associated with PDEs by replacing some of the linear layers with deep sub-networks. We demonstrate the performance of the proposed architecture by performing a comprehensive set of experiments that include (i) solving a linear problem such as the Fredholm integral equation of the second kind, (ii) solving PDEs such as the nonlinear Schrödinger equation, Burgers' equation, and the steady-state Darcy's flow equation, (iii) generalization study across varying parameter values, (iv) comparing the inference time of the proposed network with the run time of a classical numerical solver, and (v) comparing the proposed network with some of the existing neural operator learning networks.

Jashwanth Reddy Kadaru, Vaishnavi Gujjula

Department of Computer Science & Engineering, International Institute of Information Technology Bangalore (IIIT-B), India · Department of Data Science and Artificial Intelligence, International Institute of Information Technology Bangalore (IIIT-B), India

PDE Solving PDE Operator Learning Neural Operators

Jun 18, 2026cs.LG

Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.

Zhiwen Yu, Derong Yang, Liujian Zhang +5

School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China · Peng Cheng Laboratory, Shenzhen 518000, China · School of Future Technology, South China University of Technology, Guangzhou 510006, China +2

PDE Solving Universal Approximation Neural Network Approximation Theory Physics-Informed ML

Jun 17, 2026physics.comp-ph

Acceleration of an algebraic multigrid pressure solver using graph neural networks

Solving the pressure-Poisson equation remains the primary computational bottleneck in incompressible unstructured flow solvers primarily due to the inherent sensitivity of traditional linear solvers to mesh irregularities. This work introduces a data-driven algebraic multigrid (AMG) smoother that uses a modified graph convolutional isomorphism network (GCIN). The graph neural network predicts optimal polynomial coefficients to construct a sparse pseudo-inverse operator across diverse grid topologies. The coefficients are optimized to reduce the residual after each V-cycle iteration. By directly capturing the algebraic structure of the system from the sparse coefficient matrix, the proposed method maintains the solver's linearity while adapting to local anisotropies in unstructured grids. Our framework demonstrates significant performance gains by reducing the number of V-cycles required for a given tolerance and delivering wall-clock speedups from 4% to 37% across diverse benchmarks. Notably, the model exhibits robust generalization by maintaining efficiency on meshes up to 128 times larger than those seen in training, and by accelerating the solver's convergence on unseen industry-relevant problems such as the AirfRANS dataset.

Eric Chillón, Artur K. Lidtke, Nguyen Anh Khoa Doan +1

Faculty of Mechanical Engineering, Delft University of Technology, The Netherlands · Maritime Research Institute Netherlands, The Netherlands · Department of Aeronautics, Imperial College London, United Kingdom

Graph Neural Networks PDE Solving Computational Fluid Dynamics

Jun 16, 2026math.NA

A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks

We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization. The trial space, which we term Linear-in-Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics-informed neural network. The method thus replaces the nonconvex gradient-based training that limits standard PINNs with a convex per-step solve. We establish local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best-approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL-Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley-Leverett), coupled systems (plane-strain elasticity and the incompressible Navier-Stokes equations in two and three spatial dimensions), and steady-state Darcy flow with heterogeneous permeability. Across these problems, LiL-Q converges in single-digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient-based optimization.

Gbenga T. Awojinrin, Abdul-Akeem Olawoyin, Rami M. Younis

Texas A&M University, College Station, Texas, U.S.A.

PDE Solving Physics-Informed ML

Jun 16, 2026math.NA

INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities

We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN). INI-VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks-based formulations, the INI-VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

Shayan Dodge, Alessandro Formisano, Sami Barmada

DESTeC, University of Pisa, Pisa, Italy · DESTeC University of Pisa Pisa, Italy · Department of Engineering, University of Campania Luigi Vanvitelli, Aversa, Italy +1

Neural PDE Solvers PDE Solving Physics-Informed ML

Jun 15, 2026cs.LG

RepNN: Tackling spectral bias in deep neural networks for regression and PDE problems via parameter reparameterization

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with ReLU or tanh activations designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one-, two-, and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

Yong Wang, Tao Zhou, Xuhui Meng

Institute of Interdisciplinary Research for Mathematics and Applied Science, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China · Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

PDE Solving Spectral Bias Neural Network Approximation Theory

Jun 12, 2026cs.LG

Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications

This work introduces the Separable Neural Architecture (SNA), a function representational class combining neural approximation with tensor decomposition. The SNA decouples localized coordinate functions (atoms) from global interactions governed by a sparse, low-rank interaction object. This architecture possesses a compact and smooth inductive bias well-suited for solving partial differential equations (PDEs). When viewed as a Galerkin trial space under the variational SNA (VSNA) framework, the formulation satisfies classical variational guarantees under Lax-Milgram: well-posedness, quasi-optimality, convergence, and stability. In high-dimensional spatiotemporal--parametric PDEs, the VSNA mitigates the curse of dimensionality by scaling algebraically rather than exponentially. Exploiting an entirely factorized, tensor-native alternating least squares (ALS) optimization framework reduces this cost to linear in dimension. The VSNA is validated across elliptic, hyperbolic, and parabolic systems, demonstrating close alignment with predicted algebraic and spectral scaling rates. We showcase the SNA as a "solve once, query anywhere" physical world model via two engineering case studies: a 7D parametric manufacturing simulation and an experimental thermal-to-property inversion pipeline for Inconel 718. The VSNA executes a 1,000,000-query Monte Carlo sweep in 102s on a standard laptop CPU, yielding a 150,000x speedup over a full-grid finite element baseline hosted on an NVIDIA A100 GPU. It further enables real-time generative inverse-mode reconstructions under 100ms. These results demonstrate that the SNA serves as a compact mathematical substrate for continuous parameter manifolds to enable real-time inversion, optimization loops, and rapid uncertainty propagation.

Reza T Batley, Andrew Kichline, Sourav Saha

Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, 1600 Innovation Drive, Blacksburg, 24060, VA, United States

Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Jun 12, 2026math.NA

Robin-Neumann Coupling of PINN and FEM Solvers: A Steklov-Poincaré View, with Application to Fluid-Structure Interaction with Contact

Physics-informed neural networks (PINNs) are meshless and carry moving geometry and topology change through resampling of collocation points; the finite-element method (FEM) is the workhorse for boundary-fitted discretisations. Coupling the two across a shared interface promises the best of both, yet existing PINN-FEM schemes are validated only empirically. We put the coupling on a domain-decomposition footing: viewing each solver as a Steklov-Poincaré (trace-to-flux) operator, we transfer the classical Dirichlet-Neumann (DN) divergence diagnosis and its Robin-Neumann (RN) cure, including a closed-form, sweep-free interface impedance, and prove a PINN-specific contraction theorem: a trained network realises only a perturbed Steklov operator with a per-step training residual, and RN still contracts, with no shared-eigenbasis hypothesis, to a floor set by the achieved training loss. Because a PINN has no stiffness matrix, we introduce a Fourier-mode interface probe that recovers the network's resolvable Steklov eigenvalues to within 0.5% and doubles as a diagnostic of the network's spectral cap. The theory predicts measured PINN-FEM contraction rates to within 7% on 1D and 2D Poisson couplings, and a two-slab analogue of the large-added-mass regime shows RN's per-mode impedance matching winning decisively where tuned scalar relaxation saturates. We demonstrate the framework on a Stokes/rigid-disc problem with Alart-Curnier contact: the meshless PINN fluid absorbs the topology change at contact by collocation exclusion alone, no remeshing and no cut cells, and the static-equilibrium contact reaction matches the submerged weight to 0.4% under mesh refinement. We quantify remaining limitations: the warm-started PINN drifts off the Stokes manifold over long horizons, and matched FEM-FEM benchmarks attribute pre-impact squeeze-film signatures to PINN under-resolution.

Mikel Landajuela

Lawrence Livermore National Laboratory

PDE Solving Physics-Informed ML Finite Element Method Physics-Based Simulation

Jun 10, 2026cs.LG

Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources

Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS formulations other than PINN, we focus on the Radial Basis Function (RBF) expansion (also known as a single-layer RBF Network). We show that while integrating out the Dirac delta in PINNs causes residuals to fail to converge to zero, RBF-RLS consistently provides good forward and inverse solutions to transport problems. We explain this finding using the Neural Tangent Kernel (NTK) theory. We test both approaches on linear PDEs that represent groundwater flow and transport in porous media and rivers. We solve inverse problems to fit synthetic data, noisy synthetic data, and real-world measurements.

Manuel Reyna, Alexandre Tartakovsky

Department of Civil and Environmental Engineering, University of Illinois Urbana-Champaign, 205 North Mathews Avenue, Urbana, 61801, IL, USA

PDE Solving Physics-Informed ML PDE Inverse Problems Radial Basis Function Networks

Jun 9, 2026cs.CE

A Constrained Natural-Language Interface for Variational Multi-Physics Finite Element Simulations in FEniCS

Large language models can reduce the manual effort required to set up finite element simulations, but they introduce reliability risks when generated solver code lies on the critical path. We present a constrained natural-language interface for multi-physics finite element analysis in which the LLM is limited to front-end tasks: parsing prompts into structured JSON, generating Gmsh code only for non-catalog geometries, and using retry feedback for those stages. It never writes FEniCS solver templates, derives weak forms, or writes the numerical solver core. A deterministic dispatcher maps the validated specification to five human-written FEniCS/UFL templates: linear elasticity, hyperelasticity, elastoplasticity, thermo-mechanical coupling, and phase-field fracture. We validate this deterministic template layer against analytical solutions and published 2D/3D benchmarks. Smooth cases reach sub-percent agreement on adequate meshes, while harder nonlinear cases reach the 2-5 percent range. We also evaluate the LLM-facing front end directly. In a 15-prompt parser benchmark, first-pass valid parses were obtained for 9 cases, and all remaining cases were repaired after retry, giving a final valid parse rate of 100.0 percent, 100.0 percent problem-class accuracy, and 97.1 percent field-extraction accuracy. In a 10-case custom-geometry benchmark routed through the real LLM-to-Gmsh path, first-pass and final success were both 90.0 percent, with one unrecovered invalid-geometry failure. These results show that the parser and constrained prompt/validation design are effective on these benchmarks. As an end-to-end demonstration, the system generates and analyzes a 3D elastoplastic L-bracket with a fillet and bolt hole from one natural-language prompt. The contribution is a measured architecture for natural-language-driven variational simulation, not open-ended autonomous code generation.

Nilay Upadhyay, Wesley F. Reinhart

Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA · Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

PDE Solving LLM Prompting Finite Element Method Physics-Based Simulation LLM Code Generation

Jun 9, 2026cs.AI

AutoPDE: Reliable Agentic PDE Solving via Explicitly Represented Solver Strategies

Numerical solvers for partial differential equations (PDEs) are core computational tools in science and engineering. Building reliable PDE solvers requires not only executable code, but a numerical solver strategy, a set of decisions about discretization, stabilization, solver configuration, and resolution control, that matches the PDE structure. Recent LLM-based coding agents have begun to reduce the programming burden by generating and debugging solver implementations. However, they typically move directly from a PDE problem to solver code, leaving the solver strategy implicit in implementation details. Feedback from a failed solve is therefore routed back to code edits rather than to the underlying strategy, so numerical decisions remain hard to check before code is generated and hard to revise using numerical evidence when it fails. To address this limitation, we propose AutoPDE, a code agent that maintains the solver strategy as an explicitly represented object throughout the solving process: an independent, inspectable object that is built before any code is written and can be revised, using numerical evidence, whenever a solve fails. AutoPDE builds and maintains this object in three stages, all drawing from a library of reusable PDE-solving skills: PDE analysis identifies the equation type and algebraic structure; numerical method selection chooses a numerical method that matches the analysis result and commits to a discretization, stabilization, and linear solver accordingly; and adaptive tuning runs low-cost pilot solves to calibrate resolution and tolerances under the prescribed accuracy and runtime budget. We evaluate AutoPDE on the PDE Agent Bench, where experimental results show that AutoPDE achieves a pass rate of

54.5%

, improving over the strongest baseline by

14.2

percentage points.

Huanshuo Dong, Keyao Zhang, Hong Wang +6

University of Science and Technology of China · 2Zhejiang University · University of the Chinese Academy of Sciences +2

AI Coding Agents PDE Solving