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

May 11, 2026cs.LG

Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

Koichi Taniguchi, Sho Sonoda

Department of Mathematical and Systems Engineering, Faculty of Engineering Shizuoka University 3-5-1 Johoku, Chuo-ku, Hamamatsu, 432-8561, Japan · RIKEN AIP / CyberAgent 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan

PDE Solving PDE Operator Learning Generalization Bounds

May 10, 2026cs.AI

PDEAgent-Bench: A Multi-Metric, Multi-Library Benchmark for PDE Solver Generation

PDE-to-solver code generation aims to automatically synthesize executable numerical solvers from partial differential equation (PDE) specifications. This task requires not only understanding the mathematical structure of PDEs, but also selecting appropriate discretization schemes and solver configurations, and correctly implementing the resulting formulations in finite-element method (FEM) libraries. Existing code generation benchmarks mainly evaluate syntactic correctness, or success on predefined test cases. To our knowledge, there is currently no publicly available benchmark specifically for PDE-to-solver code generation, and general-purpose code benchmarks do not fully capture the unique challenges of numerical PDE solution, such as ensuring solver accuracy, efficiency, and compatibility with professional FEM libraries. We introduce PDEAgent-Bench, to the best of our knowledge, the first multi-metric, multi-library benchmark for PDE-to-solver code generation. PDEAgent-Bench contains 645 instances across 6 mathematical categories and 11 PDE families, with common FEM libraries for DOLFINx, Firedrake, and deal.II. Each instance provides an agent-facing problem specification, a reference solution on a prescribed evaluation grid, and case-specific accuracy and runtime targets. PDEAgent-Bench adopts a staged evaluation framework in which generated solvers must sequentially pass executability, numerical accuracy, and computational efficiency checks. Experiments with representative LLMs and code agents show that models can often produce runnable code, but their pass rate drops substantially once accuracy and efficiency requirements are enforced. These results indicate that current agents remain limited in producing numerically reliable and efficient PDE solvers, and that PDEAgent-Bench provides a reproducible testbed grounded in the practical requirements of numerical PDE solving.

Zhen Hang, Yushan Yashengjiang, Junhui Li +21

University of Science and Technology of China · Beijing University of Posts and Telecommunications · Tencent +10

PDE Solving Benchmark Design Scientific Code Generation Finite Element Method Code Generation Benchmarks

May 10, 2026cs.LG

HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field

u(t,\cdot)

is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time

t

and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state

u_t(θ,x)=u(t+θ,x)

,

θ\in[-τ,0]

. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction--diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from

0.241

,

0.188

, and

0.185

for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to

0.094

. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

Lennon J. Shikhman

College of Computing, Georgia Institute of Technology Atlanta, GA, United States

Fourier Neural Operator PDE Solving PDE Surrogate Modeling PDE Operator Learning Neural Operators

May 10, 2026cs.LG

MC $^2$ : Monte Carlo Correction for Fast Elliptic PDE Solving

Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC

^2

}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC

^2

matches the accuracy of solutions using over

1000\times

more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC

^2

} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs

\sim

\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.

Ethan Hsu, Hong Meng Yam, Ivan Ge

Stanford University

Neural PDE Solvers PDE Solving

May 9, 2026cs.LG

Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance

Physics-informed neural networks (PINNs) are a simple surrogate-modelling paradigm for partial differential equations, but their standard strong-form residual formulation is ill suited to the shallow water equations (SWE). It cannot enforce local conservation, handle discontinuities, or leverage the boundary-conforming unstructured meshes used in real-world applications. We introduce ``Data-Guided FVM-PINN'', a framework that replaces the strong-form residual with a differentiable, well-balanced Roe Riemann-solver finite-volume (FVM) loss evaluated on unstructured meshes. The major finding is that physics-only FVM-PINN training often fails on realistic 2D problems: the network collapses to a trivial low-momentum state that nearly satisfies the FVM-PINN residual but bears no resemblance to the true flow. A loss-landscape diagnostic shows that the FVM-PINN loss at zero momentum is only about

7\times

larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a

310\times

separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just

200

random velocity measurements drop the velocity-field

L_2

error by

22\times

versus physics-only;

50

measurements still deliver a

7\times

reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field

L_2

by

\sim$$23\%

in the sparse-data regime and is essentially neutral when dense reference data is available. On a real-world Savannah River reach (

1306

cells,

3600

~s simulation, five Manning zones), the framework constructs an accurate surrogate from SRH-2D anchor data, with time-window decomposition reducing error monotonically via progressive initial-condition handoff.

Xiaofeng Liu

Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA, USA · Institute for Computational and Data Sciences, Pennsylvania State University, University Park, PA, USA

PDE Solving PDE Surrogate Modeling Physics-Informed ML Computational Fluid Dynamics

May 9, 2026cs.LG

Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow

Deep learning paradigms, such as PINNs and neural operators, have significantly advanced the solving of PDEs. However, they often struggle to capture the continuous integral nature of physical systems, relying either on pointwise residuals that ignore the integral perspective or on pre-discretized temporal grids. Drawing inspiration from MeanFlow, a continuous-time integrator recently developed to efficiently solve generative ODEs, we introduce Spatio-Temporal MeanFlow, which functions as a novel PDE solver learning the finite-interval evolution of physical states. By substituting the generative velocity field with the physical PDE operator, we transform multi-step numerical integration into an efficient prediction with a freely controllable integration length. Crucially, we extend the original MeanFlow constraint from the temporal to the spatio-temporal domain, coupling time evolution with spatial consistency. This yields a unified framework naturally accommodating both time-dependent and stationary PDEs. Comprehensive experiments on benchmarks demonstrate that our approach achieves superior accuracy and inference efficiency over representative baselines. Furthermore, the proposed integral constraint enables excellent generalization to out-of-distribution initial conditions and varying spatial resolutions.

Hanru Bai, Yuncheng Zhou, Difan Zou

1Fudan University · 2The University of Hong Kong.

Flow Matching Neural PDE Solvers PDE Solving MeanFlow PDE Surrogate Modeling Physics-Informed ML

May 8, 2026cs.LG

When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains

We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators. We introduce the \textbf{Multi-Scale Attention Transformer} (\msat{}), a deep learning architecture that encodes spatiotemporal solution histories as token sequences and trains end-to-end via a composite supervised objective with optional physics-informed regularization terms. We conduct a comprehensive empirical evaluation against nine baselines -- including physics-informed neural networks (PINNs), neural operators (FNO, DeepONet, GNOT), and state-space models (Mamba-NO) -- across five benchmark problems from the PINNacle suite, using identical train/test splits and reference data for all methods. \msat{} achieves state-of-the-art generalization on complex geometry problems (

L^2_\mathrm{rel} = 0.0101

on Heat2D-CG, a

3.7\times

improvement over FNO) at

34\,\mathrm{s}

total inference vs.\

120{,}812\,\mathrm{s}

for Mamba-NO. Ablation studies over the physics regularization component reveal a precise inductive bias tradeoff: physics priors reduce test error on diffusion-dominated problems but degrade generalization on chaotic and recirculating-flow regimes, directly characterizing the prior misspecification boundary. Approximation error bounds as a function of domain boundary complexity

κ

provide a theoretical basis for these empirical findings and a principled rule for architecture selection.

Brandon Yee, Pairie Koh, Jack Rodriguez +1

Physics Lab, Yee Collins Research Group

Neural PDE Solvers PDE Solving PDE Operator Learning Physics-Informed ML

May 8, 2026math.NA

NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces

The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs.

Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak +1

Synopsys UK Ltd, UK · Division of Applied Mathematics, Brown University, Rhode Island, USA · Synopsys Inc, California, USA

PDE Solving Structural Mechanics Proper Orthogonal Decomposition Deep Operator Networks Krylov Subspace Methods

May 7, 2026cs.LG

INEUS: Iterative Neural Solver for High-Dimensional PIDEs

In this paper, we introduce INEUS, a meshfree iterative neural solver for partial integro-differential equations (PIDEs). The method replaces the explicit evaluation of nonlocal jump integrals with single-jump sampling and reformulates PIDE solving as a sequence of recursive regression problems. Like Physics-Informed Neural Networks (PINNs), INEUS learns global solutions over the entire space-time domain, yet it offers a more efficient treatment of nonlocal terms and avoids the computationally expensive differentiation of full PIDE residuals. These features make INEUS particularly well suited for high-dimensional PDEs and PIDEs. Supported by a contraction-based convergence proof for linear PIDEs, our numerical experiments show that INEUS delivers accurate and scalable solutions for various high-dimensional linear and nonlinear examples.

Jean-Loup Dupret, Davide Gallon, Patrick Cheridito

Department of Mathematics and RiskLab, ETH Zurich, Switzerland · Applied Mathematics: Institute for Analysis and Numerics, University of Münster, Germany

Neural PDE Solvers PDE Solving

May 6, 2026cs.LG

A Robust Foundation Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers

We propose an architecture that augments the Flux Neural Operator (Flux NO), which combines the classical finite volume method (FVM) with neural operators, with ViT-based context injection. Our model is formulated as a hypernetwork: it extracts solution dynamics over a finite temporal window, encodes them with a recurrent Vision Transformer, and generates the parameters of a context-conditioned neural operator. This enables the model to infer and solve conservation laws without explicit access to the governing equation or PDE coefficients. Experimentally, we show that the proposed method preserves the robustness, generalization ability, and long-time prediction advantages of Flux NO over standard neural operators, while delivering reliable numerical solutions across a broad range of conservative systems, including previously unseen fluxes. Our code is available at https://github.com/xx257xx/CONTEXT_FLUX_NO.

Taeyoung Kim, Joon-Hyuk Ko

Center for AI and Natural Sciences Korea Institute for Advanced Study Seoul, South Korea 02455

PDE Solving Neural Operators

May 6, 2026cs.LG

Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations

We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains

\mathcal{O}(τ)

, yielding a Gronwall stability factor bounded uniformly in the step size and independent of the spatial resolution. The network is trained end-to-end through a solver-in-the-loop objective that unrolls the full iteration and penalises trajectory error in a Bourgain-type norm, aligning learning with multi-step solver dynamics rather than isolated one-step targets. Under stated assumptions, the global error satisfies

C(\varepsilon_{net}+δ)\,τ^γ\ln(1/τ)

, where

\varepsilon_{net}

measures the network approximation quality and

δ

the training shortfall. Experiments on three dispersive benchmarks with rough data show that HIN-LRI improves accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.

Zhangyong Liang, Huanhuan Gao

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

Neural PDE Solvers PDE Solving Neural Operators

May 6, 2026cs.LG

From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics

Inferring continuum models directly from video is hampered by two facts: the recorded field is uncalibrated image intensity rather than a physical state, and direct numerical differentiation of noisy frames is unstable. We develop a video-to-PDE pipeline that converts grayscale recordings of an ink plume into a normalised scalar field

u(x,y,t)

, isolates a bulk drift

\mathbf{v}(t)

from intrinsic spreading via the intensity-weighted centroid, and identifies an effective transport law by weak-form sparse regression. Conditioning, threshold-sweep and random-centre diagnostics show that overcomplete libraries are strongly collinear; the search is therefore restricted to compact gradient-based libraries. Coefficients are refined by an inverse physics-informed network and recalibrated against forward rollouts, with a chronological block bootstrap quantifying uncertainty. The selected reduced model

u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu

outperforms advection--diffusion baselines on held-out frames, retains a positive Laplacian coefficient, and admits a Cole--Hopf reduction to a linear advection--diffusion equation. The framework demonstrates that uncalibrated visual data can yield compact, predictive and structurally interpretable continuum models when discovery, calibration and uncertainty are treated as distinct stages.

Cesar Acosta-Minoli, Sayantan Sarkar

aGEDES, Universidad del Quindio, Cra 15 Cl 12-00, Armenia, 630004, Quindio, Colombia · bDepartment of Mathematics, State University of New York at Buffalo, Mathematics Building, Buffalo, 14260, New York, USA

PDE Solving Sparse Identification of Nonlinear Dynamics PDE Discovery

May 5, 2026cs.LG

Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

This paper develops convolutional neural network (CNN) methods for simultaneous Sobolev approximation and elliptic boundary value problems on compact Riemannian manifolds. We prove approximation estimates for single- and multichannel CNNs, with rates governed by the intrinsic dimension and the smoothness gap. Motivated by elliptic stability, we propose a physics-informed CNN framework with a spectral boundary loss. The boundary residual is expanded in boundary Laplace--Beltrami eigenmodes and penalized by Sobolev trace weights, matching the natural

\mathcal H^{2s-1/2}(\partial\mathcal M^d)

trace norm for

2s

-order elliptic problems. This avoids smooth auxiliary constructions for exact boundary enforcement and singular Sobolev--Slobodeckij double integrals, while allowing FFT-based or precomputed spectral implementations. We also derive an error decomposition separating approximation, generalization, and spectral truncation errors, showing that the proposed loss is aligned with localized fast-rate generalization analysis. Numerical experiments on the upper hemisphere and upper half-torus demonstrate improved accuracy, convergence, and stability over standard PINNs, with one to two orders of magnitude gains for high-frequency boundary data.

Hanfei Zhou, Lei Shi

School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

PDE Solving Neural Network Approximation Theory Physics-Informed ML Convolutional Neural Networks

May 5, 2026math.NA

Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks

The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the

H^{-1}

norm of the residual requires a supremum over an infinite-dimensional test space. We prove that the

H^{-1}

norm of any functional is equivalent to its expected squared evaluation against a random test function whose probability distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for

d \geq 2

, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently of the differential operator, and no supremum evaluation is necessary. This equivalence introduces the notion of stochastically weak solutions, which coincide with classical weak solutions, and motivates stochastic variational physics-informed neural networks (SV-PINNs): neural networks trained by minimising an empirical approximation of the stochastic norm of the PDE residual. Although instantiated here with neural networks, the underlying principle is independent of the trial space and suggests a broader paradigm for numerical methods based on stochastic rather than deterministic test spaces. The framework extends naturally to higher-order elliptic, parabolic and hyperbolic equations and to abstract operator equations on Hilbert spaces. As a proof of concept, we present numerical experiments on eight challenging second-order linear elliptic problems spanning high-frequency and multi-scale solutions, indefinite operators, variable coefficients, and non-standard domains, in which SV-PINNs consistently and significantly outperform standard PINNs, recovering solutions to within one percent relative error in hundreds of L-BFGS steps.

Diego Marcondes

Neural PDE Solvers PDE Solving Physics-Informed ML

May 4, 2026cs.LG

Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation

Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator

\mathcal{G}: H^{s}(D)\to H^{t}(D')

with

s > d/2

and inputs restricted to a compact subset of

H^{s}(D)

, we prove that

\mathcal{G}

can be uniformly approximated in

H^{t}

-norm by a neural operator with

\mathcal{O}(\varepsilon^{-d/s})

trainable parameters. This yields an explicit complexity--error relation of the form

\|\mathcal{G}-\mathcal{G}_θ\|_{H^{t}} \lesssim C N^{-s/d}

. We then study the one-dimensional viscous Burgers solution operator

\mathcal{G}: u_{0}\mapsto u(\cdot,1)

on a bounded

H^{1}

-ball and train FNOs with an

H^{1}

-loss. Across a sweep of model sizes, we obtain test

H^{1}

-errors down to

\mathcal{O}(10^{-7})

and relative errors of order

10^{-3}

, with predictions accurately matching both solutions and spatial derivatives on held-out data. A log-log plot of Sobolev error versus parameter count exhibits an approximate power law

\|\mathcal{G}-\mathcal{G}_θ\|_{H^{1}} \approx C N^{-α}

with empirical exponent

α\approx 1.4

, and long-horizon training reveals optimization instabilities in large FNOs, providing quantitative evidence that Sobolev-space approximation theory meaningfully predicts neural-operator scaling behavior.

Nicole Hao

Department of Mathematics, Cornell University

Fourier Neural Operator PDE Solving PDE Operator Learning Neural Network Approximation Theory

May 2, 2026cs.LG

Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations

Physics-Informed Neural Networks (PINNs) offer a flexible paradigm for solving differential equations by embedding governing laws into the training objective. A persistent limitation is instance specificity: standard PINNs typically require retraining for each new forcing term, boundary/initial condition, or parameter setting. One-shot transfer learning (OTL) addresses this bottleneck for linear operators by freezing a pretrained latent representation and computing optimal output weights in closed form, but for nonlinear problems closed-form adaptation is generally unavailable because the loss is nonconvex in the output layer. In this paper we substantially broaden the class of nonlinearities amenable to one-shot PINN transfer by combining OTL with Chebyshev polynomial surrogates. We approximate general smooth weakly nonlinear terms by truncated Chebyshev expansions over a prescribed solution range, yielding a polynomial nonlinearity that can be handled by a perturbative decomposition into linear subproblems. A multi-head PINN learns a reusable latent space associated with the dominant linear operator; at test time, solutions to new instances are obtained via a sequence of closed-form linear solves in the output layer, without retraining the network body. We provide a unified derivation of the framework for ODEs and PDEs and demonstrate accuracy and fast online adaptation on nonlinear benchmarks, including non-polynomial and singular ODE nonlinearities as well as a reaction-diffusion PDE with saturating kinetics, demonstrating the method's utility in many-query regimes.

Yiqi Rao, Pavlos Protopapas

Harvard University

PDE Solving Transfer Learning Physics-Informed ML

May 1, 2026cs.LG

Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries

This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. This inclusion acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion, evaluated at several points in the domain, is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture, while local information is used as input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries is compared with its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM-generated training data.

Rodolphe Barlogis, Ferhat Tamssaouet, Quentin Falcoz +1

PROMES-CNRS, Université Perpignan Via Domitia (UPVD), Perpignan, France · LAAS-CNRS, Université de Toulouse (UT), Toulouse, France

PDE Solving PDE Operator Learning Deep Operator Networks Physics-Informed Neural Operators

Apr 30, 2026cs.LG

A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions

Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager's minimum-dissipation principle, we introduce a history variable -- interpretable as momentum -- and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.

Matteo Raviola, Benjamin Peherstorfer

EPFL, Lausanne, Switzerland · Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

Neural PDE Solvers PDE Solving

Apr 30, 2026cs.LG

An adaptive wavelet-based PINN for problems with localized high-magnitude source

In recent years, physics-informed neural networks (PINNs) have gained significant attention for solving differential equations, although they suffer from two fundamental limitations, namely, spectral bias inherent in neural networks and loss imbalance arising from multiscale phenomena. This paper proposes an adaptive wavelet-based PINN (AW-PINN) to address the extreme loss imbalance characteristic of problems with localized high-magnitude source terms. Such problems frequently arise in various physical applications, such as thermal processing, electro-magnetics, impact mechanics, and fluid dynamics involving localized forcing. The proposed framework dynamically adjusts the wavelet basis function based on residual and supervised loss. This adaptive nature makes AW-PINN handle problems with high-scale features effectively without being memory-intensive. Additionally, AW-PINN does not rely on automatic differentiation to obtain derivatives involved in the loss function, which accelerates the training process. The method operates in two stages, an initial short pre-training phase with fixed bases to select physically relevant wavelet families, followed by an adaptive refinement that adapts scales and translations without populating high-resolution bases across entire domains. Theoretically, we show that under certain assumptions, AW-PINN admits a Gaussian process limit and derive its associated NTK structure. We evaluate AW-PINN on several challenging PDEs featuring localized high-magnitude source terms with extreme loss imbalances having ratios up to

10^{10}:1

. Across these PDEs, including transient heat conduction, highly localized Poisson problems, oscillatory flow equations, and Maxwell equations with a point charge source, AW-PINN consistently outperforms existing methods in its class.

Himanshu Pandey, Ratikanta Behera

Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, 560012, India

PDE Solving Wavelet Methods Physics-Informed ML

Apr 28, 2026cs.LG

Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations

Neural operators have emerged as powerful data-driven surrogates for learning solution operators of parametric partial differential equations (PDEs). However, widely used Fourier Neural Operators (FNOs) rely on global Fourier representations, which can be inefficient for resolving anisotropic structures, sharp gradients, and spatially localized discontinuities that arise in shock-dominated and multiscale regimes. To address these limitations, we introduce the Shearlet Neural Operator (SNO), a neural operator architecture that replaces the Fourier transform with a shearlet-based representation. Shearlets offer directional, multiscale, and spatially localized atoms with near-optimal sparse approximation of anisotropic features, providing an inductive bias aligned with PDE solutions containing edges, fronts, and shocks. SNO learns in the shearlet domain and reconstructs predictions via the inverse transform, retaining efficient spectral computation while improving locality and directional selectivity. Across seven benchmark PDE families, including strongly anisotropic advection, anisotropic diffusion, and nonlinear conservation laws with straight, curved, interacting, spiral, and polygonal shock structures, SNO consistently improves predictive accuracy and feature fidelity over FNO baselines, with the largest gains observed in anisotropic and discontinuity-dominated settings.

Fabio Pereira dos Santos, Julio de Castro Vargas Fernandes, Adriano Mauricio de Almeida Cortes

Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing Av. Getúlio Vargas, 333 - Quitandinha, Petrópolis - RJ, 25651-075 · Systems and Computational Engineering Program, COPPE/UFRJ, Federal University of Rio de Janeiro, P.O. Box 68542, Rio de Janeiro, RJ, 21941-909, Brasil

PDE Solving PDE Operator Learning Neural Operators

Apr 28, 2026quant-ph

Quantum Dynamics via Score Matching on Bohmian Trajectories

We solve the time-dependent Schrödinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schrödinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.

Lei Wang

Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

Score Matching PDE Solving Score-Based Generative Modeling

Apr 26, 2026eess.IV

A Coupled Fourth Order Telegraph Diffusion Framework Using Grayscale Indicators for Image Despeckling

Speckle noise severely limits the quality of images acquired from coherent imaging systems such as Synthetic Aperture Radar (SAR) and medical ultrasound. Traditional second-order PDE-based despeckling approaches, although popular, often introduce staircase artifacts and blur fine details. To overcome these limitations, we present a nonlinear, fourth-order coupled hyperbolic-parabolic PDE model that effectively reduces noise while preserving the structure. The framework consists of two evolution equations: one governing fourth-order diffusion for effective speckle reduction and smooth intensity transitions, and another refining an edge indicator to protect textures and structural features. The diffusion coefficient is adaptively constructed using both the image intensity variable u and a grayscale-based indicator function, ensuring structure-aware denoising while avoiding blocky artifacts and preserving fine details. We also prove the existence of a weak solution to the proposed model by applying Schauder fixed-point theorem. A finite-difference scheme with Gauss Seidel iteration is employed for efficient implementation. We compare the proposed model with the existing coupled second-order PDE model (HPCPDE) and the fourth-order telegraph diffusion model (TDFM). The results show that our model consistently outperforms these approaches. Experiments on standard grayscale images, real SAR and ultrasound data, as well as speckle-corrupted color images, demonstrate that the proposed method achieves superior performance over conventional PDE-based techniques in terms of PSNR, MSSIM, and Speckle Index.

Manish Kumar, Rajendra K. Ray

School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India

PDE Solving Image Denoising

Apr 26, 2026cs.CV

Single Image Defogging Using a Fourth-Order Telegraph PDE Guided by Physical Haze Modeling

In real-world scenarios, image defogging is an inverse problem due to unknown scene depth, atmospheric scattering, and the common absence of ground truth . To resolve the issue, we propose a hybrid defogging model that integrates a fourth-order nonlinear PDE with a physical haze formation model. We used Dark Channel Prior to estimate atmospheric parameters and to generate a guidance image, while the final restoration is performed via a fourth-order PDE-based evolution. A fourth-order PDE of the type telegraph is then evolved, incorporating an edge-adaptive diffusion coefficient and a fidelity term weighted by the transmission map. Fourth-order diffusion effectively suppresses haze while preserving structural details, and the hyperbolic formulation improves numerical stability and convergence behavior. We use relative error norm criteria for the convergence of our PDE. The proposed method is compared with Dark Channel prior, modified Dark Channel prior, and variational-based single-image defogging techniques. When we have ground truth available, we use MSE and SSIM for quantitative evaluation, whereas no-reference metrics, including FADE, Contrast Restoration Index, Average Gradient, and Entropy, are applied to real-world foggy images. Experimental results demonstrate that the proposed hybrid PDE-based method provides comparable visual quality and maintains structural details.

Manish Kumar, Rajendra K. Ray

School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India

PDE Solving Physics-Guided Image Restoration Image Restoration Image Dehazing

Apr 26, 2026cs.LG

When PINNs Go Wrong: Pseudo-Time Stepping Against Spurious Solutions

Physics-informed neural networks (PINNs) provide a promising machine learning framework for solving partial differential equations, but their training often breaks down on challenging problems, sometimes converging to physically incorrect solutions despite achieving small residual losses. This failure, we argue, is not merely an optimization difficulty. Rather, it reflects a fundamental weakness of the empirical PDE residual loss, which can admit trivial or spurious solutions during training. From this perspective, we revisit pseudo-time stepping, a technique that has recently shown strong empirical success in PINNs. We show that its main benefit is not simply to ease optimization; instead, when combined with collocation-point resampling, it helps reveal and avoid spurious solutions. At the same time, we find that the effectiveness of pseudo-time stepping depends critically on the choice of step size, which cannot be tuned reliably from the training loss alone. To overcome this limitation, we propose an adaptive pseudo-time stepping strategy that selects the step size from a finite-difference surrogate of the local residual Jacobian, yielding the largest step permitted by local stability without per-problem tuning. Across a diverse set of PDE benchmarks, the proposed method consistently improves both accuracy and robustness. Together, these findings provide a clearer understanding of why PINNs fail and suggest a practical pathway toward more reliable physics-informed learning. All code and data accompanying this manuscript are available at https://github.com/sifanexisted/jaxpi2.

Sifan Wang, Shawn Koohy, Yiping Lu +1

Institute for Foundations of Data Science, Yale University · Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania · Industrial Engineering & Management Science, Northwestern University

PDE Solving Physics-Informed ML

Apr 23, 2026cs.LG

Transferable Physics-Informed Representations via Closed-Form Head Adaptation

Physics-informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi-PINN). Pi-PINN learns a transferable physics-informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed-form head adaptation using a least-squares-optimal pseudoinverse under PDE constraints. We further investigate the synergies between data-driven multi-task learning loss and physics-informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi-PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics-informed solutions without requiring any data for unseen instances. Pi-PINN can produce predictions 100-1000 times faster than a typical PINN, while producing predictions with 10-100 times lower relative error than a typical data-driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed-form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.

Jian Cheng Wong, Isaac Yin Chung Lai, Pao-Hsiung Chiu +3

Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore · National University of Singapore (NUS), Singapore · Indian Institute of Technology Goa (IIT Goa), India +1

Representation Learning PDE Solving Physics-Informed ML

Apr 23, 2026cs.LG

A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization

Standard physics-informed neural networks (PINNs) struggle to simulate highly oscillatory Helmholtz solutions in heterogeneous media because pointwise minimization of second-order PDE residuals is computationally expensive, biased toward smooth solutions, and requires artificial absorbing boundary layers to restrict the solution. To overcome these challenges, we introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation. It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation that imposes a nonlocal constraint. Oscillatory behavior and outgoing radiation are encoded directly through the integral kernel, eliminating second-order spatial derivatives and enforcing physical solutions without additional boundary layers. Theoretically, optimizing this GI loss via a neural network acts as a spectrally tuned preconditioned iteration, enabling convergence in heterogeneous media where the classical Born series diverges. By exploiting FFT-based convolution to accelerate the GI loss evaluation, our approach substantially reduces GPU memory usage and training time. However, this efficiency relies on a fixed regular grid, which can limit local resolution. To improve local accuracy in strong scattering regions, we also propose a hybrid GI+PDE loss, enforcing a lightweight Helmholtz residual at a small number of nonuniformly sampled collocation points. We evaluate our method on seismic benchmark models characterized by structural contrasts and subwavelength heterogeneity at frequencies up to 20Hz. GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten. In models with localized scattering, the hybrid loss yields the most accurate reconstructions, providing a stable, efficient, and physically grounded alternative.

Mohammad Mahdi Abedi, David Pardo, Tariq Alkhalifah

University of the Basque Country (UPV/EHU), Department of Mathematics, Spain · Basque Center for Applied Mathematics (BCAM), Bilbao, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1

Neural PDE Solvers PDE Solving Physics-Informed ML

Apr 22, 2026math.NA

CO $_2$ sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)

This paper presents the hybrid solver for a

CO_2

sequestration problem. The solver uses the IGA-ADS (IsoGeometric Analysis Alternating Directions solver) to compute the saturation scalar field update using the explicit method, and CRVPINN (Collocation-based Robust Variational Physics Informed Neural Networks solver) to compute the pressure scalar field. The study focuses on simulating the physical behavior of

CO_2

in porous structures, excluding chemical reactions. The mathematical model is based on Darcy's Law. The CRVPINN is pretrained on the initial pressure configuration, and the time step pressure updates require only 100 iterations of the Adam method per time step. We compare our hybrid IGA-ADS solver, coupled with the CRVPINN method, with a baseline of the IGA-ADS solver coupled with the MUMPS direct solver. Our hybrid solver is over 3 times faster on a single computational node from the ARES cluster of ACK CYFRONET. Future work includes extensive testing, inverse problem solving, and potential application to

H_2

storage problems.

Askold Vilkha, Tomasz Służalec, Marcin Łoś +1

AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 2026 30, Kraków, 30-059, Poland

Neural PDE Solvers PDE Solving Physics-Informed ML Computational Fluid Dynamics

Apr 19, 2026cs.SD

Virtual boundary integral neural network for three-dimensional exterior acoustic problems

This paper presents a virtual boundary integral neural network (VBINN) for exterior acoustic problems in three dimensions. The method introduces a virtual boundary inside the scatterer or vibrating body and represents the associated source density with a neural network. Coupled with the acoustic fundamental solution, this representation satisfies the Sommerfeld radiation condition by construction and enables direct evaluation of the acoustic pressure and its normal derivative at arbitrary field points. Because the integration surface is separated from the physical boundary, the formulation avoids the singular and near singular kernel evaluations associated with coincident source and collocation points in conventional boundary integral learning methods. To reduce sensitivity to boundary placement, the geometric parameters of the virtual boundary are optimized jointly with the source density during training. Numerical examples for acoustic scattering, multiple body interaction, and underwater acoustic propagation show close agreement with analytical solutions and COMSOL results, and the Burton Miller extension further improves stability near characteristic frequencies. These results demonstrate the potential of VBINN for exterior acoustic analysis in three dimensions.

Jiahao Li, Qiang Xi, Ilia Marchevskiy +1

College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China · Department of Applied Mathematics, Bauman Moscow State Technical University, Moscow 105005, Russia

Neural PDE Solvers PDE Solving

Apr 18, 2026math.NA

Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches

This thesis develops numerical and theoretical approaches for understanding and analyzing singularity formation in Partial Differential Equations (PDEs). The singularity formation in the Navier-Stokes Equation (NSE) is famously challenging as one of the seven Clay Prize problems. Unlike simpler equations such as the Nonlinear Heat (NLH) or Keller-Segel (KS) equations, where formal asymptotics near blowup are better understood, the intrinsic complexity of NSE makes quantitative analytical treatment difficult, if not impossible, without numerical guidance. Building on numerical insights, we introduce a robust analytical framework to simplify and systematize pen-and-paper proofs for simpler singular PDEs. We present a novel approach based on enforcing vanishing modulation conditions for perturbations around approximate blowup profiles, complemented by singularly weighted energy estimates. We demonstrate the efficacy of our method on PDEs with complicated asymptotics, such as NLH and the Complex Ginzburg-Landau (CGL) equation, and address the open problem of singularity formation in the 3D KS equation with logistic damping. We develop and refine numerical approaches that facilitate deeper insights into singularity formation. We demonstrate that machine learning methods significantly enhance our capability to identify and characterize potential blowup solutions with high precision. We improve on existing Physics-Informed Neural Network (PINN) and Neural Operator (NO) frameworks. Moreover, we present a novel machine learning paradigm, the Kolmogorov-Arnold Network (KAN) architecture, whose interpretability and excellent scaling properties are achieved through learnable nonlinearities.

Yixuan Wang

CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]

PDE Solving Kolmogorov-Arnold Networks PDE Operator Learning Physics-Informed ML

Apr 17, 2026cs.LG

A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions

Efficient and stable solution of partial differential equations (PDEs) is central to scientific and engineering applications, yet existing numerical solvers rely heavily on matrix based discretizations, while learning based methods require costly training and often suffer from limited generalization. In this work, we proposes a PDE energy driven framework that solves PDEs through physically constrained diffusion iterations, without relying on classical matrix based finite element assembly or data driven neural network training. The proposed method evolves arbitrary random initial fields through PDE energy driven implicit iterations combined with Gaussian smoothing, while strictly enforcing boundary conditions at each iteration. The proposed formulation is applied to representative one dimensional Poisson, Heat, and viscous Burgers equations, covering both steady state and transient problems. Numerical results demonstrate stable convergence to the unique physical solution from random initializations, with accurate resolution of sharp gradients and controlled Mean Squared Error (MSE) across a wide range of discretization parameters. Detailed comparisons with analytical solutions indicate that the framework achieves competitive accuracy and stability. Overall, the proposed framework provides a fast, flexible, and physically consistent alternative to traditional numerical solvers, offering a potential pathway for scalable PDE solutions in both research and engineering applications.

Yi Bing, Zheng Ran, Fu Jinyang +2

School of Traffic and Transportation Engineering, Central South University, Changsha, China · School of Civil Engineering, Central South University, Changsha, China · National Engineering Research Center of High-speed Railway Construction Technology, Changsha, China +2