Neural PDE Solvers

PDE: Partial Differential Equation

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23 papers in the last four weeks, up 109% on the four weeks before. 0.2% of all new papers.

Jul 13Week of Sep 28

Latest papers 166

Oct 7, 2026cs.LG

Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative

L^2

error on every problem, reducing error by

10.7\%

--

27.5\%

relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

Chun-Wun Cheng, Bingcheng Hu, Angelica I. Aviles-Rivero

Shanghai Normal University · YMSC, Tsinghua University

Neural PDE Solvers PDE Solving Physics-Informed ML Dynamic Neural Networks

Oct 7, 2026cs.LG

An extended deep energy method for thermo-mechanical crack propagation

Thermo-mechanical fracture couples transient heat conduction on a cracked domain with a crack that grows as the temperature and the displacement evolve. Neural energy solvers have been proposed for phase-field fracture and later extended to represent a sharp crack through the network input, but heat conduction on the cracked domain and crack propagation under the resulting thermal stresses have not yet been treated together in these solvers. We present an extended deep energy method for thermo-mechanical crack propagation in which the crack remains a sharp polyline. Two networks represent the temperature and the displacement and receive the crack through a scalar embedding function, discontinuous across the crack and smooth elsewhere, so that both fields can jump across it without a regularization length, and the displacement is enriched near the tip by the Williams expansion with trainable amplitudes. The two fields are obtained by minimizing an incremental conduction functional and the thermoelastic potential energy in a staggered sequence, with Monte Carlo integration on points stratified over background elements, densified near the tip and redrawn during training. The stress intensity factors are extracted by the interaction integral with the area term of Wilson and Yu and checked by a sweep of the contour radius, and the crack advances at the maximum hoop stress angle when the energy release rate of the kink reaches the critical value at the crack-tip temperature. On a stationary thermal edge crack the extracted stress intensity factor agrees with the published value to 0.11%, in a functionally graded shear test initiation agrees with an independent sharp-crack finite element solution to within one load step, and on a notched cruciform specimen the crack paths follow the published solutions under mechanical, thermal and combined loading.

Han Zhang, Mehrisadat Makki Alamdari, Babak Shahbodagh +4

University of New South Wales, Sydney, NSW, Australia · Central Queensland University, Melbourne, VIC, Australia · Bauhaus-Universität Weimar, Weimar, Germany

Neural PDE Solvers Materials Science

Oct 6, 2026math.NA

FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equations

We construct neural approximation classes native to the graph spaces H(div) and H(curl), in two and three dimensions and, for H(div), in any dimension. Every realization lies in the space for all parameter values, and with kinked potentials, such as ReLU networks, the admissible jumps appear at finite width. The classes are images of scalar and componentwise networks under fixed operators of the de Rham complex, and do not involve a mesh or finite element emulation. For H(div) in R^n two native classes are given on an equal footing, with a skew-symmetric potential

A

:

\mathrm{Div}\,A+R_nq+\mathbf{h}

, with the divergence

q

as an explicit unknown, and

\mathrm{Div}\,A+\mathbf{z}

with an

H^1

field

\mathbf{z}

; for H(curl) the analogous classes are

\mathrm{grad}\,φ+Sr+\mathbf{h}

in two dimensions and

\mathrm{grad}\,φ+\mathbf{z}

in two and three dimensions. In all of them every interface jump of the field is carried by the potential term,

\mathrm{Div}\,A

or

\mathrm{grad}\,φ

, while the remaining part has no interface jump (it is an

H^1

field in the regular-decomposition classes); the classes with

\mathbf{z}

are the componentwise approach enriched by this term. Known or learned interface geometry enters the potential through factors with trainable amplitudes, and the remaining part if the divergence jumps. The classes lead to the FOSLS-deRhaNN method, first-order system least squares with de Rham neural networks, whose loss is the least-squares functional posed in the natural spaces of the weak formulation; for elliptic equations this includes

H^{-1}

right-hand sides and

H^{1/2}

Dirichlet data. Elliptic equations with discontinuous coefficients and curl-curl problems are treated as instances, with the functional equivalent to the error; linear transport with discontinuous solutions and conservation laws with shocks use the same flux classes.

Shun Zhang

Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong, China

Neural PDE Solvers Neural Network Approximation Theory

Oct 6, 2026math.NA

A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs

We develop a mesh-free neural JKO scheme for advection-reaction-diffusion equations with a gradient-flow structure in the Hellinger-Kantorovich (HK) geometry of unbalanced optimal transport. Each update is parametrized by a spatial map and a mass-changing factor, allowing spatial redistribution and local mass creation or loss to be treated jointly within a single variational step. Their cone action bounds the squared HK distance from above, yielding a sufficient condition for discrete energy dissipation through comparison with the identity pair. Minimizing the pair objective over all admissible pairs recovers the exact JKO minimum when the source and a minimizer have positive densities. We establish existence and mass bounds for JKO minimizers and, under additional assumptions, obtain positivity and regularity together with a discrete Euler-Lagrange equation and a metric-dissipation identity. The self-consistent chemical potential is then nonincreasing along an optimal map. There exist parametric pairs whose endpoint densities and objective values converge to those of an exact JKO minimizer, provided a regular-pair approximation hypothesis holds. Finally, we show that a primal-dual gap controls objective suboptimality and, for Boltzmann entropy, the

L^1

density error, assuming exact-step regularity, positive-semidefinite interactions, and global dual feasibility. Numerical experiments examine pointwise agreement with the PDE, energy dissipation, and the roles of transport, reaction, and fully implicit interactions.

Geuntaek Seo, Cheolhyeong Kim, Hwijae Son +1

Department of Mathematics, Pohang University of Science and Technology · Samsung Electronics · Department of Mathematics, Konkuk University

Neural PDE Solvers PDE Solving Optimal Transport

Oct 5, 2026math.NA

Singular parameters and missing limits in neural PDE solvers

Neural solvers for partial differential equations (PDEs) can approach an accurate solution while their parameters grow without bound. In such cases, the limiting solution may have no finite representation in the chosen model, leaving the best loss unattained. Our analysis connects missing limits in deep neural tanh- networks to unbounded hidden parameters or increasingly redundant neurons. For a class of models built from translated kernels, we describe the missing functions and recover them by adding kernel derivatives to the model. This completion makes the best approximation attainable under standard assumptions. Numerical studies follow the associated parameter growth and explore how completion affects PDE optimization.

Daniel Fernández

Chair for Dynamics, Control, Machine Learning, and Numerics (Alexander von Humboldt Professorship), Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany.

Neural PDE Solvers Neural Network Optimization Neural Network Approximation Theory

Oct 5, 2026math.NA

IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs

Isogeometric analysis (IGA) solves partial differential equations accurately on exact NURBS geometry, whereas neural solvers are mesh-free but often orders of magnitude less accurate and typically trained by non-convex optimization without error control. We propose IGA-KAN, which uses local Kolmogorov-Arnold networks, fitted in closed form, to improve the IGA solution instead of replacing it. An IGA Galerkin solve produces u_h; on every knot-vertex patch a Kolmogorov-Arnold ridge model is fitted to the strong form of the equation, the exact boundary data and u_h, and the models are blended by IGA hat functions. With fixed inner functions the fit is one batched linear least-squares problem, without optimizer, learning rate or initialization. An a posteriori safeguard, motivated by a maximum-principle bound, decides where local models are used, keeping the IGA solution elsewhere. On eight benchmarks with exact solutions, five from the literature and one also posed on a domain fitted to a brain slice from MRI, the method reduces the error of IGA, at an unchanged number of Galerkin unknowns, by factors of 4.2 to 90 in L^2 and 4.1 to 220 in H^1 on the reference meshes, and its L^2 error is 6 to 6x10^4 times smaller than that of the best Kolmogorov-Arnold network trained from scratch on the same equations with a fixed budget. In an inverse problem it recovers an unknown constant source from one noise-free observation 167 times more accurately than IGA. The gain is attributed to the superconvergence of local averages of the Galerkin solution.

Sima Naraghi, Kourosh Parand, Amirhossein Sadr +1

Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran · Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran · Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran +1

Neural PDE Solvers Kolmogorov-Arnold Networks Physics-Informed ML PDE Inverse Problems

Oct 4, 2026cs.LG

Component-Level Evaluation of Adaptive PINN Training for CFD-Oriented Crystal Growth Simulation

Physics-informed neural network (PINN) training minimizes a weighted combination of partial differential equation (PDE), boundary-condition, and initial-condition losses. Because adaptive methods modify these weights during training, their weighted total losses are not always directly comparable. We compare fixed-weight PINN, gradient-normalized PINN (GNPINN), and a rule-based adaptive controller (AgenticPINN) under matched settings on a heat-equation benchmark and a simplified Czochralski-oriented thermal-fluid problem. In the crystal-growth MLP experiment, adaptive control reduced the PDE residual from the order of

10^{-5}

to

10^{-6}

, while the boundary-condition loss increased from the order of

10^{-5}

to

10^{-2}

. On the heat-equation benchmark, GNPINN achieved the lowest relative

L_2

field error (0.054), whereas AgenticPINN obtained the smallest PDE residual but a relative

L_2

error of 1.368. Gaussian-process surrogates were additionally evaluated using case-wise holdout tests on corrected Czochralski CFD parameter sweeps. The temperature-field error for the temperature sweep was approximately 6%, whereas the axial-velocity error for the crystal-rotation sweep was approximately 42%. These findings show that adaptive control can improve equation satisfaction while weakening other physical constraints. PINN training should therefore be evaluated using separate PDE, boundary-condition, and solution-error metrics rather than weighted total loss alone.

Niruta Chapagain, Rohit Raj, Bertwin Kurisinkal Shine +1

University of Europe for Applied Sciences Potsdam, Germany · Vienna, Austria · Technische Universität Chemnitz Chemnitz, Germany +1

Neural PDE Solvers PDE Surrogate Modeling Adaptive Loss Weighting Physics-Informed ML

Oct 2, 2026cs.AI

Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers

Neural surrogate models for Partial Differential Equations (PDEs) on unstructured 3D geometries are often limited by poor generalization and the high cost of generating large-scale training datasets. Consequently, pre-training on massive datasets of related PDE dynamics has emerged as a critical alternative to enhance the robustness and scalability of these models. However, this strategy is neither compute- nor data-efficient, as it relies on massive pre-computed data that is very costly to generate. In this work, we introduce a disk-data-free pre-training framework tailored to both steady-state and transient regimes. For steady-state problems, we propose a geometry-driven strategy that leverages intrinsic shape descriptors to learn representations of complex 3D domains. For transient problems, we introduce a physics-driven approach based on online generation of synthetic PDE data, enabling scalable pre-training without reliance on expensive datasets. Across multiple experiments, our approach achieves faster convergence, greater data efficiency, and higher accuracy during fine-tuning, particularly under realistic low-data regimes. This methodology provides a practical pathway toward data-efficient neural emulators for large-scale simulations.

Luis Medrano-Navarro, Giacomo Baldan, Qiang Liu +4

Technical University of Munich · University of Stuttgart

Synthetic Data Pretraining Neural PDE Solvers PDE Solving PDE Surrogate Modeling Data-Efficient Learning

Oct 1, 2026math.NA

Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations

We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear

p

-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative

L^2

and

L^\infty

errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.

Tan Phuong Dong Le

Department of Applied Mathematics, University of Waterloo, Canada · Department of Statistics and Actuarial Science, University of Waterloo, Canada

Neural PDE Solvers PDE Solving Kolmogorov-Arnold Networks

Sep 30, 2026cs.LG

Physics-Informed Method of Group Data Handling: Adaptive Construction of Functional Representations with an Application to the Navier-Stokes Equations

Physics-informed computational methods usually optimize parameters within a functional representation whose structure is fixed in advance. This work proposes a Physics-Informed Method of Group Data Handling (PI-GMDH), in which representations of coupled physical fields are progressively constructed during solution. Candidate functional directions are evaluated through the first variation of the complete physical and observational objective, introduced in packages, and followed by block-coordinate damped Gauss-Newton coefficient optimization. The framework is demonstrated with tensor-product Chebyshev functions on the incompressible Navier-Stokes equations using a two-dimensional time-dependent Taylor-Green benchmark. Under the tested configuration, adaptive PI-GMDH reached validation and held-out test losses of 5.299e-19 and 5.296e-19 with 204, 201, and 175 active functions for u, v, and p. Complete degree-by-degree and all-terms PI-GMDH variants, together with selected PINN and KAN reference configurations, are used to examine the effect of structural construction policy. The results show that, for this controlled synthetic benchmark, selective progressive construction can provide a favorable combination of accuracy, representation size, and wall-clock time. The comparison is illustrative rather than a claim of universal superiority over alternative physics-informed approaches.

Mykhailo Minin

JSC KIEP, Kyiv, Ukraine

Neural PDE Solvers Physics-Informed ML Computational Fluid Dynamics

Sep 30, 2026stat.ML

Warm-starting PDE solvers with any-dimensional machine learning

Any-dimensional machine learning models, such as graph neural networks (GNNs), can be naturally trained and evaluated on inputs of different sizes and dimensions. Inspired by the GNN transferability literature, we show mathematical conditions under which a partial differential equation (PDE) learning-based solver can be trained in small dimensions and directly applied to solve a higher dimensional PDE in a zero-shot fashion. These conditions are based on symmetries in both the partial differential equation and the initial data. When the equations satisfy the symmetries but the data does not, which is the case for many PDEs arising from physics, we show that our theory gives a principled way of warm-starting low-dimensional PDE solvers for higher dimensional PDEs. We apply this method on the heat equation, Burgers' equation, and the compressible Navier--Stokes equations, improving the performance in both zero-shot and typical training regimes on high dimensional data. For example, we train a surrogate model on 2D Navier--Stokes data and achieve better results on 3D test data than a baseline surrogate model trained on 3D data, while only using 12

\%

of the flops and 20

\%

of the total data size.

Wilson G. Gregory, George A. Kevrekidis, Ben Blum-Smith +1

Department of Statistics and Irving Institute for Cancer Dynamics, Columbia University · Division of Applied Mathematics, Brown University · Department of Applied Mathematics and Statistics, Johns Hopkins University

Neural PDE Solvers PDE Surrogate Modeling Transfer Learning Computational Fluid Dynamics

Sep 29, 2026cs.AI

PE-EK-PINN: Physics Embedding with Evolving Kernel for Scalable Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) embed governing equations into deep learning, but enforce them only through loss residuals, leaving highly oscillatory wave behavior to be discovered by optimization. As a result, methods that achieve relative

L_2

errors below

10^{-3}

on standard manufactured Helmholtz benchmarks can fail on practical radiation problems involving singular excitations, absorbing boundaries, and wave fields spanning tens of wavelengths. Architectural physics embedding addresses this limitation by factorizing the field into analytically derived oscillatory kernels and learnable envelopes. However, the kernel dictionary must be manually constructed and scales with the number of elementary units, growing exponentially with the depth of hierarchically structured systems such as antenna arrays and metasurfaces. We propose PE-EK-PINN (Physics Embedded with Evolving Kernels), which treats physics kernels as reusable learned representations rather than fixed analytical inputs. A converged subsystem field is frozen and promoted to an evolved kernel, whose transformed copies are reused to represent higher-level configurations without deriving new governing equations. The resulting hierarchy makes the peak number of active kernels independent of system size and reduces cumulative training cost from

O(N)

to

O(\log N)

. Experiments on dipole arrays, composite line-source geometries, and cross arrays demonstrate the dramatic training cost reduction, while achieving a reduced or comparable relative

L_2

error. One notable example is PE-EK-PINN solves a

256

-dipole array more than 30 times faster than direct PE-PINN.

Huiwen Zhang, Feng Ye, Chu Ma

Department of Electrical & Computer Engineering Madison, WI, 53706, USA

Neural PDE Solvers Physics-Informed ML

Sep 29, 2026cs.AI

Transolver- $σ$ : Joint Spectral-Physical Subspace Modeling for Neural PDE Solving

Neural solvers offer efficient surrogates for numerical simulation of partial differential equations (PDEs). For time-dependent problems, strong one-step accuracy does not necessarily translate into reliable autoregressive rollout. We observe that a solver based only on physical-state modeling can achieve lower one-step error, whereas its spectral-only counterpart can become more accurate at later rollout steps. Motivated by this observation, we present Transolver-

σ

, a neural PDE solver based on joint spectral--physical subspace modeling. Within each block, adaptive physical-state interactions and spectral transformations are modeled in dedicated latent subspaces, whose responses are recomposed to enable information exchange between the two representations. Within the physical subspace, we introduce Slice-Residual Physics-Attention (SRPA), which preserves an explicit slice-space identity path while retaining learnable cross-slice interaction. In parallel, an axis-factorized Fourier operator captures global spectral structure. Across five well-established PDE benchmarks spanning steady-state prediction and time-dependent dynamics, Transolver-

σ

achieves state-of-the-art with a benchmark-averaged relative error reduction of 33.4% over the strongest baseline for each metric, while consistently improving autoregressive rollout over single-operator counterparts. Transolver-

σ

further delivers strong gains on coupled multiphysics systems and real-world fluid and combustion measurements from RealPDEBench, demonstrating its effectiveness beyond standard simulation benchmarks.

Haonan Shangguan, Hang Zhou, Haixu Wu +3

School of Software, BNRist, Tsinghua University · MIT CSAIL

Fourier Neural Operator Neural PDE Solvers PDE Surrogate Modeling PDE Operator Learning

Sep 28, 2026cs.LG

Neural Harmonic Measure Operator

We introduce Neural Harmonic Measure Operator (NHMO), a neural solver for elliptic PDE problems on variable-shape domains. The harmonic measure of a domain is the boundary probability distribution that, integrated against any boundary data, returns the Dirichlet Laplace solution. It depends only on the geometry, not on the boundary data. NHMO parameterizes the density of this measure as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples, so one trained kernel handles different boundary values on a shape with no retraining. We extend it to Poisson via a classical decomposition, with an auxiliary network amortizing the source-induced correction and avoiding the singular volume quadrature that breaks direct evaluation. At inference, new boundary values and new sources both yield PDE solutions by re-integration against the fitted kernel and lift, with no retraining. NHMO improves over four prior baselines on the MCB-B 3D variable-shape Poisson benchmark across all five categories, and is competitive with major neural-operator baselines on a controlled 2D testbed.

Jinjin He, Sinan Wang, Yuchen Sun +1

Georgia Institute of Technology

Neural PDE Solvers PDE Solving PDE Operator Learning

Sep 28, 2026math.NA

GAC-PINN: Geometry-Adaptive and Constraint-Enhanced Physics-Informed Neural Networks

For systems with steep gradients, sharp interfaces, or severe spatio-temporal coupling, Physics-informed neural networks (PINNs) suffer from spectral bias, geometric inflexibility, and boundary constraint conflicts, which undermine accuracy and convergence. To overcome these issues, we propose a geometry-adaptive and constraint-enhanced PINN (GAC-PINN). The framework comprises four components: a gradient-driven adaptive grid mapping (AGM) for diffeomorphic point concentration with Jacobian regularization, an adaptive bandwidth hard-constraint ansatz with spatially-varying boundary transition widths, a Gaussian Fourier feature mapping as a spectral preconditioner to further enhance high-wavenumber representation, and an operator-aware router that automatically selects the appropriate hard-constraint construction based on whether the governing PDE contains temporal derivatives. An AGM callback mechanism and a three-stage training strategy ensure stable coordination. Benchmarks including the viscous Burgers equation, a sharp-peaked 2D Poisson problem, and the Allen-Cahn phase-transition equation show that GAC-PINN attains relative (L^2) errors of ((1.747\pm 0.450)\times 10^{-4}), ((2.868\pm 0.947)\times 10^{-5}), and ((1.756 \pm 0.712)\times 10^{-3}), respectively, consistently outperforming the baselines. Ablation studies further reveal that AGM alone yields a substantially lower error than residual-based adaptive refinement (RAR), while RAR becomes beneficial only when combined with FFM, demonstrating a context-dependent module interaction. Convergence analysis verifies rapid error reduction and saturation with increasing resolution, establishing a practical adaptive framework for high-fidelity simulation of problems with localized sharp features in applied mechanics and computational physics.

Yanxin Zhang, Yong Zhang, Houbiao Li

University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

Neural PDE Solvers Physics-Informed ML

Sep 28, 2026physics.flu-dyn

Physics-Informed Neural Networks for Depth-Averaged Avalanche Dynamics

Accurate prediction of avalanche motion is essential for hazard assessment in mountainous terrain. This study develops and evaluates a physics-informed neural network (PINN) framework for the Savage-Hutter model of depth-averaged granular flow, progressing from 1D analytical verification to 2D experimental validation. First, three 1D problems of increasing complexity were verified against the analytical solution: height prediction with prescribed velocity, velocity prediction with prescribed height, and coupled prediction of both fields using the conservative formulation. The decoupled tests accurately reconstructed the spatio-temporal evolution of each field when the other was prescribed. The coupled formulation learned both fields without prescribed data, achieving mean height and velocity RMSEs of 0.043 and 0.079 in non-dimensional units. A hyperparameter sensitivity study evaluated the effects of network depth, width, collocation density, learning rate, and epochs. The framework was then extended to 2D and validated against laboratory experiments of a cylindrical granular pile collapsing on an inclined plane, with TITAN2D providing numerical comparisons. Purely physics-based training converged to the trivial zero solution; augmenting the loss with 10 sparse training points from final deposit profiles produced a physics-informed, data-assisted hybrid framework. Peak flow depth, depth-averaged velocity, RMSE, and wetted-area IoU evaluated global and local agreement. Global height RMSE ranged from 2.7 to 6.7 mm across four experimental cases, while mean wetted-area IoU ranged from 69 to 81 %, demonstrating consistent performance across variations in pile mass and slope angle.

Pradyumn Singh Sikarwar, Vishal Sharma, Gaurav Bhutani

School of Mechanical and Materials Engineering, Indian Institute of Technology Mandi, India

Neural PDE Solvers Physics-Informed ML

Sep 27, 2026cs.LG

Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning

Accurate numerical solutions of partial differential equations (PDEs) are crucial in numerous science and engineering applications. In this work, we introduce a novel neural PDE solver named AFDONet, which incorporates neural operator learning and adaptive Fourier decomposition (AFD) theory for the first time into a specifically designed variational autoencoder (VAE) structure, to solve a general class of nonlinear PDEs on smooth manifolds. AFDONet is the first neural PDE solver whose architectural and component design is fully guided by an established mathematical framework (in this case, AFD theory), turning neural operator design from an art to a science. Thus, AFDONet also exhibits exceptional mathematical explainability and groundness, and enjoys several desired properties. Furthermore, AFDONet achieves outstanding solution accuracy and competitive computational efficiency in several benchmark problems. In particular, thanks to its deep connections with AFD theory, AFDONet shows superior performance in solving PDEs on i) arbitrary (Riemannian) manifolds, and ii) datasets with sharp gradients. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Zeyuan Song, Zheyu Jiang

Oklahoma State University Stillwater, OK 74078

Neural PDE Solvers Neural Network Interpretability PDE Operator Learning

Sep 23, 2026math.NA

A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems

In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

Tianhao Hu, Bangti Jin, Fengru Wang +1

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong · Department of Mathematics & Scientific Computing Key Laboratory of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China.

Neural PDE Solvers PDE Solving

Sep 21, 2026math.NA

Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver

Neural operators are data-driven models that learn mappings from inputs that parameterize partial differential equations, such as spatially varying coefficients, initial conditions, forcing terms, boundary conditions, or geometries, to solution fields or quantities of interest. Once trained, they can serve as surrogates for classical numerical solvers in many-query settings that require repeated evaluations for varying inputs. We address the question of when, and then why, neural operator surrogates outperform classical numerical solvers, in terms of cost for a given accuracy. We focus on the post-training, many-query limit, in which data-acquisition and training costs are treated as fixed and fully amortized. Even in this deliberately favorable regime for neural operators, there are regimes in which classical solvers outperform the surrogate models. We compare the cost-accuracy performance of neural operator surrogates and classical numerical solvers through a reproducible benchmark study comparing neural operators with problem-matched classical solvers on representative problems in computational science and engineering, focusing on prediction error, per-query floating-point cost, and wall-clock runtime. Neural operators are most competitive at low-to-moderate accuracy requirements. Their floating-point cost advantage depends strongly on the problem structure, arising when they avoid temporal or nonlinear iterations or predict a reduced quantity of interest rather than a full solution field. Additional wall-clock speedups result from dense tensor operations that are well suited to modern hardware. As the target accuracy is tightened, achieving the required accuracy with neural operators becomes increasingly challenging, and classical solvers outperform surrogates in this regime; thus classical solvers will remain important for verification and high-accuracy computation.

Daniel Zhengyu Huang, Andrew M. Stuart

Beijing International Center for Mathematical Research, Center for Machine Learning Research, Peking University, Beijing, China · California Institute of Technology, Pasadena, CA

Neural Surrogate Modeling Neural PDE Solvers PDE Solving Neural Operators

Sep 17, 2026math.NA

Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers

Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.

Nilo Schwencke, Roland Maier

Gauss-Newton Optimization Neural PDE Solvers PDE Solving Second-Order Optimization Finite Element Method

Sep 17, 2026physics.flu-dyn

Well-posedness of neural turbulence closures and tangent dissipation

A neural turbulence closure defines a new boundary-value problem,

R(U)=N(U)+F(U)=0

, with a coupled Jacobian

J(U)=N'(U)+F'(U)

, where

N

is the original mean-flow operator and

F

the learned closure. We establish two consequences of global tangent dissipation. For a monotone original operator, a positive uniform margin supplied by the original operator and closure together guarantees existence, uniqueness and a global inverse-sensitivity bound relating a posteriori solution error to the a priori residual. For a general original operator, a dissipative closure cannot worsen tangent dissipation, but this alone does not guarantee uniqueness. Tangent dissipation depends on both diffusion and reaction. We study two complementary ways to promote it: (1) an exact-integral construction enforcing non-negative tangent diffusion while leaving reaction unconstrained, and (2) a penalty on tangent-reaction violations at sampled states. Tangent diffusion enters the Jacobian, and non-negative secant eddy viscosity alone does not control its coercivity. We conduct tests with channel flow at

Re_τ=180

--

5200

, which provides a strongly monotone baseline. Both constrained closures reach accurate solutions in all 50 training-seed/Reynolds-number cases. At

Re_τ=1000

, we conduct tests with 10,000 starts for one fixed network per closure and we find one root for each constrained closure and multiple roots for the other closures. Although this does not prove uniqueness, it provides strong empirical evidence for uniqueness of the tested constrained closures. At

Re_τ=5200

, the construction and penalty reduce the reported inverse sensitivity relative to the original operator by approximately

372\times

and

11\times

, respectively.

Zhen Zhang, George Em Karniadakis

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

Neural PDE Solvers Computational Fluid Dynamics

Sep 15, 2026cs.LG

Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method

We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.

George Chumbipuma, Irina Tezaur, Alejandro Diaz +1

Rice University · Sandia National Laboratories

Neural PDE Solvers PDE Solving

Sep 14, 2026physics.flu-dyn

Physics Informed Neural Network model for the dynamical study of Abdominal Aortic Aneurysm

We present the development and application of a three-dimensional Physics-Informed Neural Network (PINN) framework for the investigation of haemodynamic behaviour in the human aorta. The model incorporates a time-resolved simulation of pulsatile blood flow over a two-minute interval, enabling the extraction of pressure and velocity fields with high temporal fidelity. The mechanical stress exerted on the aortic wall was quantified through Laplace's law, with temporal averaging applied to derive representative stress distributions. This approach circumvents the computational overhead associated with conventional computational fluid dynamics (CFD) methods by eliminating mesh generation and exploiting the automatic differentiation capabilities inherent to neural networks. The proposed methodology demonstrates that PINNs can serve as an efficient and accurate alternative for modelling complex vascular flow phenomena, offering significant advantages in scalability and computational cost reduction while maintaining physical consistency.

Adrián Robles Arques, Martín Ruiz Fernandez, Javier Sanchis +2

Lucentia Research; Instituto Universitario de Investigación en Informática, Universidad de Alicante, Carr. San Vicent del Raspeig, Alicante, 03690, Comunidad Valenciana, España

Neural PDE Solvers Physics-Informed ML Computational Fluid Dynamics

Sep 14, 2026physics.comp-ph

Physics-enriched neural solvers for transient ice-flow simulation

Transient glacier simulations with higher-order ice flow require the repeated solution of a nonlinear problem as the geometry evolves. In the online mode of the Instructed Glacier Model, the velocity field is represented by a neural network whose weights are warm-started from the previous time step and updated with a few optimizer iterations. We show that supplying the network with inexpensive input fields derived from low-order ice-flow balances improves this online solver. Unlike residual-based physics-informed neural networks, which incorporate physics through governing-equation penalties in the loss, our approach leaves the governing energy objective unchanged, adding physical structure through the network inputs. Across three real-world glacier configurations, the enriched solver is markedly more robust to solver settings. On the two alpine cases, it also improves the tuned accuracy--runtime trade-off, reducing surface-velocity errors by factors of two to four at fixed runtime and reaching few-percent relative errors with only

10^4

--

10^5

trainable parameters, far fewer than comparable raw-input baselines. A 300-year Aletsch simulation then completes in under one minute, and the larger Valais domain in about two minutes, on a single GPU---a budget once reserved for much simpler shallow-ice models. Gains are smaller for the fast marine-terminating glacier, where nonlocal stress coupling favors larger or spectral networks. More broadly, the results suggest that enriching a neural solver's inputs with reduced-order physics can make repeated higher-order solves much cheaper, with no training data and no offline training.

Thomas Gregov, Sebastian Rosier, Brandon Finley +2

Institute of Earth Surface Dynamics, Universit´e de Lausanne, Lausanne, Switzerland · Department of Geography, Universität Zürich, Zürich, Switzerland

Neural Surrogate Modeling Neural PDE Solvers Physics-Informed ML Computational Fluid Dynamics

Sep 11, 2026cs.LG

Attention Is All You Need (to Avoid Spurious Oscillations)

Can attention move a shock across several cells in one update without breaking it? We develop a conservative, fixed grid finite-volume scheme in which a CFL-conditioned attention flux selects upstream information according to the transport required by the current time step. One-dimensional inviscid Burgers transport is used as the central mechanism test: the same learned flux remains reliable in the conventional small-step regime and, with a time step four times larger, preserves sharp shocks while using one stage per update. A standard fifth-order WENO scheme with third-order strong-stability-preserving Runge-Kutta time integration (WENO-5+SSP-RK3) is included alongside controlled Forward Euler comparisons to separate flux selection from time integration. The learned attention shifts upstream with the local transport reach and becomes more selective near shocks; inference-time interventions and retrained ablations show that transport-scale information and state-dependent selection contribute directly to performance. Directional two-dimensional scalar Burgers transport and the one-dimensional shallow-water system then test whether the conservation-scale-selection principle transfers beyond the original scalar setting. The results support attention as a learnable information stencil for conservative large-step shock transport, while identifying finite candidate reach and problem-dependent robustness as the present limits.

Jinyoung Jeong, Joseph B. Choi, Xinlun Cheng +3

School of Mechanical Engineering & IEDT, Kyungpook National University, Daegu, South Korea · School of Data Science, University of Virginia, Charlottesville, VA 22903, USA · Department of Mechanical Engineering, University of Iowa, Iowa City, Iowa 52242, USA +1

Neural PDE Solvers PDE Solving Computational Fluid Dynamics

Sep 10, 2026math.NA

A variational physics-informed graph neural network for heterogeneous solid mechanics

Stress localization in heterogeneous solids is governed by the bimaterial interface, where the displacement field remains

C^0

-continuous, while in-plane stresses jump due to the stiffness mismatch. Coordinate-based physics-informed neural networks (PINNs) represent this jump via a prescribed regularization width or a weighted interface penalty, making their accuracy sensitive to how phase-contrast changes are handled. This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the heterogeneity is carried by the discretization rather than by the trial field. The solver operates on a conforming adaptive mesh graph, assigns constitutive behavior per element, and minimizes the discrete total potential energy as a single unweighted objective in which only first derivatives appear. The discrete energy on piecewise-linear elements coincides with the finite element (FE) Ritz functional. Dirichlet conditions are enforced by construction, with no penalty term, no interface weight, and no prescribed transition width. Using one fixed architecture, optimizer, and loss across small-strain elasticity and finite-strain Neo-Hookean hyperelasticity in two and three dimensions, the von Mises error remains below

3.58\%

across a stiffness-contrast sweep spanning

(E_{\mathrm{inc}}/E_{\mathrm{mat}}\in[10^{-2},10^{2}])

, where a strong-form PINN degrades to

5.58\%

, and its displacement error reaches

7.66\%

against

0.49\%

for the PI-GNN. A trained network halves the (

σ_{xx}

) error of an energy-based PINN (

5.01\%

versus

10.94\%

). Training cost exceeds a single FE solve by more than an order of magnitude, so the construction is a variationally consistent, penalty-free interface representation for parametric surrogates and inverse identification rather than a replacement for a one-off FE analysis.

Aashay Rajan Yadav, Amiya Prakash Das, Ratna Kumar Annabattula

Mechanics of Materials Lab, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, Tamil Nadu, India

Neural PDE Solvers Structural Mechanics Physics-Informed ML Physics-Informed GNNs

Sep 7, 2026cs.LG

Solving the Elastic Wave Equation with Physics-Informed Neural Networks: A Robust and Critical Assessment

Physics-Informed Neural Networks (PINNs) have recently emerged as a promising approach for solving Partial Differential Equations (PDEs), offering a meshfree alternative that integrates physical principles into the learning process. This presents a new paradigm compared to traditional discretization methods and purely data-driven machine learning techniques. While promising, PINNs are not a panacea; they inherit challenges such as spectral bias and unstable convergence. Moreover, their potential in seismology remains largely unexplored. In this work, we provide a robust and critical assessment of PINNs for solving the elastic wave equation in seismology. We investigate the performance of PINNs on problems with varying degrees of complexity across various seismic sources and parameter models, from constant to highly heterogeneous settings. A pivotal aspect of our work involves investigating whether embedding physical principles directly into the network architecture enhances convergence and accuracy. We test an extensive range of neural architecture designs, from unrestricted, uninformed PINNs to highly specialized ones. We find that integrating an understanding of wave physics into the network design significantly improves accuracy. For instance, introducing a custom wavelet or plane wave layer, coupled with encoder and decoder layers, consistently yields a relative

L_2

error approximately half that of the standard PINN, as evidenced across numerous experiments. We further demonstrate that this novel architecture enhances accuracy when applied to the acoustic wave equation, underlying the versatility of our network. Another key contribution of our research is the successful conditioning of PINNs on seismic source locations. This signifies a considerable advancement towards rapid seismic hazard detection and seismic analysis.

Davide Staub, Ben Moseley

Department of Mathematics, ETH Zürich

Neural PDE Solvers Physics-Informed ML Seismology

Sep 7, 2026cs.LG

Latent-MoE: Domain-Aware Mixture-of-Experts for PDEs with Multi-Regime Physics

Physics-informed neural networks (PINNs) struggle on PDEs whose governing physics varies across the domain. We trace this to a structural property of standard coordinate networks: their neural tangent kernel (NTK) is translation-variant and lets training points of large coordinate magnitude disproportionately influence predictions elsewhere, producing long-range coupling and gradient conflict during training. We show analytically and empirically that mixture-of-experts (MoE) architectures with centered, compact-support routers yield a uniformly banded NTK whose kernel-regression weights decay exponentially with distance, localizing the learning. Building on this, we propose \emph{Latent-MoE}, which interleaves domain-aware MoE blocks within a shared backbone. Unlike FB-PINNs or X-PINNs, which rigidly partition both the domain and the parameters so that the parameters on different subdomains are updated independently, Latent-MoE is designed to preserve the localization benefit of domain-aware routing while allowing capacity to flow across regions through the shared backbone. On standard homogeneous-physics benchmarks Latent-MoE is competitive with established baselines; on benchmarks with multi-stage time-variable physics, where global models and rigid domain decompositions both fall into spurious solutions, it improves over them by more than an order of magnitude, with markedly reduced gradient conflict during training.

Hanwen Wang, Paris Perdikaris

Graduate Group of Applied Mathematics and Computational Science University of Pennsylvania Philadelphia, PA 19104 · Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, PA 19104

Neural PDE Solvers Physics-Informed ML Mixture-of-Experts Models Neural Tangent Kernel

Sep 7, 2026math.NA

A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers

Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.

Xing Guo, Hongwei Tang, Zewei Meng +3

School of Systems Science and Engineering, Sun Yat-sen University, Guangzhou, China

Automatic Differentiation Neural PDE Solvers PDE Solving Physics-Informed ML

Sep 3, 2026math.NA

Residual neural networks overcome the curse of dimensionality for semilinear heat equations

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist

η\in(0,\infty)

and ResNets

Ψ_{d,\varepsilon}

,

d\in\mathbb{N}

,

\varepsilon\in(0,1]

, with at most

ηd^η\varepsilon^{-η}

parameters whose realizations approximate the solution in dimension

d

with an

L^2

-error of at most

\varepsilon

. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every

ξ>0

, the explicit bound

C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}

on the number of parameters.

Ilkhom Mukhammadiev, Diyora Salimova

Department of Applied Mathematics, University of Freiburg, Germany

Neural PDE Solvers PDE Solving Residual Learning Neural Network Approximation Theory