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

• URL: http://arxiv.org/abs/2604.21411v1
• Date: Thu, 23 Apr 2026 08:24:10 GMT
• Title: A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization
• Abstract summary: We introduce a Green-Integral (GI) neural solver for the acoustic Helmholtz equation.<n>It departs from the PDE-residual-based formulation by enforcing wave physics through an integral representation.<n>We show that GI-based training consistently outperforms PDE-based PINNs, reducing computational cost by over a factor of ten.
• Score: 3.491999371287298
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• An adaptive framework for the axisymmetric pulsar magnetosphere using physics-informed Kolmogorov-Arnold networks [4.725935825821886]
  The pulsar magnetosphere has only recently been addressed using Physics-Informed Neural Networks (PINNs)<n>We introduce domain-specific neural architectures based on Kolmogorov-Arnold networks, an automated adaptive training pipeline and a physics-based convergence criterion.<n>The proposed methodology delivers self-consistent axisymmetric magnetosphere solutions with mean squared errors of the PDE residuals at O(1e-6) in double precision.
  arXiv  Detail & Related papers  (2026-06-09T10:44:10Z)
• Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations [0.0]
  HIN-LRI is a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error.<n>A lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close.<n>Experiments show that HIN-LRI improves accuracy over analytical, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.
  arXiv  Detail & Related papers  (2026-05-06T12:50:36Z)
• Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs [0.0]
  We introduce a framework that learns a probability density function representing the location of physics'' for adaptively sampling kernels of PIELMs.<n>By employing a weighted Expectation-Maximization (EM) algorithm, GMM-PIELM autonomously concentrates radial basis function centers in regions of high numerical error.
  arXiv  Detail & Related papers  (2026-03-06T13:46:38Z)
• PhyG-MoE: A Physics-Guided Mixture-of-Experts Framework for Energy-Efficient GNSS Interference Recognition [49.955269674859004]
  This paper introduces PhyG-MoE (Physics-Guided Mixture-of-Experts), a framework designed to align model capacity with signal complexity.<n>Unlike static architectures, the proposed system employs a spectrum-based gating mechanism that routes signals based on their spectral feature entanglement.<n>A high-capacity TransNeXt expert is activated on-demand to disentangle complex features in saturated scenarios, while lightweight experts handle fundamental signals to minimize latency.
  arXiv  Detail & Related papers  (2026-01-19T07:57:52Z)
• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids [73.28614344779076]
  We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
  arXiv  Detail & Related papers  (2026-01-15T18:59:57Z)
• Adaptive Mesh-Quantization for Neural PDE Solvers [51.26961483962011]
  Graph Neural Networks can handle the irregular meshes required for complex geometries and boundary conditions, but still apply uniform computational effort across all nodes.<n>We propose Adaptive Mesh Quantization: spatially adaptive quantization across mesh node, edge, and cluster features, dynamically adjusting the bit-width used by a quantized model.<n>We demonstrate our framework's effectiveness by integrating it with two state-of-the-art models, MP-PDE and GraphViT, to evaluate performance across multiple tasks.
  arXiv  Detail & Related papers  (2025-11-23T14:47:24Z)
• Neural PDE Solvers with Physics Constraints: A Comparative Study of PINNs, DRM, and WANs [1.131316248570352]
  Partial equations (PDEs) underpin models across science and engineering, yet analytical solutions are atypical and classical mesh-based solvers can be costly in high dimensions.<n>This dissertation presents a unified comparison of three mesh-free neural PDE solvers, physics-informed neural networks (PINNs), the deep Ritz method (DRM), and weak adversarial networks (WANs), on Poisson problems (up to 5D) and the time-independent Schr"odinger equation in 1D/2D.
  arXiv  Detail & Related papers  (2025-10-09T13:41:51Z)
• A Residual Guided strategy with Generative Adversarial Networks in training Physics-Informed Transformer Networks [8.614387766858496]
  We propose a novel Residual Guided Training strategy for Physics-In Transformer via Generative Adrative Network (GAN)<n>Our framework integrates a Transformer to inherently capture temporal correlations through autoregressive processing, coupled with a residual-aware GAN.<n>Experiments on the Allen-Cahn-Gordon, and Navier-Stokes equations demonstrate significant improvements, relative relative MSE reductions of up three orders of magnitude compared to baseline methods.
  arXiv  Detail & Related papers  (2025-07-15T03:45:42Z)
• PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations [4.7903561901859355]
  We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step.<n>Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase.<n>Across three representative PDE systems, PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time.
  arXiv  Detail & Related papers  (2025-07-03T01:22:57Z)
• Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction [49.66486092259376]
  The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. We provide a framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and gradient approximation.
  arXiv  Detail & Related papers  (2023-06-12T16:28:11Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions [68.8204255655161]
  We develop a physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. Localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. We propose criteria for weights in the loss function that produce a more accurate PINN solution than those obtained with the weights selected via other methods.
  arXiv  Detail & Related papers  (2022-08-18T05:00:24Z)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.