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

2026-04-23

AI Summary

This work addresses the limitations of conventional physics-informed neural networks (PINNs) in solving high-frequency Helmholtz equations in heterogeneous media, which include high computational cost, a bias toward smooth solutions, and reliance on artificial absorbing boundary conditions. The authors propose a novel neural solver based on the Green’s integral representation, which, for the first time, incorporates nonlocal integral constraints into the neural network framework, replacing pointwise PDE residuals. This formulation inherently embeds wave physics, eliminates the need for second-order derivative computations and artificial boundary layers, and leverages FFT-accelerated convolutions, stochastic physical regularization, a hybrid Green’s integral–PDE loss, and nonuniform sampling. The method reduces computational cost by over an order of magnitude on a 20 Hz seismic model and maintains convergence with significantly improved reconstruction accuracy—even in strongly scattering regimes where the classical Born series diverges—outperforming existing PINN approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Efficient ML / Green AI
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Web data integration and cleaning

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.

Problem

• Helmholtz equation
• physics-informed neural networks
• highly oscillatory solutions
• heterogeneous media
• computational efficiency

Innovation

• Green-Integral
• Physics-Informed Neural Networks
• Helmholtz Equation
• FFT-based Convolution
• Hybrid Loss