Physics Informed Random Feature Neural Networks for Solving PDEs

2026-09-14

AI Summary

本文提出了一种基于物理信息的随机特征方法，用于解决PINN求解某些PDE时遇到的频谱偏置问题，并通过减少配置点数量降低计算复杂度。

Abstract

Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method (such as physics-informed Gaussian Processes). We introduce a physics-informed random feature method for countering part of the spectral bias which PINN-based solvers are facing for a certain class of PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a specialized randomized neural network. Compared to other state-of-the-art PINN-based solvers which require a large number of collocation points, our proposed method reduces the computational complexity. In this paper, we develop a rigorous approximation error analysis and derive high-probability error bounds on the $H^1$ norm. We provide extensive numerical tests for verifying our theoretical guarantees on error decay rates, as well as several comparison tests to showcase our claimed capability for combating spectral bias in these deep learning based methods.

Problem

• Physics Informed Neural Networks
• PDEs
• Spectral Bias
• Random Feature Method
• Computational Complexity

Innovation

• Physics Informed Random Feature
• Spectral Bias
• Computational Complexity
• Error Analysis
• H^1 Norm