A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations

TL;DR

This paper introduces a kernel-weighted residual framework that combines kernel methods with neural networks to improve the robustness and performance of physics-informed machine learning models for solving nonlinear PDEs.

Contribution

The paper presents a novel kernel-weighted residual approach that enhances neural network-based PDE solvers by reducing sensitivity to architecture and training factors, with theoretical support.

Findings

Outperforms existing methods on benchmark problems

Reduces sensitivity to network initialization and architecture

Simplifies training while maintaining low inference costs

Abstract

Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose architecture and training process are designed such that the network satisfies the PDE system. While such PIML models have substantially advanced over the past few years, their performance is still very sensitive to the NN's architecture and loss function. Motivated by this limitation, we introduce kernel-weighted Corrective Residuals (CoRes) to integrate the strengths of kernel methods and deep NNs for solving nonlinear PDE systems. To achieve this integration, we design a modular and robust framework which consistently outperforms competing methods in solving a broad range of benchmark problems. This performance improvement has a theoretical justification and is particularly attractive since we simplify the training process while negligibly increasing the inference costs. Additionally, our studies on solving multiple PDEs indicate that kernel-weighted CoRes considerably decrease the sensitivity of NNs to factors such as random initialization, architecture type, and choice of optimizer. We believe our findings have the potential to spark a renewed interest in leveraging kernel methods for solving PDEs.