Neural Evolutionary Kernel Method: A Knowledge-Guided Framework for Solving Evolutionary PDEs

TL;DR

The paper introduces the Neural Evolutionary Kernel Method (NEKM), a neural network framework that embeds mathematical knowledge into DNN architectures to efficiently solve various time-dependent PDEs with high accuracy.

Contribution

It presents a novel kernel-based neural network approach that integrates boundary integral techniques with operator learning for solving PDEs more effectively.

Findings

Achieves high accuracy on heat, wave, and Schrödinger equations.

Demonstrates computational efficiency and ability to handle multiple PDEs.

Supports solving random PDEs with different coefficients.

Abstract

Numerical solution of partial differential equations (PDEs) plays a vital role in various fields of science and engineering. In recent years, deep neural networks (DNNs) have emerged as a powerful tool for solving PDEs, leveraging their approximation capabilities to handle complex domains and high-dimensional problems. Among these, operator learning has gained increasing attention by learning mappings between function spaces using DNNs. This paper proposes a novel approach, termed the Neural Evolutionary Kernel Method (NEKM), for solving a class of time-dependent partial differential equations (PDEs) via deep neural network (DNN)-based kernel representations. By integrating boundary integral techniques with operator learning, prior mathematical information of time-dependent partial differential equations (PDEs) is embedded into the design of neural network architectures for predicting their solutions, enhancing both computational efficiency and solution accuracy. Numerical experiments on the heat, wave, and Schr\"{o}dinger equations demonstrate that the Neural Evolutionary Kernel Method (NEKM) achieves high accuracy and favorable computational efficiency. Furthermore, the operator learning framework inherently supports the simultaneous prediction of solutions to multiple PDEs with different coefficients, rendering its capability for solving random PDEs.