Physics-Informed Kernel Function Neural Networks for Solving Partial Differential Equations

TL;DR

This paper introduces a novel neural network approach using physics-informed kernel functions as activation functions to efficiently solve various PDEs with high accuracy, requiring only boundary or initial data for training.

Contribution

The paper presents a new RBF neural network model that incorporates physics-informed kernel functions into activation functions, differing from PINNs which embed physics in the loss function.

Findings

Effective in solving high-wavenumber wave problems

Capable of handling infinite domain and nonhomogeneous PDEs

Validated on multiple benchmark problems with high accuracy

Abstract

This paper proposed a novel radial basis function neural network (RBFNN) to solve various partial differential equations (PDEs). In the proposed RBF neural networks, the physics-informed kernel functions (PIKFs), which are derived according to the governing equations of the considered PDEs, are used to be the activation functions instead of the traditional RBFs. Similar to the well-known physics-informed neural networks (PINNs), the proposed physics-informed kernel function neural networks (PIKFNNs) also include the physical information of the considered PDEs in the neural network. The difference is that the PINNs put this physical information in the loss function, and the proposed PIKFNNs put the physical information of the considered governing equations in the activation functions. By using the derived physics-informed kernel functions satisfying the considered governing equations of homogeneous, nonhomogeneous, transient PDEs as the activation functions, only the boundary/initial data are required to train the neural network. Finally, the feasibility and accuracy of the proposed PIKFNNs are validated by several benchmark examples referred to high-wavenumber wave propagation problem, infinite domain problem, nonhomogeneous problem, long-time evolution problem, inverse problem, spatial structural derivative diffusion model, and so on.