Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks

TL;DR

This paper introduces Fourier-based mixed physics informed neural networks (FMPINN) for solving multi-scale elliptic PDEs, effectively handling high-frequency oscillations and avoiding ill-conditioning issues present in classical PINNs.

Contribution

The study develops a novel FMPINN approach with dual variables and Fourier features, improving multi-scale PDE solutions over traditional PINNs.

Findings

FMPINN accurately solves multi-scale elliptic PDEs in various dimensions.

The method effectively handles high-frequency oscillations.

Numerical examples demonstrate robustness and efficiency of FMPINN.

Abstract

Deep neural networks have garnered widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this study, we probe into solving a class of elliptic partial differential equations(PDEs) with multiple scales by utilizing Fourier-based mixed physics informed neural networks(dubbed FMPINN), its solver is configured as a multi-scale deep neural network. In contrast to the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix caused by the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, then obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, to validate the efficiency and robustness of the proposed FMPINN algorithm, we present several numerical examples of multi-scale problems in various dimensional Euclidean spaces. These examples cover both low-frequency and high-frequency oscillation cases, demonstrating the effectiveness of our approach. All code and data accompanying this manuscript will be made publicly available at \href{https://github.com/Blue-Giant/FMPINN}{https://github.com/Blue-Giant/FMPINN}.