Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Bases

TL;DR

Fourier PINNs enhance traditional PINNs by integrating adaptive Fourier bases, enabling better learning of high-frequency solutions in PDEs without domain restrictions, demonstrated through systematic experiments.

Contribution

The paper introduces Fourier PINNs, a novel method combining neural networks with adaptive Fourier bases to improve high-frequency solution learning in PDEs.

Findings

Strong BC PINNs outperform standard PINNs in error reduction.

Fourier PINNs effectively learn high-frequency components.

Adaptive basis selection improves solution accuracy.

Abstract

Interest is rising in Physics-Informed Neural Networks (PINNs) as a mesh-free alternative to traditional numerical solvers for partial differential equations (PDEs). However, PINNs often struggle to learn high-frequency and multi-scale target solutions. To tackle this problem, we first study a strong Boundary Condition (BC) version of PINNs for Dirichlet BCs and observe a consistent decline in relative error compared to the standard PINNs. We then perform a theoretical analysis based on the Fourier transform and convolution theorem. We find that strong BC PINNs can better learn the amplitudes of high-frequency components of the target solutions. However, constructing the architecture for strong BC PINNs is difficult for many BCs and domain geometries. Enlightened by our theoretical analysis, we propose Fourier PINNs -- a simple, general, yet powerful method that augments PINNs with pre-specified, dense Fourier bases. Our proposed architecture likewise learns high-frequency components better but places no restrictions on the particular BCs or problem domains. We develop an adaptive learning and basis selection algorithm via alternating neural net basis optimization, Fourier and neural net basis coefficient estimation, and coefficient truncation. This scheme can flexibly identify the significant frequencies while weakening the nominal frequencies to better capture the target solution's power spectrum. We show the advantage of our approach through a set of systematic experiments.