Spectral-Prior Guided Multistage Physics-Informed Neural Networks for Highly Accurate PDE Solutions

Abstract

Physics-Informed Neural Networks (PINNs) are becoming a popular method for solving PDEs, due to their mesh-free nature and their ability to handle high-dimensional problems where traditional numerical solvers often struggle. Despite their promise, the practical application of PINNs is still constrained by several fac- tors, a primary one being their often-limited accuracy. This paper is dedicated to enhancing the accuracy of PINNs by introducing spectral-prior guided multistage strategy. We propose two methods: Spectrum- Informed Multistage Physics-Informed Neural Networks (SI-MSPINNs) and Multistage Physics-Informed Neural Networks with Spectrum Weighted Random Fourier Features (RFF-MSPINNs). The SI-MSPINNs integrate the core mechanism of Spectrum-Informed Multistage Neural Network (SI-MSNNs) and PINNs, in which we extract the Dominant Spectral Pattern (DSP) of residuals by the discrete Fourier transform. This DSP guides the network initialization to alleviate spectral bias, and gradually optimizes the resolution accuracy using a multistage strategy. The RFF-MSPINNs combines random Fourier features with spectral weighting methods, dynamically adjusting the frequency sampling distribution based on the residual power spectral density, allowing the network to prioritize learning high-energy physical modes. Through experimental verification of the Burgers equation and the Helmholtz equation, we show that both models significantly improve the accuracy of the original PINNs.