A Hybrid Iterative Neural Solver Based on Spectral Analysis for Parametric PDEs

TL;DR

This paper introduces a spectral analysis-based neural solver, FNS, that efficiently solves parametric PDEs by learning eigenvalues and eigenvectors, achieving convergence rates independent of grid size and PDE parameters.

Contribution

It designs a neural network from an eigen perspective, integrating spectral analysis with iterative methods to improve convergence for parametric PDEs.

Findings

FNS achieves convergence rates independent of grid size and PDE parameters.

The spectral neural approach effectively learns error components difficult for traditional methods.

Performance verified on five types of linear parametric PDEs.

Abstract

Deep learning-based hybrid iterative methods (DL-HIM) have emerged as a promising approach for designing fast neural solvers to tackle large-scale sparse linear systems. DL-HIM combine the smoothing effect of simple iterative methods with the spectral bias of neural networks, which allows them to effectively eliminate both high-frequency and low-frequency error components. However, their efficiency may decrease if simple iterative methods can not provide effective smoothing, making it difficult for the neural network to learn mid-frequency and high-frequency components. This paper first conducts a convergence analysis for general DL-HIM from a spectral viewpoint, concluding that under reasonable assumptions, DL-HIM exhibit a convergence rate independent of grid size $h$ and physical parameters $\boldsymbol{\mu}$. To meet these assumptions, we design a neural network from an eigen perspective, focusing on learning the eigenvalues and eigenvectors corresponding to error components that simple iterative methods struggle to eliminate. Specifically, the eigenvalues are learned by a meta subnet, while the eigenvectors are approximated using Fourier modes with a transition matrix provided by another meta subnet. The resulting DL-HIM, termed the Fourier Neural Solver (FNS), can be trained to achieve a convergence rate independent of PDE parameters and grid size within a local neighborhood of the training scale by designing a loss function that ensures the neural network complements the smoothing effect of the damped Jacobi iterative methods. We verify the performance of FNS on five types of linear parametric PDEs.