Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networks

TL;DR

This paper presents a spectral-inspired neural network approach for solving PDEs on manifolds, demonstrating improved performance over standard physics-informed neural networks through theoretical proofs and extensive experiments.

Contribution

It introduces spectral method-aligned neural network architectures for PDEs on manifolds, with theoretical validation and superior empirical results.

Findings

Spectral-inspired neural networks outperform standard physics-informed architectures.

The method is validated for the heat equation on intervals, spheres, and tori.

Networks generalize well to unseen initial conditions.

Abstract

We introduce an approach for solving PDEs over manifolds using physics informed neural networks whose architecture aligns with spectral methods. The networks are trained to take in as input samples of an initial condition, a time stamp and point(s) on the manifold and then output the solution's value at the given time and point(s). We provide proofs of our method for the heat equation on the interval and examples of unique network architectures that are adapted to nonlinear equations on the sphere and the torus. We also show that our spectral-inspired neural network architectures outperform the standard physics informed architectures. Our extensive experimental results include generalization studies where the testing dataset of initial conditions is randomly sampled from a significantly larger space than the training set.