Multiscale Neural Networks for Approximating Green's Functions

TL;DR

This paper introduces multiscale neural networks to efficiently learn Green's functions for PDEs, reducing network size and training time through theoretical insights and experimental validation.

Contribution

It presents a novel multiscale neural network architecture for approximating Green's functions, improving efficiency over traditional methods.

Findings

Multiscale NNs require smaller network sizes.

Training times are significantly reduced.

Theoretical analysis confirms improved approximation capabilities.

Abstract

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Green's functions. However, Green's functions are notoriously difficult to learn due to their poor regularity, which typically requires larger NNs and longer training times. In this paper, we address these challenges by leveraging multiscale NNs to learn Green's functions. Through theoretical analysis using multiscale Barron space methods and experimental validation, we show that the multiscale approach significantly reduces the necessary NN size and accelerates training.