Machine-learning custom-made basis functions for partial differential equations

TL;DR

This paper introduces a novel method combining deep neural networks with spectral techniques to create custom basis functions for solving PDEs, enhancing approximation and solution expansion capabilities.

Contribution

It presents a new approach using DeepONet to generate basis functions that are orthonormal, hierarchical, and tailored for spectral methods in PDEs.

Findings

Custom basis functions improve PDE solution approximation.

The method applies to both linear and nonlinear PDEs.

Enhanced spectral method performance with deep learning-derived bases.

Abstract

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet), to identify candidate functions on which to expand the solution of PDEs. We have devised an approach which uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions which have the following properties: i) they constitute a basis, 2) they are orthonormal, and 3) they are hierarchical i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs.