What do physics-informed DeepONets learn? Understanding and improving training for scientific computing applications

TL;DR

This paper investigates what physics-informed DeepONets learn, analyzing their basis functions, and introduces transfer learning techniques to enhance training and model reduction in scientific computing applications.

Contribution

It provides insights into the universality of learned basis functions and proposes transfer learning methods to improve DeepONet training across related PDEs.

Findings

Decays of singular values and expansion coefficients measure DeepONet performance.

Transfer learning significantly reduces errors in training DeepONets.

Learned basis functions become more effective in representing PDE solutions.

Abstract

Physics-informed deep operator networks (DeepONets) have emerged as a promising approach toward numerically approximating the solution of partial differential equations (PDEs). In this work, we aim to develop further understanding of what is being learned by physics-informed DeepONets by assessing the universality of the extracted basis functions and demonstrating their potential toward model reduction with spectral methods. Results provide clarity about measuring the performance of a physics-informed DeepONet through the decays of singular values and expansion coefficients. In addition, we propose a transfer learning approach for improving training for physics-informed DeepONets between parameters of the same PDE as well as across different, but related, PDEs where these models struggle to train well. This approach results in significant error reduction and learned basis functions that are more effective in representing the solution of a PDE.