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

TL;DR

This work assesses the universality of the extracted basis functions and demonstrates their potential toward model reduction with spectral methods, and proposes 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.

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.

Figures (29)

• Figure 1: DeepONet architecture and training (adapted from lu_learning_2021).
• Figure 2: Physics-informed DeepONet (adapted from wang_learning_2021).
• Figure 3: (Left) Fourier reference solution field, (middle) data-driven, (right) physics-informed DeepONet predictions for advection-diffusion. The absolute error is calculated as $\hat{s} - s$.
• Figure 4: (Left) Singular values and (right) expansion coefficients $e^{\sin(x)}$ for data-driven and physics-informed DeepONets for advection-diffusion.
• Figure 5: Custom basis functions, plotting with consistent boundaries $\Tilde{\phi}(x = 0 \text{ or } 2\pi)$ for advection-diffusion.