Derivative-enhanced Deep Operator Network

TL;DR

This paper introduces DE-DeepONet, a neural operator that incorporates derivative information and dimension reduction to improve accuracy and efficiency in learning mappings for parametric PDEs, especially with limited data.

Contribution

The paper proposes a derivative-enhanced DeepONet that leverages derivative information and linear dimension reduction to improve accuracy and reduce training costs for neural operators.

Findings

DE-DeepONet improves solution prediction accuracy.

Derivative loss reduces the number of training samples needed.

Extension to Fourier neural operator enhances other neural operators.

Abstract

The deep operator networks (DeepONet), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages derivative information to enhance the solution prediction accuracy and provides a more accurate approximation of solution-to-parameter derivatives, especially when training data are limited. DE-DeepONet explicitly incorporates linear dimension reduction of high dimensional parameter input into DeepONet to reduce training cost and adds derivative loss in the loss function to reduce the number of required parameter-solution pairs. We further demonstrate that the use of derivative loss can be extended to enhance other neural operators, such as the Fourier neural operator (FNO). Numerical experiments validate the effectiveness of our approach.