Physics-informed Discretization-independent Deep Compositional Operator Network

TL;DR

This paper introduces a physics-informed neural operator architecture capable of handling irregular domain shapes and various discrete PDE parameter representations, improving efficiency and generalization in solving parametric PDEs.

Contribution

The proposed model is discretization-independent and integrates parameter embeddings with response embeddings through compositional layers, enhancing expressivity and generalization.

Findings

Demonstrates high accuracy in solving PDEs across different domain shapes.

Shows improved efficiency over traditional neural operators.

Validates effectiveness through numerical experiments.

Abstract

Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE parameter inputs}, have been successfully used. However, the training of neural operators typically demands large training datasets, the acquisition of which can be prohibitively expensive. To address this challenge, physics-informed training can offer a cost-effective strategy. However, current physics-informed neural operators face limitations, either in handling irregular domain shapes or in in generalizing to various discrete representations of PDE parameters. In this research, we introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes. Particularly, inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly, and this parameter embedding is integrated with the response embeddings through multiple compositional layers, for more expressivity. Numerical results demonstrate the accuracy and efficiency of the proposed method. All the codes and data related to this work are available on GitHub: https://github.com/WeihengZ/PI-DCON.