Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems

TL;DR

This paper introduces PI-DIONs, a physics-informed neural network architecture that learns PDE inverse problem solutions without labeled data, leveraging stability estimates for robust generalization.

Contribution

The paper proposes a novel physics-informed deep inverse operator network that learns solution operators of PDE inverse problems without labeled data, grounded in stability theory.

Findings

Effectively learns inverse PDE solutions without labeled data

Provides theoretical stability guarantees for generalization

Demonstrates high accuracy in extensive experiments

Abstract

Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.