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

2024-12-04

arXiv.org

AI Summary

Inverse problems for partial differential equations (PDEs) suffer from insufficient labeled data, leading to difficulties in operator learning, weak physical consistency, and poor generalization. Method: We propose the first unsupervised, physics-informed inverse operator learning framework. Our approach integrates PDE prior constraints into a deep operator network, introduces a physics-based regularization loss derived from the governing equations, and establishes theoretical stability guarantees for operator learning across heterogeneous grids and functional spaces. Contribution/Results: This work achieves, for the first time, label-free PDE inverse operator learning; provides rigorous theoretical guarantees on strong generalization and physical consistency; and demonstrates high accuracy and robustness on canonical inverse PDE problems—including inverse conductivity and inverse scattering—significantly reducing reliance on labeled data. The framework bridges data-driven learning with first-principles physics, enabling reliable surrogate modeling in low-data regimes.

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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.

Problem

• Learn PDE inverse problems without labeled data
• Capture underlying physical principles accurately
• Generalize effectively across domain and function space

Innovation

• Physics-informed deep networks
• No labeled training data
• Robust theoretical foundation