Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems
This addresses a practical limitation for researchers and engineers in fields like physics and engineering by enabling inverse problem solving without costly labeled data, though it builds on existing operator learning frameworks.
The paper tackles the problem of solving PDE inverse problems without labeled training data by proposing Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which learn solution operators effectively and accurately as demonstrated in experiments.
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.