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.

翻译：涉及偏微分方程（PDE）的逆问题可视为从测量数据到未知量的映射发现过程，通常被置于算子学习框架中。然而，现有方法通常依赖大量带标签的训练数据，这在大多数实际应用中并不现实。此外，这些监督模型可能无法准确捕捉潜在的物理原理。为克服这些局限性，我们提出一种称为物理信息深度逆算子网络（PI-DIONs）的新型架构，该架构能够在无需带标签训练数据的情况下学习基于PDE的逆问题的解算子。我们将逆问题文献中建立的稳定性估计推广至算子学习框架，从而为该方法提供了坚实的理论基础。这些估计保证了所提出的模型在有限样本和网格上训练后，能在整个定义域和函数空间中实现有效泛化。大量实验表明，PI-DIONs能够在无需标签数据的情况下，有效且准确地学习逆问题的解算子。