无需标注数据，用物理规律训练模型求解偏微分方程反问题。
Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems
- 基于物理规律设计网络，不依赖标签数据学习反问题解映射。
- 理论证明模型在有限样本和网格下可泛化到整个函数空间。
- 适合缺乏标注数据的科学计算场景，如地质反演、医学成像。
偏微分方程（PDE）相关的反问题可视为从观测数据到未知量的映射发现，常以算子学习框架建模。然而现有方法通常依赖大量标注数据，这在大多数真实场景中不现实，且监督模型可能无法准确捕捉底层物理规律。为此，我们提出一种新架构——物理信息深度反算子网络（PI-DIONs），可在无标签训练数据情况下学习PDE反问题的解算子。我们将反问题文献中的稳定性估计扩展至算子学习框架，为方法提供稳健的理论基础，确保模型在有限采样和网格条件下，仍能在整个定义域和函数空间中有效泛化。大量实验表明，PI-DIONs能无需标注数据即高效准确地学习反问题解算子。
原文摘要 · Abstract (English)
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.
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