In this work, we propose an Operator Learning (OpL) method for solving boundary value inverse problems in partial differential equations (PDEs), focusing on recovering diffusion coefficients from boundary data. Inspired by the classical Direct Sampling Method (DSM), our operator learner, named $\gamma$-deepDSM, has two key components: (1) a data-feature generation process that applies a learnable fractional Laplace-Beltrami operator to the boundary data, and (2) a convolutional neural network that operates on these data features to produce reconstructions. To facilitate this workflow, leveraging FEALPy \cite{wei2024fealpy}, a cross-platform Computed-Aided-Engineering engine, our another contribution is to develop a set of finite element method (FEM) modules fully integrated with PyTorch, called Learning-Automated FEM (LA-FEM). The new LA-FEM modules in FEALPy conveniently allows efficient parallel GPU computing, batched computation of PDEs, and auto-differentiation, without the need for additional loops, data format conversions, or device-to-device transfers. With LA-FEM, the PDE solvers with learnable parameters can be directly integrated into neural network models.

翻译：本文提出了一种用于求解偏微分方程边值反问题的算子学习方法，其重点在于从边界数据中恢复扩散系数。受经典直接采样法的启发，我们的算子学习器（称为γ-deepDSM）包含两个关键组成部分：(1) 一个数据特征生成过程，将可学习的分数阶拉普拉斯-贝尔特拉米算子应用于边界数据；(2) 一个在这些数据特征上运行以生成重构结果的卷积神经网络。为支持此工作流程，我们借助跨平台计算机辅助工程引擎FEALPy，另一项贡献是开发了一套与PyTorch完全集成的有限元法模块，称为学习自动化有限元法。FEALPy中这些新的LA-FEM模块能够方便地实现高效的并行GPU计算、偏微分方程的批量计算以及自动微分，而无需额外的循环、数据格式转换或设备间数据传输。借助LA-FEM，具有可学习参数的偏微分方程求解器可以直接集成到神经网络模型中。