标题： $γ$- 深度DSM用于界面重构：算子学习和一个学习自动有限元方法软件包 显示英文标题

标题： $γ$-deepDSM for interface reconstruction: operator learning and a Learning-Automated FEM package

摘要： 在这项工作中，我们提出了一种用于求解偏微分方程（PDEs）中的边界值反问题的算子学习（OpL）方法，重点是从边界数据中恢复扩散系数。 受经典直接采样方法（DSM）的启发，我们的算子学习器，命名为$\gamma$-deepDSM，有两个关键组件：(1) 一个数据特征生成过程，该过程对边界数据应用可学习的分数拉普拉斯-贝尔特拉米算子，以及 (2) 一个在这些数据特征上操作的卷积神经网络，以产生重建结果。 为了促进这一工作流程，利用 FEALPy\cite{wei2024fealpy}，一个跨平台的计算机辅助工程引擎，我们的另一个贡献是开发一组与 PyTorch 完全集成的有限元方法（FEM）模块，称为学习自动化有限元方法（LA-FEM）。 FEALPy 中的新 LA-FEM 模块方便地允许高效的并行 GPU 计算、PDE 的批量计算和自动微分，而无需额外的循环、数据格式转换或设备间传输。 借助 LA-FEM，具有可学习参数的 PDE 求解器可以直接集成到神经网络模型中。 显示英文摘要

摘要： 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.