$\gamma$-deepDSM for interface reconstruction: operator learning and a Learning-Automated FEM package

TL;DR

This paper introduces $b3$-deepDSM, an operator learning approach for inverse boundary value problems in PDEs, utilizing a novel FEM package integrated with PyTorch for efficient, GPU-accelerated computations.

Contribution

It presents a new operator learning method for PDE inverse problems and develops a fully integrated FEM module within PyTorch for efficient neural network training.

Findings

Effective boundary data feature extraction using fractional Laplace-Beltrami operator.

Seamless integration of PDE solvers with learnable parameters into neural networks.

Enhanced computational efficiency with GPU acceleration and auto-differentiation.

Abstract

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.