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

detalle del documento

IDENTIFICACIÓN

oai:arXiv.org:2411.05341

Tema

Mathematics - Numerical Analysis

Autor

Zheng, Yangyang Wei, Huayi Cao, Shuhao Guo, Ruchi

Categoría

Computer Science

Año

2024

fecha de cotización

13/11/2024

Palabras clave

la-fem boundary operator

Métrico

Resumen

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.

Zheng, Yangyang,Wei, Huayi,Cao, Shuhao,Guo, Ruchi, 2024, $\gamma$-deepDSM for interface reconstruction: operator learning and a Learning-Automated FEM package