Détail du document
Identifiant
oai:arXiv.org:2411.05341
Sujet
Mathematics - Numerical AnalysisAuteur
Zheng, Yangyang Wei, Huayi Cao, Shuhao Guo, RuchiCatégorie
Computer Science
Année
2024
Date de référencement
13/11/2024
Mots clés
la-fem boundary operatorRésumé
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
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