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

TL;DR

An Operator Learning (OpL) method for solving boundary value inverse problems in partial differential equations (PDEs), focusing on recovering diffusion coefficients from boundary data, and a set of finite element method (FEM) modules fully integrated with PyTorch, called Learning-Automated FEM (LA-FEM).

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 $γ$-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.

Figures (14)

• Figure 1: The outline of the $\gamma$-deepDSM, where the PDE model is more like a "white box" containing only a few parameters with very clear mathematical and physical meaning, and the OpL model is a "black box" containing a large number of network parameters. All these parameters are data-driven.
• Figure 1: The architecture of $\gamma$-deepDSM: in the first module the learnables are the fractional orders, while in the second module the learnables are the network parameters.
• Figure 1: The architecture of FEALPy can be broadly categorized into four levels: Tensor level, Common level, Algorithm level, and Field level, ranging from low-level to high-level functionality. The modules in the small dotted box are stil in the process of development.
• Figure 1: The time spent on each item shown in the table \ref{['torch_exp']}. The time and Dof are in log-scale.
• Figure 2: Left: spectrum of channels in $\xi_l$ and $\mathfrak{L}^{\gamma}_{\partial\Omega} \xi_l$, where $g_{D,l}=\cos(l\theta({\boldsymbol x}))$, ${\boldsymbol x}\in\partial\Omega$, $l=1,...,6,8$ and $16$. Right: the fLB operator with $\gamma=0.75$ effectively increases the energy of high-frequency information.