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Related Experiment Videos

Solver-in-the-loop joint operator learning: Fractional Laplace-Beltrami features for interface reconstruction.

Yangyang Zheng¹, Huayi Wei², Shuhao Cao³

¹School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, China.

Neural Networks : the Official Journal of the International Neural Network Society

|May 29, 2026

Summary

This summary is machine-generated.

This study introduces a novel joint operator learning method for conductivity image reconstruction. The approach uses a "solver-in-the-loop" training mechanism with a learnable fractional Laplace-Beltrami operator to enhance accuracy.

Keywords:

Direct sampling methods Electrical impedance tomography Finite element package Inverse problems Learnable fractional order Operator learning

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Area of Science:

Computational Mathematics
Image Reconstruction
Inverse Problems

Background:

Reconstructing conductivity coefficient images from boundary data is a challenging inverse problem.
Traditional methods often struggle with accuracy and computational efficiency.
Partial differential equation (PDE) solvers can be used as preconditioners for inverse problems.

Purpose of the Study:

To propose a joint operator learning method for accurate conductivity image reconstruction.
To investigate a "solver-in-the-loop" training mechanism for inverse problems.
To develop a flexible computational framework for this approach.

Main Methods:

Implemented a "solver-in-the-loop" training mechanism integrating PDE solvers and neural networks.
Utilized a fractional Laplace-Beltrami operator with a learnable fractional order to process boundary data.
Developed a Learning-Automated Finite Element Method (LA-FEM) package with PyTorch for auto-differentiation.

Main Results:

The proposed method significantly improved the accuracy of conductivity image reconstruction.
The learnable fractional order operator effectively transformed boundary data into high-dimensional features.
The LA-FEM package enabled seamless auto-differentiation through the PDE solver and neural networks.

Conclusions:

Joint operator learning with a "solver-in-the-loop" mechanism is effective for conductivity image reconstruction.
The fractional Laplace-Beltrami operator and LA-FEM package offer a powerful tool for inverse problems.
This approach enhances reconstruction accuracy and computational efficiency in solving inverse problems.