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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.
Summary
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.
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.
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