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A pretraining domain decomposition method using artificial neural networks to solve elliptic PDE boundary value

Jeong-Kweon Seo¹

¹Institute of Data Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, South Korea. seojksc@korea.ac.kr.

Scientific Reports

|August 17, 2022

Summary

This study introduces a novel pretraining method combining domain decomposition (DDM) with artificial neural networks (ANNs) for solving partial differential equations (PDEs). The approach enhances approximation quality and speeds up solutions for machine learning tasks.

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Area of Science:

Numerical computation
Artificial intelligence
Machine learning

Background:

Domain decomposition methods (DDM) are established for solving partial differential equations (PDEs).
Applying artificial neural networks (ANNs) to solve PDEs is an emerging area with demonstrated feasibility.

Purpose of the Study:

To develop and implement a novel pretraining scheme for ANNs using DDM to enhance PDE solutions.
To improve the accuracy, smoothness, and approximation speed of NN-based PDE solvers.

Main Methods:

A pretraining scheme named 'smoothing with a basis reconstruction process' was devised for ANNs.
This pretraining was integrated with the established domain decomposition (DDM) concept.
Numerical experiments were conducted to validate the proposed DDM-based ANN method.

Main Results:

The pretraining process ensures a well-posed approximation basis, enhancing solution quality.
The DDM-based ANN approach demonstrated accelerated approximation and improved solution smoothness.
Effectiveness was verified through numerical experiments for PDE solutions.

Conclusions:

The proposed DDM method with ANN pretraining is effective for estimating PDE solutions.
This approach offers a valuable tool for general machine learning applications involving PDEs.