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

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.