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Two-level deep domain decomposition method

Victorita Dolean, Serge Gratton, Alexander Heinlein, Valentin Mercier

TL;DR

The paper addresses the scalability limitations of the Deep-DDM when solving PDEs with neural networks by introducing a two-level architecture that adds a coarse global network. This coarse network is coupled to local subdomain PINNs via extension operators and a partition of unity, enabling efficient global information exchange while maintaining local fidelity. On a Poisson problem with Dirichlet conditions, the two-level method achieves convergence that is largely independent of the number of subdomains for low-frequency modes and outperforms the one-level approach for higher-frequency modes given adequate training epochs. The approach offers a scalable, parallelizable ML-based solver for large-scale PDEs, with modest overhead from the coarse network and clear practical benefits for complex domain decompositions.

Abstract

This study presents a two-level Deep Domain Decomposition Method (Deep-DDM) augmented with a coarse-level network for solving boundary value problems using physics-informed neural networks (PINNs). The addition of the coarse level network improves scalability and convergence rates compared to the single level method. Tested on a Poisson equation with Dirichlet boundary conditions, the two-level deep DDM demonstrates superior performance, maintaining efficient convergence regardless of the number of subdomains. This advance provides a more scalable and effective approach to solving complex partial differential equations with machine learning.

Two-level deep domain decomposition method

TL;DR

The paper addresses the scalability limitations of the Deep-DDM when solving PDEs with neural networks by introducing a two-level architecture that adds a coarse global network. This coarse network is coupled to local subdomain PINNs via extension operators and a partition of unity, enabling efficient global information exchange while maintaining local fidelity. On a Poisson problem with Dirichlet conditions, the two-level method achieves convergence that is largely independent of the number of subdomains for low-frequency modes and outperforms the one-level approach for higher-frequency modes given adequate training epochs. The approach offers a scalable, parallelizable ML-based solver for large-scale PDEs, with modest overhead from the coarse network and clear practical benefits for complex domain decompositions.

Abstract

This study presents a two-level Deep Domain Decomposition Method (Deep-DDM) augmented with a coarse-level network for solving boundary value problems using physics-informed neural networks (PINNs). The addition of the coarse level network improves scalability and convergence rates compared to the single level method. Tested on a Poisson equation with Dirichlet boundary conditions, the two-level deep DDM demonstrates superior performance, maintaining efficient convergence regardless of the number of subdomains. This advance provides a more scalable and effective approach to solving complex partial differential equations with machine learning.
Paper Structure (5 sections, 11 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 5 sections, 11 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Deep Domain Decomposition Method
  3. Extension via a Coarse Network
  4. Numerical Results
  5. Conclusion

Figures (3)

  • Figure 1: (1) Strong scalability test on the Deep-DDM method (2) Sampling points for the two-level Deep-DDM method
  • Figure 2: (1) Strong scalability test for test problem with $\omega_1=1,\omega_2=3$ : Test with 2 500 epoch (2) an example of sampling with a $6\times 6$ decomposition
  • Figure 3: Strong scalability test for test problem with $\omega_1=1,\omega_2=6$ : (1) Test with 2 500 epoch (2) Test on $6\times 6$ decomposition with variation in the number of epochs