Domain decomposition based physics and equality constrained neural networks for the Helmholtz equation

Sungmin Won; Evan Olson; Xuemin Tu; Sungmin Won; Evan Olson; Xuemin Tu

doi:10.3934/era.2025265

Electronic Research Archive

2025, Volume 33, Issue 10: 5965-5989. doi: 10.3934/era.2025265

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Domain decomposition based physics and equality constrained neural networks for the Helmholtz equation

Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045, USA

Received: 30 June 2025 Revised: 28 September 2025 Accepted: 09 October 2025 Published: 15 October 2025

Overlapping and non-overlapping domain decomposition methods are proposed for the physics and equality constrained neural networks (PECANN) to solve the Helmholtz equation. A coarse component is introduced using a small neural network defined on the whole domain with carefully designed loss functions. This coarse component supplies both function values and normal derivatives on the subdomain interfaces to the local solvers, ensuring the scalability of the algorithms; the number of outer iterations remains nearly constant as the number of subdomains increases. Numerical experiments conducted with a wide range of wavenumbers over both standard and enlarged domains demonstrate the efficiency and scalability of the proposed algorithms.
- PINN,
- PECANN,
- domain decomposition,
- overlapping,
- non-overlapping,
- coarse component,
- Helmholtz
Citation: Sungmin Won, Evan Olson, Xuemin Tu. Domain decomposition based physics and equality constrained neural networks for the Helmholtz equation[J]. Electronic Research Archive, 2025, 33(10): 5965-5989. doi: 10.3934/era.2025265

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Abstract

Overlapping and non-overlapping domain decomposition methods are proposed for the physics and equality constrained neural networks (PECANN) to solve the Helmholtz equation. A coarse component is introduced using a small neural network defined on the whole domain with carefully designed loss functions. This coarse component supplies both function values and normal derivatives on the subdomain interfaces to the local solvers, ensuring the scalability of the algorithms; the number of outer iterations remains nearly constant as the number of subdomains increases. Numerical experiments conducted with a wide range of wavenumbers over both standard and enlarged domains demonstrate the efficiency and scalability of the proposed algorithms.

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