# Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks

arXiv: https://arxiv.org/abs/2409.13644 · HTML: https://mlsanity.com/paper/2409.13644/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 55 references that are in the map.
- First posted on arXiv 20 Sep 2024.

## Builds on (most cited first, 11 of 11)

- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)

## Built on by (most cited first, 3 of 3)

- [Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems](https://mlsanity.com/paper/2508.15695/) (2025)
- [A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machines](https://mlsanity.com/paper/2503.10032/) (2025)
- [Unsupervised simulation of incompressible flows with physics- and equality- constrained artificial neural networks](https://mlsanity.com/paper/2511.18820/) (2025)

## Abstract (arXiv)

We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition, designed for physics-informed machine learning of partial differential equations (PDEs) in both forward and inverse contexts. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. Unlike the original PECANN method, which relies solely on initial and boundary conditions to constrain PDEs, our method uses both boundary conditions and the governing PDE to constrain a unique interface loss function for each subdomain. This modification improves the learning of subdomain-specific interface parameters while reducing communication overhead by delaying information exchange between neighboring subdomains. To address the constrained optimization in each subdomain, we apply an augmented Lagrangian method with a conditionally adaptive update strategy, transforming the problem into an unconstrained dual optimization. A distinct advantage of our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations, even in cases with high-wavenumber and complex-valued solutions. Through numerical experiments with up to 64 subdomains, we demonstrate that our method consistently generalizes well as the number of subdomains increases.