# Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains

arXiv: https://arxiv.org/abs/2606.31342 · HTML: https://mlsanity.com/paper/2606.31342/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 43 references that are in the map.
- First posted on arXiv 30 Jun 2026.

## Builds on (most cited first, 9 of 9)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [TransNet: Transferable Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2301.11701/) (2023)
- [Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework](https://mlsanity.com/paper/2604.23999/) (2026)
- [Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces](https://mlsanity.com/paper/2603.01689/) (2026)

## Abstract (arXiv)

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov--Galerkin or collocation formulations. We develop a Petrov--Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schr"odinger equations demonstrate the accuracy and flexibility of the proposed method.