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

2026-06-30

AI Summary

This work addresses the challenges of solving partial differential equations on unbounded domains, where conventional domain truncation incurs large errors and global spectral methods struggle with local features and multiscale behavior. The authors propose a domain-decomposition-based stochastic neural network framework that models near- and far-field regions with separate subnetworks, coupled through interface conditions. By optimizing only the output-layer coefficients, the method yields a linear least-squares system. The approach innovatively integrates domain decomposition with stochastic neural networks and, for the first time, establishes a provably bounded parametric approximation theory under broken Sobolev norms, along with a complete error decomposition. Combined with Petrov–Galerkin formulations for semi-infinite elliptic problems and collocation strategies for fully unbounded, multiply connected, or time-dependent settings, the method demonstrates high accuracy, strong adaptivity, and effective handling of complex geometries in numerical experiments on Poisson and time-dependent Schrödinger equations.

Abstract

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 Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

Problem

• unbounded domains
• partial differential equations
• truncation error
• artificial boundary conditions
• near-field and far-field behavior

Innovation

• domain decomposition
• randomized neural networks
• unbounded domains
• Petrov–Galerkin method
• far-field decay