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

• URL: http://arxiv.org/abs/2606.31342v1
• Date: Tue, 30 Jun 2026 08:41:54 GMT
• Title: Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains
• Abstract summary: Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error.<n>We propose a domain-decomposed randomized neural network framework for such problems.
• Score: 5.423324761643612
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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