{"ID":5438734,"CreatedAt":"2026-07-01T01:17:58.482524686Z","UpdatedAt":"2026-07-03T08:38:05.6384997Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.31342","arxiv_id":"2606.31342","title":"Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains","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.","short_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, irregu...","url_abs":"https://arxiv.org/abs/2606.31342","url_pdf":"https://arxiv.org/pdf/2606.31342v1","authors":"[\"Haixin Wang\",\"Haoning Dang\",\"Fei Wang\",\"Shimin Guo\"]","published":"2026-06-30T08:41:54Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.LG\"]","methods":"[]","has_code":false}