Random Neural Network Expressivity for Non-Linear Partial Differential Equations

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.

Randomized neural networks (RaNNs) are attractive for partial differential equations (PDEs) because they replace expensive end-to-end training with a linear least-squares solve over randomized hidden features. Their practical performance, however, depends strongly on the sampling distribution of the hidden-layer parameters, which is usually chosen heuristically and problem by problem. This distribution sensitivity is a central bottleneck in randomized neural PDE solvers. In this work, we propose Adaptive-Distribution Randomized Neural Networks (AD-RaNN), a framework that promotes randomized feature generation from a fixed heuristic choice to a low-dimensional adaptive optimization problem. Instead of training all hidden weights and biases, AD-RaNN parameterizes the hidden-feature sampling distribution by a low-dimensional vector p and optimizes only p, thereby preserving the least-squares structure of RaNNs while reducing manual distribution tuning. The method uses a two-stage strategy: ridge-regularized reduced training for stable distribution-parameter optimization, followed by an unregularized least-squares refit for final solution recovery. We develop two adaptive mechanisms, PDE-Driven Adaptive Distribution (PDAD) and Data-Driven Adaptive Distribution (DDAD), and deploy them in space-time solvers, discrete-time solvers, and operator-learning models. We also incorporate an adaptive layer-growth enhancement for localized structures. For the reduced optimization problem, we establish well-posedness of the reduced objectives, consistency of ridge-regularized minimizers, an efficient gradient formula, and a practical lower-bound estimate for the ridge parameter. Numerical experiments on benchmark problems show that AD-RaNN provides an effective distribution-level adaptation mechanism, reduces reliance on hand-crafted hidden-feature distributions, and achieves strong empirical accuracy.

We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The decoupled linearized subproblems are solved iteratively by randomized neural networks in a space-time framework. For the concentration variables, a pointwise cut-off is used to enforce positivity at the value level, and discrete mass-scaling factors are computed at selected correction instants and interpolated in time, so as to ensure exact mass matching at those instants and to promote approximate mass preservation between them. To introduce an auxiliary discrete dissipation mechanism, we further employ an SAV-type post-processing correction, which yields monotonicity of the SAV auxiliary variable under the ideal SAV update. For the PNP-NS system, a structure-preserving randomized neural network (SP-RaNN) is used for the velocity field, so that the velocity approximation satisfies the incompressibility constraint pointwise by construction. On the theoretical side, we derive residual-based estimates for the raw, uncorrected RaNN solvers of the linearized subproblems, formulate a conditional local-in-time convergence result for the raw outer Picard iteration of the PNP system, and analyze the value-level positivity correction together with the mass-correction and SAV post-processing steps. For the PNP-NS system, we establish an approximation result for the SP-RaNN space and provide a conditional error statement for the corresponding linearized Oseen-type problem. Numerical experiments demonstrate approximation accuracy in the source-driven manufactured tests and illustrate the intended value-level positivity correction, selected-time mass matching, computed free-energy curves based on the final gauge-fixed potential, and divergence-free approximation in benchmark tests.