Random Neural Network Expressivity for Non-Linear Partial Differential Equations
摘要
Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, showing that the obtained convergence rates extend beyond the considered setting.
引用
@article{arxiv.2605.25057,
title = {Random Neural Network Expressivity for Non-Linear Partial Differential Equations},
author = {Muhammed Ali Mehmood and Lukas Gonon},
journal= {arXiv preprint arXiv:2605.25057},
year = {2026}
}