Random Neural Network Expressivity for Non-Linear Partial Differential Equations

2026-05-24

AI Summary

This work addresses the lack of rigorous theoretical characterization of the capability of randomized neural networks (RaNNs) to solve nonlinear partial differential equations (PDEs). By integrating function approximation theory, Sobolev space analysis, and numerical experiments, the study establishes, for the first time, a dimension-independent error bound of order $1/2$ for RaNN approximations of solutions to time-dependent nonlinear PDEs—such as the porous medium equation and the compressible Navier–Stokes equations. This result provides a rigorous guarantee of the expressive efficiency of RaNNs for complex nonlinear PDE solutions and demonstrates through extensive numerical experiments that the derived convergence rate holds across a broad range of scenarios, thereby significantly expanding the theoretical applicability of RaNNs in computational PDEs.

Abstract

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 $\frac{1}{2}$ 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.

Problem

• Random Neural Networks
• Non-linear PDEs
• Expressivity
• Approximation Theory
• Sobolev Functions

Innovation

• Random Neural Networks
• Non-linear PDEs
• Approximation Theory
• Dimension-free Rate
• Sobolev Functions