Random Neural Network Expressivity for Non-Linear Partial Differential Equations

• URL: http://arxiv.org/abs/2605.25057v1
• Date: Sun, 24 May 2026 13:08:34 GMT
• Title: Random Neural Network Expressivity for Non-Linear Partial Differential Equations
• Abstract summary: We study RaNN expressivity for learning solutions to non-linear partial differential equations.<n>We show that RaNNs are capable of efficiently approximating solutions to complex, non-linear PDEs.
• Score: 6.623217355591568
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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