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.
| Subjects: | Numerical Analysis (math.NA); Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.25057 [math.NA] |
| (or arXiv:2605.25057v1 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25057 arXiv-issued DOI via DataCite (pending registration) |
From: Muhammed Mehmood [view email]
[v1]
Sun, 24 May 2026 13:08:34 UTC (137 KB)
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