# Random Neural Network Expressivity for Non-Linear Partial Differential Equations

arXiv: https://arxiv.org/abs/2605.25057 · HTML: https://mlsanity.com/paper/2605.25057/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 72 references that are in the map.
- First posted on arXiv 24 May 2026.

## Builds on (most cited first, 30 of 40)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Neural Network with Unbounded Activation Functions is Universal Approximator](https://mlsanity.com/paper/1505.03654/) (2015)
- [HYDRA: Competing convolutional kernels for fast and accurate time series classification](https://mlsanity.com/paper/2203.13652/) (2022)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Learning with SGD and Random Features](https://mlsanity.com/paper/1807.06343/) (2018)
- [Neural networks-based algorithms for stochastic control and PDEs in finance](https://mlsanity.com/paper/2101.08068/) (2021)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Approximation and Estimation for High-Dimensional Deep Learning Networks](https://mlsanity.com/paper/1809.03090/) (2018)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)

## Abstract (arXiv)

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 frac12 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.