We present approximation results and numerical experiments for the use of randomized neural networks within physics-informed extreme learning machines to efficiently solve high-dimensional PDEs, demonstrating both high accuracy and low computational cost. Specifically, we prove that RaNNs can approximate certain classes of functions, including Sobolev functions, in the H2-norm at dimension-independent convergence rates, thereby alleviating the curse of dimensionality. Numerical experiments are provided for the high-dimensional heat equation, the Black-Scholes model, and the Heston model, demonstrating the accuracy and efficiency of randomized neural networks.
@article{arxiv.2501.12145,
title = {Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs},
author = {T. De Ryck and S. Mishra and Y. Shang and F. Wang},
journal= {arXiv preprint arXiv:2501.12145},
year = {2025}
}