Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs

TL;DR

This paper introduces randomized neural networks as an efficient method for solving high-dimensional PDEs, achieving high accuracy with low computational cost and overcoming the curse of dimensionality.

Contribution

It provides theoretical approximation guarantees for RaNNs on Sobolev functions and demonstrates their practical effectiveness on various high-dimensional PDEs.

Findings

RaNNs approximate Sobolev functions in $H^2$-norm with dimension-independent rates

Numerical experiments show high accuracy on high-dimensional heat, Black-Scholes, and Heston models

RaNNs significantly reduce computational costs compared to traditional methods

Abstract

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 $H^2$-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.