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arXiv · 2501.12145

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

T. De Ryck·S. Mishra·Y. Shang·F. Wang
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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.

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T. De Ryck, S. Mishra, Y. Shang, F. Wang. 2025-01-21. Approximation Theory and Applications of Randomized Neural Networks for Solving High-Dimensional PDEs. https://arxiv.org/abs/2501.12145

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