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

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

TL;DR

It is proved 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.

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.

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

TL;DR

It is proved that RaNNs can approximate certain classes of functions, including Sobolev functions, in the -norm at dimension-independent convergence rates, thereby alleviating the curse of dimensionality.

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 -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.
Paper Structure (14 sections, 5 theorems, 93 equations, 5 tables)

This paper contains 14 sections, 5 theorems, 93 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Randomized neural networks
  4. Extreme learning machines
  5. Physics-informed machine learning
  6. Physics-informed extreme learning machines
  7. Approximation results for randomized NNs in Sobolev norms
  8. Function approximation with generalized randomized neural networks
  9. Approximating Sobolev functions with randomized neural networks with uniformly distributed weights
  10. Numerical results
  11. Heat equation
  12. Black-Scholes equations
  13. Heston model
  14. Conclusion

Key Result

Proposition 3.4

Let $u \colon \mathbb{R}^d \to \mathbb{R}$, let $M \geq 1$ and assume there exists $G \colon \mathbb{R}^d \to \mathbb{C}$ such that for all $x \in [-M,M]^d$. For any $\varepsilon>0$ we define $u_\varepsilon:\mathbb{R}\to\mathbb{R}$ as where $\alpha$ is defined in terms of $\widetilde{g}(\xi) := 2\mathrm{Re}[G](\xi)-\mathrm{Im}[G](\xi)$ and $G$ as If Assumption ass:piA-radial is satisfied, then

Theorems & Definitions (14)

  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • Lemma 3.8
  • proof
  • Theorem 3.9
  • proof
  • ...and 4 more