Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations

2024/03/14 by Ariel Neufeld, Tuan Anh Nguyen, Neufeld, Ariel +1 · 2 citations
Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2403.09200

openalex publication_date 2024/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension d∈ ℕ and the reciprocal of the prescribed accuracy ε∈ (0,1). The main contribution of this paper is to rigorously prove for the first time that deep neural networks can also overcome the curse dimensionality in the approximation of a certain class of nonlinear PDEs with gradient-dependent nonlinearities.

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• Error analysis for the deep Kolmogorov method
• Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality

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