Error Analysis for Empirical Risk Minimization over Clipped ReLU Networks in Solving Linear Kolmogorov Partial Differential Equations
Year: 2024
Author: Jichang Xiao, Xiaoqun Wang
Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 4 : pp. 996–1017
Abstract
Deep learning algorithms have been successfully applied to numerically solve linear Kolmogorov partial differential equations (PDEs). A recent research shows that if the initial functions are bounded, the empirical risk minimization (ERM) over clipped ReLU networks generalizes well for solving the linear Kolmogorov PDE. In this paper, we propose to use a truncation technique to extend the generalization results for polynomially growing initial functions. Specifically, we prove that under an assumption, the sample size required to achieve an generalization error within $\varepsilon$ with a confidence level $\varrho$ grows polynomially in the size of the clipped neural networks and $(\varepsilon^{−1}, \varrho^{−1}),$ which means that the curse of dimensionality is broken. Moreover, we verify that the required assumptions hold for Black-Scholes PDEs and heat equations which are two important cases of linear Kolmogorov PDEs. For the approximation error, under certain assumptions, we establish approximation results for clipped ReLU neural networks when approximating the solution of Kolmogorov PDEs. Consequently, we establish that the ERM over artificial neural networks indeed overcomes the curse of dimensionality for a larger class of linear Kolmogorov PDEs.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2024-0026
Numerical Mathematics: Theory, Methods and Applications, Vol. 17 (2024), Iss. 4 : pp. 996–1017
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Linear Kolmogorov PDE curse of dimensionality empirical risk minimization generalization error.