A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

Martin Hutzenthaler (), Arnulf Jentzen (), Thomas Kruse () and Tuan Anh Nguyen ()
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Martin Hutzenthaler: University of Duisburg-Essen
Arnulf Jentzen: ETH Zurich
Thomas Kruse: University of Gießen
Tuan Anh Nguyen: University of Duisburg-Essen

Partial Differential Equations and Applications, 2020, vol. 1, issue 2, 1-34

Abstract: Abstract Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs it has also been proved mathematically that deep neural networks overcome the curse of dimensionality in the numerical approximation of solutions of such linear PDEs. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations for semilinear PDEs.

Keywords: Curse of dimensionality; High-dimensional PDEs; Deep neural networks; Information based complexity; Tractability of multivariate problems; Multilevel Picard approximations; 65C99; 68T05 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (13)

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DOI: 10.1007/s42985-019-0006-9

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