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In recent years deep artificial neural networks (DNNs) have been successfullyemployed in numerical simulations for a multitude of computational problemsincluding, for example, object and face recognition, natural languageprocessing, fraud detection, computational advertisement, and numericalapproximations of partial differential equations (PDEs). These numericalsimulations indicate that DNNs seem to possess the fundamental flexibility toovercome the curse of dimensionality in the sense that the number of realparameters used to describe the DNN grows at most polynomially in both thereciprocal of the prescribed approximation accuracy $ \varepsilon > 0 $ and thedimension $ d \in \mathbb{N}$ of the function which the DNN aims to approximatein such computational problems. There is also a large number of rigorousmathematical approximation results for artificial neural networks in thescientific literature but there are only a few special situations where resultsin the literature can rigorously justify the success of DNNs inhigh-dimensional function approximation. The key contribution of this paper isto reveal that DNNs do overcome the curse of dimensionality in the numericalapproximation of Kolmogorov PDEs with constant diffusion and nonlinear driftcoefficients. We prove that the number of parameters used to describe theemployed DNN grows at most polynomially in both the PDE dimension $ d \in\mathbb{N}$ and the reciprocal of the prescribed approximation accuracy $\varepsilon > 0 $. A crucial ingredient in our proof is the fact that theartificial neural network used to approximate the solution of the PDE is indeeda deep artificial neural network with a large number of hidden layers.

communications in mathematical sciences

A proof that deep artificial neural networks overcome the curse of  dimensionality in the numerical approximation of Kolmogorov partial  differential equations with constant diffusion and nonlinear drift  coefficients