In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). These numerical simulations indicate that DNNs seem to have the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon \gt 0$ and the dimension $d \in \mathbb{N}$ of the function which the DNN aims to approximate in such computational problems. There is also a large number of rigorous mathematical approximation results for artificial neural networks in the scientific literature but there are only a few special situations where results in the literature can rigorously justify the success of DNNs to approximate high-dimensional functions. The key contribution of this article is to reveal that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients. We prove that the number of parameters used to describe the employed DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon \gt 0$ and the PDE dimension $d \in \mathbb{N}$. A crucial ingredient in our proof is the fact that the artificial neural network used to approximate the solution of the PDE is indeed a deep artificial neural network with a large number of hidden layers.

中文翻译：

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证明深度人工神经网络克服了具有常数扩散和非线性漂移系数的 Kolmogorov 偏微分方程的数值逼近中的维数诅咒

近年来，深度人工神经网络 (DNN) 已成功应用于多种计算问题的数值模拟，例如，对象和人脸识别、自然语言处理、欺诈检测、计算广告和偏微分方程的数值近似（偏微分方程）。这些数值模拟表明 DNN 似乎具有克服维数灾难的基本灵活性，因为用于描述 DNN 的实际参数的数量最多以多项式增长，并且在规定的近似精度的倒数 $\varepsilon\gt 0$ 和 DNN 旨在在此类计算问题中近似的函数的维度 $d \in \mathbb{N}$。科学文献中也有大量关于人工神经网络的严格数学逼近结果，但只有少数特殊情况，文献中的结果可以严格证明 DNN 能够成功逼近高维函数。本文的主要贡献是揭示 DNN 确实克服了具有恒定扩散和非线性漂移系数的 Kolmogorov PDE 数值近似中的维数诅咒。我们证明，用于描述所采用的 DNN 的参数数量在规定的近似精度 $\varepsilon \gt 0$ 和 PDE 维度 $d \in \mathbb{N}$ 的倒数中最多以多项式增长。