Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in \mathbb{N}$. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.

翻译：人工神经网络（ANN）已成功应用于一系列计算问题的数值模拟中，包括图像分类/识别、语音识别、时间序列分析、游戏智能、计算广告以及偏微分方程（PDE）的数值逼近。这些数值模拟表明，ANN具有高效逼近高维函数的能力，尤其表明ANN似乎在逼近上述计算问题中出现的多维函数时，具有克服维度灾厄的根本能力。科学文献中已有关于ANN的一系列严谨数学逼近结果。其中一些证明了收敛性但未给出收敛速率，另一些甚至严格建立了收敛速率，但仅有少数特殊情形中，数学结果能严谨解释ANN在逼近高维函数时的实证成功。本文的关键贡献在于揭示了ANN在布莱克-斯科尔斯PDE数值逼近情形下能够高效逼近高维函数。更精确地说，本工作发现，用于逼近布莱克-斯科尔斯PDE解的ANN所需参数数量，在预设逼近精度$\varepsilon > 0$的倒数以及PDE维度$d \in \mathbb{N}$上均最多呈多项式增长。由此，我们首次证明了ANN在布莱克-斯科尔斯PDE的数值逼近中确实克服了维度灾厄。