Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature, which examine the approximation capabilities of such deep learning-based approximation algorithms for PDEs. These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs. In these mathematical results from the scientific literature, usually the error between the solution of the PDE and the approximating ANN is measured in the $L^p$-sense, with respect to some $p \in [1,\infty )$ and some probability measure. In many applications it is, however, also important to control the error in a uniform $L^\infty $-sense. The key contribution of the main result of this article is to develop the techniques to obtain error estimates between solutions of PDEs and approximating ANNs in the uniform $L^\infty $-sense. In particular, we prove that the number of parameters of an ANN to uniformly approximate the classical solution of the heat equation in a region $ [a,b]^d $ for a fixed time point $ T \in (0,\infty ) $ grows at most polynomially in the dimension $ d \in {\mathbb {N}} $ and the reciprocal of the approximation precision $ \varepsilon> 0 $. This verifies that ANNs can overcome the curse of dimensionality in the numerical approximation of the heat equation when the error is measured in the uniform $L^\infty $-norm.

中文翻译：

---

热方程人工神经网络近似的均匀误差估计

最近，人工神经网络 (ANN) 与随机梯度下降优化方法相结合，已被用于近似计算可能相当高维的偏微分方程 (PDE) 的解。最近，科学文献中也出现了许多严格的数学结果，这些结果检验了这种基于深度学习的 PDE 逼近算法的逼近能力。这些来自科学文献的数学结果部分证明了基于 ANN 的算法能够克服高维 PDE 数值逼近中的维数灾难。在来自科学文献的这些数学结果中，通常 PDE 的解和近似 ANN 之间的误差是在 $L^p$-sense 中测量的，关于一些 $p \in [1,\infty )$ 和一些概率测度。然而，在许多应用中，控制统一 $L^\infty $-sense 中的误差也很重要。本文主要成果的主要贡献是开发了在均匀 $L^\infty $-sense 中获得 PDE 解与近似 ANN 之间的误差估计的技术。特别是，我们证明了一个人工神经网络的参数数量一致地近似热方程的经典解在一个区域 $ [a,b]^d $ 对于一个固定的时间点 $ T \in (0,\infty ) $ 在维度 $ d \in {\mathbb {N}} $ 和近似精度的倒数 $ \varepsilon> 上最多以多项式增长。0 美元。