In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.

中文翻译：

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具有边界条件的高维椭圆偏微分方程的深度神经网络逼近

在最近的工作中，已经确定深度神经网络 (DNN) 能够逼近一大类抛物线偏微分方程的解，而不会导致维度灾难。然而，所有这些工作都仅限于在整个欧几里得域上制定的问题。另一方面，工程和科学中的大多数问题都是在有限域上制定的，并受到边界条件的影响。本论文考虑了一个重要的模型问题，即域$D\subset \mathbb {R}^d$ 上的泊松方程，其服从Dirichlet 边界条件。结果表明，DNN 能够表示该问题的解决方案，而不会引起维度灾难。