论文标题
具有边界条件的高维椭圆PDE的深神经网络近似
Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions
论文作者
论文摘要
在最近的工作中，已经确定了深层神经网络能够将解决方案近似于大类抛物线偏微分方程，而不会产生维度的诅咒。但是，所有这些工作都仅限于在整个欧几里得领域提出的问题。另一方面，工程和科学方面的大多数问题都是在有限域中提出的，并遭受边界条件。本文考虑了一个重要的这种模型问题，即域上的泊松方程$ d \ subset \ mathbb {r}^d $受dirichlet边界条件的约束。结果表明，深度神经网络能够代表该问题的解决方案，而不会产生维度的诅咒。这些证明是基于泊松方程解的概率表示以及合适的采样方法。
In recent work it has been established that deep neural networks 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 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 deep neural networks 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.
