In recent years, deep learning has led to impressive results in many fields. In this paper, we introduce a multiscale artificial neural network for high-dimensional nonlinear maps based on the idea of hierarchical nested bases in the fast multipole method and the \(\mathcal {H}^2\)-matrices. This approach allows us to efficiently approximate discretized nonlinear maps arising from partial differential equations or integral equations. It also naturally extends our recent work based on the generalization of hierarchical matrices (Fan et al. arXiv:1807.01883), but with a reduced number of parameters. In particular, the number of parameters of the neural network grows linearly with the dimension of the parameter space of the discretized PDE. We demonstrate the properties of the architecture by approximating the solution maps of nonlinear Schrödinger equation, the radiative transfer equation and the Kohn–Sham map.

中文翻译：

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基于层次嵌套基的多尺度神经网络

近年来，深度学习在许多领域取得了令人瞩目的成果。在本文中，我们基于快速多极点方法中的分层嵌套库和\（\ mathcal {H} ^ 2 \）的思想，为高维非线性地图引入了多尺度人工神经网络。-矩阵。这种方法使我们能够有效地近似由偏微分方程或积分方程产生的离散非线性映射。它自然也扩展了我们基于层次矩阵泛化的最新工作（Fan等人arXiv：1807.01883），但参数数量减少了。特别地，神经网络的参数数量随着离散化PDE的参数空间的尺寸线性增长。我们通过近似非线性Schrödinger方程，辐射传递方程和Kohn-Sham映射的解图来证明该体系结构的特性。