A Multiscale Neural Network Based on Hierarchical Matrices

By Yuwei Fan, Lin Lin, Lexing Ying, Leonardo Zepeda-Núñez. 2018

83 citations recorded by the source corpus.

In this work we introduce a new multiscale artificial neural network based on the structure of $\mathcal{H}$-matrices. This network generalizes the latter to the nonlinear case by introducing a local deep neural network at each spatial scale. Numerical results indicate that the network is able to efficiently approximate discrete nonlinear maps obtained from discretized nonlinear partial differential equations, such as those arising from nonlinear Schr\"odinger equations and the Kohn-Sham density functional theory.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

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Citations in the corpus

Physics-Informed Neural Operator for Learning Partial Differential Equations (2021, background): “They are even able to tackle previously intractable problems such as higher-dimensional, multiscale, high-contrast, and chaotic PDE systems [33, 3, 8, 22, 12, 2].”
Fourier Neural Operator for Parametric Partial Differential Equations (2020, methodology): “Recently, some spectral methods for PDEs have been extended to neural networks (Fan et al., 2019a;b; Kashinath et al., 2020).”
Multipole Graph Neural Operator for Parametric Partial Differential Equations (2020, methodology): “Multipole and multi-resolution methods: The works [18, 17, 25] propose a similar multipole expansion for solving parametric PDEs on structured grids.”
Neural Operator: Learning Maps Between Function Spaces (2021, methodology): “Recently, some spectral methods for PDEs have been extended to neural networks (Fan et al., 2019a,c; Kashinath et al., 2020).”
Integrating Physics-Based Modeling with Machine Learning: A Survey (2020, unknown): “Solve PDEs [147] [65] [140] [9] [215] [42] [264] [106] [235] [107] [131] [209] [234] [279] [281] [60] [177] [208] [231] [301] [71] [89] [129] [268] [194] [176] [16] [48] [218] [44] [60] [172] [235] [19] [51] [131] [76] [278] [194] [176] [206] [166]”
Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems (2020, unknown): “5) [226] [251] [301] [303] [70] [196] [228] [248] [318] [82] [102] [144] [292] [216] [195] [23] [90] [320] [154] [58] [236] [53] [70] [192] [252] [26] [61] [149] [88] [299] [216] [195] [225] [145] [198] [171] [44] [188]”
Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks (2019, background): “In application, how the F-Principle can help devise effective DNN-based approaches to solve scientiﬁc computing problems is an important issue (E et al., 2017; Khoo et al., 2017; He et al., 2018; Fan et al., 2018).”
Neural Jump Stochastic Differential Equations (2019, background): “More broadly, learning parameters of differential equations from data has been successful for physics-based problems with deterministic dynamics [33, 34, 35, 36, 37].”