Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs

We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation...

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Bibliographic Details
Published inAdvances in computational mathematics Vol. 47; no. 1
Main Authors Laakmann, Fabian, Petersen, Philipp
Format Journal Article
LanguageEnglish
Published New York Springer US 01.02.2021
Springer Nature B.V
Subjects
Approximation
Artificial neural networks
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Initial conditions
Mathematical analysis
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Neural networks
Transport equations
Visualization
41A46
Deep neural networks
35Q49
41A25
Transport equations
68T05
35A35
Parametric PDEs
Approximation rates
Curse of dimension
65N30
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Summary:We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-020-09834-7