Volume 56, pp. 187-208, 2022.

A deep learning based nonlinear upscaling method for transport equations

Tak Shing Au Yeung, Eric T. Chung, and Simon See

Abstract

We will develop a nonlinear upscaling method for nonlinear transport equations. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local downscaling operator is constructed. This downscaling operation recovers fine scale properties using cell averages. This is achieved by solving the equation on an oversampling region with the given cell average as constraint. Due to the nonlinearity, one needs to compute these downscaling operations on the fly and cannot pre-compute these quantities. In order to give an efficient downscaling operation, we apply a deep learning approach. We will use a deep neural network to approximate the downscaling operation. Our numerical results show that the proposed scheme can achieve good accuracy and efficiency.

Full Text (PDF) [5.1 MB], BibTeX

Key words

nonlinear upscaling, transport equations, deep learning

AMS subject classifications

65N30, 65N40

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