A Neural Network Approach for Homogenization of Multiscale Problems
- URL: http://arxiv.org/abs/2206.02032v1
- Date: Sat, 4 Jun 2022 17:50:00 GMT
- Title: A Neural Network Approach for Homogenization of Multiscale Problems
- Authors: Jihun Han and Yoonsang Lee
- Abstract summary: We propose a neural network-based approach to the homogenization of multiscale problems.
The proposed method incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution.
We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems.
- Score: 1.6244541005112747
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We propose a neural network-based approach to the homogenization of
multiscale problems. The proposed method uses a derivative-free formulation of
a training loss, which incorporates Brownian walkers to find the macroscopic
description of a multiscale PDE solution. Compared with other network-based
approaches for multiscale problems, the proposed method is free from the design
of hand-crafted neural network architecture and the cell problem to calculate
the homogenization coefficient. The exploration neighborhood of the Brownian
walkers affects the overall learning trajectory. We determine the bounds of
micro- and macro-time steps that capture the local heterogeneous and global
homogeneous solution behaviors, respectively, through a neural network. The
bounds imply that the computational cost of the proposed method is independent
of the microscale periodic structure for the standard periodic problems. We
validate the efficiency and robustness of the proposed method through a suite
of linear and nonlinear multiscale problems with periodic and random field
coefficients.
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