A Neural Network Approach for Homogenization of Multiscale Problems
This work addresses computational challenges in multiscale modeling for fields like materials science or engineering, offering a more automated approach, though it appears incremental as it builds on existing neural network methods for homogenization.
The authors tackled the homogenization of multiscale PDEs by proposing a neural network method that uses Brownian walkers and a derivative-free loss, eliminating the need for hand-crafted architectures or cell problems, and demonstrated its efficiency and robustness on linear and nonlinear problems with periodic and random coefficients.
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.