Learning computational upscaling models for a class of convection-diffusion equations

In this paper, we develop a nonlinear upscaling method for the nonlinear convection-diffusion equation based on a carefully designed deep learning framework. The proposed scheme solves the equation on a coarse grid with the cell average as the solution obtained from finite volume method. A local downscaling operator is constructed in order to compute the parameters in the coarse scale equation. This downscaling operation produces a fine scale local solution, and the combined local solutions are used to solve the coarse scale equation. Using the cell averages as a constraint, solving the equation on an oversampling region is able to recover the fine scale properties. Because of the nonlinearity of the convection term, the downscaling operations cannot be precomputed. As a result, the bottleneck in runtime of this scheme lies in solving numerous local problems to compute these downscaling operations on the fly. In order to improve the computational efficiency regarding the downscaling operation, we apply a deep learning approach. We will use a stacked neural network to approximate the downscaling operation with the cell average as input, and use a physics informed-like loss function. Extensive numerical simulations are performed to validate the performance of the proposed scheme, and our numerical results show that the proposed scheme can achieve a good accuracy and efficiency.