Deep learning solver for solving advection–diffusion​ equation in comparison to finite difference methods

Abstract

In numerical modeling, the advection–diffusion equation describes the long-range transport of atmospheric pollutants. Most numerical models in the atmospheric science community are based on finite difference methods (FDM). In this study, we conduct a comprehensive comparative analysis of standard FDM-based numerical solvers with a deep learning-based solver, the objective of which is to solve the 2D unsteady advection–diffusion equation. The performance is compared on key performance aspects accuracy, stability, and interpolation. In the analysis, we find that despite being trained with a coarse resolution, the DNN solver is the most accurate among all the solvers. For the DNN solver, the mean absolute error and maximum absolute error of fluid concentration are lowered up to 2 orders of magnitude than the FDM-based method, which corresponds to 95% and 97% relative error reduction, respectively. The analysis also shows that the DNN solver is more stable in coarse spatial–temporal domains. Owing to its continuous nature, the DNN can interpolate a solution with consistent accuracy in a resampled spatial and temporal domain magnified up to 5 and 16 times, respectively. This study highlights the fundamental differences in the partial differential equation solving methods by comparing the DNN and FDM-based solvers and presents the DNN solver as a potential alternative to the FDM-based solvers in atmospheric numerical modeling.