Modeling subgrid effects and temporal splitting in machine learning

In this talk, we will start with some main concepts in multiscale modeling and temporal splitting. Our goal is to model processes in multiscale media without scale separation and with high contrast. We assume that the coarse grid doesn’t resolve the scales and the contrast. To deal with these problems, I will introduce multiscale methods that use multicontinua approaches. These approaches use additional macroscopic variables. I will discuss the convergence of these approaches and show that these methods converge independent of the contrast. The multicontinua approaches can benefit from machine learning techniques, which I will discuss. I will also consider how multiscale methods can be used for temporal splitting. High contrast brings stiffness to the system, which requires small time steps. We will introduce partial explicit methods that construct time discretizations with the time stepping that is independent of the contrast. Numerical results will be shown to back up our theories. We will discuss how these approaches are used in machine learning.