Partial learning using partially explicit discretization for multicontinuum/multiscale problems with limited observation: Language interactions simulation

In this paper, we propose a new mathematical model of language interactions considering bilingualism. We assume diffusive and convective language spreads with language exchange terms. The resulting model consists of a coupled system of partial differential equations for three functions. We develop a finite element approximation of the mathematical model using implicit and partially explicit time schemes. In addition, we present a partial learning approach for this problem using a partially explicit discretization. In this method, we train a Deep Neural Network to predict the values of the difficult-to-compute (implicit) part of the solution at some observed points. We then perform linear temporal interpolation and spatial interpolation using the Proper Orthogonal Decomposition (POD) and the Discrete Empirical Interpolation Method (DEIM).