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

Abstract

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).To test the proposed approach and the model, we consider two model problems. Each of them simulates different language situations. For each problem, we compute the relative errors of the proposed approach and define the different parts of the error. The numerical results show that the proposed approach can provide good accuracy while reducing computational costs.