2024/03/21 by Denis Spiridonov, С. А. Степанов, Spiridonov, Denis +3 · 1 citation
Engineering · Materials Science · #65M12 #65M60 #68T07 #FOS: Mathematics #Magnetic Properties and Applications #Material Properties and Failure Mechanisms #Non-Destructive Testing Techniques #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2403.14177
openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online GMsFEM) together with deep learning. A novelty of this approach is that local online multiscale basis functions are computed rapidly and frequently by utilizing deep neural networks (DNNs). More precisely, we employ the training set of stochastic permeability realizations and the computed relating online multiscale basis functions to train neural networks. The nonlinear map between such permeability fields and online multiscale basis functions is developed by our proposed deep learning algorithm. That is, in a new way, the predicted online multiscale basis functions incorporate the nonlinearity treatment of the Richards equation and refect any time-dependent changes in the problem's properties. Multiple numerical experiments in two-dimensional model problems show the good performance of this technique, in terms of predictions of the online multiscale basis functions and thus finding solutions.