A Variational Neural Network Approach for Glacier Modelling with Nonlinear Rheology (Communications in Computational Physics, 2023)
We propose a mesh-free method to solve the full Stokes equation for modeling the glacier dynamics with nonlinear rheology. Inspired by the Deep-Ritz methodproposed in [13], we first formulate the solution to the non-Newtonian Stokes equation as the minimizer of a variational problem with boundary constraints. Then, weapproximate its solution space by a deep neural network. The loss function for training the neural network is a relaxed version of the variational form, in which penaltyterms are used to present soft constraints due to mixed boundary conditions. Insteadof introducing mesh grids or basis functions to evaluate the loss function, our methodonly requires uniform sampling from the physical domain and boundaries. Furthermore, we introduce a re-normalization technique in the neural network to address thesignificant variation in the scaling of real-world problems. Finally, we illustrate theperformance of our method by several numerical experiments, including a 2D modelwith the analytical solution, the Arolla glacier model with realistic scaling and a 3Dmodel with periodic boundary conditions. Numerical results show that our proposedmethod is efficient in solving the non-Newtonian mechanics arising from glacier modeling with nonlinear rheology.
Keywords: Deep learning method; variational problems; mesh-free method; non-Newtonian mechanics; nonlinear rheology; glacier modelling.