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In this paper, we propose a mesh-free method to solve full stokes equationwhich models the glacier movement with nonlinear rheology. Our approach isinspired by the Deep-Ritz method proposed in [12]. We first formulate thesolution of non-Newtonian ice flow model into the minimizer of a variationalintegral with boundary constraints. The solution is then approximated by a deepneural network whose loss function is the variational integral plus softconstraint from the mixed boundary conditions. Instead of introducing meshgrids or basis functions to evaluate the loss function, our method onlyrequires uniform samplers of the domain and boundaries. To address instabilityin real-world scaling, we re-normalize the input of the network at the firstlayer and balance the regularizing factors for each individual boundary.Finally, we illustrate the performance of our method by several numericalexperiments, including a 2D model with analytical solution, Arolla glaciermodel with real scaling and a 3D model with periodic boundary conditions.Numerical results show that our proposed method is efficient in solving thenon-Newtonian mechanics arising from glacier modeling with nonlinear rheology.

A variational neural network approach for glacier modelling with nonlinear rheology