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