A variational neural network approach for glacier modelling with nonlinear rheology

TL;DR

This paper introduces a mesh-free deep learning approach to solve complex glacier flow models governed by nonlinear rheology, demonstrating efficiency and accuracy through various numerical experiments.

Contribution

It presents a novel variational neural network method for glacier modeling that avoids mesh generation and handles nonlinear rheology effectively.

Findings

Accurately solves 2D and 3D glacier flow models.

Efficiently handles nonlinear rheology without mesh grids.

Demonstrates robustness with real-world glacier data.

Abstract

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.