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.
翻译:在本文中,我们提出一种无网状方法来解决完全的斯托克方程式,该方程式以非线性风学为冰川运动的模型。我们的方法受[12]中提议的深利方法的启发。我们首先将非纽顿冰流模型的解决方案制定成一个最小的、具有边界限制的变式整体的最小体。然后,这一解决方案被一个深神经网络所近似,其损失功能是混合边界条件的变异整体和软约束。我们的方法不是采用网状网格或基函数来评价损失功能,而是只要求域和边界的统一取样员。为了解决现实世界规模的不稳定,我们调整了网络在第一层的投入,平衡每个边界的常规因素。最后,我们通过几个数字实验来说明我们的方法表现,包括一个具有分析解决方案的2D模型、一个具有实际缩放的亚罗拉冰川模型和一个具有定期边界条件的3D模型。数字结果显示,我们提出的方法在解决由非线性冰川模型产生的非纽顿机械方面是有效的。