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Emulators of ice flow models have shown promise for speeding up simulations of glaciers and ice sheets. Existing ice flow emulators have relied primarily on convolutional neural networks (CNN’s), which assume that model inputs and outputs are discretized on a uniform computational grid. However, many existing finite element-based ice sheet models such as the Ice-Sheet and Sea-level System model (ISSM) benefit from their ability to use unstructured computational meshes. Unstructured meshes allow for greater flexibility and computational efficiency in many modeling scenarios. In this work, we present an emulator of a higher order, finite element ice flow model based on a graph neural network (GNN) architecture. In this architecture, an unstructured finite element mesh is represented as a graph, with inputs and outputs of the ice flow model represented as variables on graph nodes and edges. An advantage of this approach is that the ice flow emulator can interface directly with a standard finite element –based ice sheet model by mapping between the finite element mesh and a graph suitable for the GNN emulator. We test the ability of the GNN to predict velocity fields on complex mountain glacier geometries and show how the emulated velocity can be used to solve for mass continuity using a standard finite element approach. 

Abstract. Numerical simulations of ice sheets rely on the momentum balance to determine how ice velocities change as the geometry of the system evolves. Ice is generally assumed to follow a Stokes flow with a nonlinear viscosity. Several approximations have been proposed in order to lower the computational cost of a full-Stokes stress balance. A popular option is the Blatter–Pattyn or higher-order model (HO), which consists of a three-dimensional set of equations that solves the horizontal velocities only. However, it still remains computationally expensive for long transient simulations. Here we present a depth-integrated formulation of the HO model, which can be solved on a two-dimensional mesh in the horizontal plane. We employ a specific polynomial function to describe the vertical variation in the velocity, which allows us to integrate the vertical dimension using a semi-analytic integration. We assess the performance of this MOno-Layer Higher-Order (MOLHO) model to compute ice velocities and simulate grounding line dynamics on standard benchmarks (ISMIP-HOM and MISMIP3D). We compare MOLHO results to the ones obtained with the original three-dimensional HO model. We also compare the time performance of both models in time-dependent runs. Our results show that the ice velocities and grounding line positions obtained with MOLHO are in very good agreement with the ones from HO. In terms of computing time, MOLHO requires less than 10 % of the computational time of a typical HO model, for the same simulations. These results suggest that the MOno-Layer Higher-Order formulation provides improved computational time performance and a comparable accuracy compared to the HO formulation, which opens the door to higher-order paleo simulations.