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Inferring ice sheet damage models from limited observations using CRIKit: the Constitutive Relation Inference Toolkit

We examine the prospect of learning ice sheet damage models from observational data. Our approach, implemented in CRIKit (the Constitutive Relation Inference Toolkit), is to model the material time derivative of damage as a frame-invariant neural network, and to optimize the parameters of the model from simulations of the flow of an ice dome. Using the model of Albrecht and Levermann as the ground truth to generate synthetic observations, we measure the difference of optimized neural network models from that model to try to understand how well this process generates models that can then transfer to other ice sheet simulations. The use of so-called "deep-learning" models for constitutive equations, equations of state, sub-grid-scale processes, and other pointwise relations that appear in systems of PDEs has been successful in other disciplines, yet our inference setting has some confounding factors. The first is the type of observations that are available: we compare the quality of the inferred models when the loss of the numerical simulations includes observation misfits throughout the ice, which is unobtainable in real settings, to losses that include only combinations of surface and borehole observations. The second confounding factor is the evolution of damage in an ice sheet, which is advection dominated. The non-local effect of perturbations in a damage models results in loss functions that have both many local minima and many parameter configurations for which the system is unsolvable. Our experience suggests that basic neural networks have several deficiencies that affect the quality of the optimized models. We suggest several approaches to incorporating additional inductive biases into neural networks which may lead to better performance in future work.