  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.
