Geoscientific models are facing increasing challenges to exploit growing datasets coming from remote sensing. Universal differential equations (UDEs), aided by differentiable programming, provide a new scientific modelling paradigm enabling both complex functional inversions to potentially discover new physical laws and data assimilation from heterogeneous and sparse observations. We demonstrate an application of UDEs as a proof of concept to learn the creep component of ice flow, i.e. a nonlinear diffusivity differential equation, of a glacier evolution model. By combining a mechanistic model based on a two-dimensional shallow-ice approximation partial differential equation with an embedded neural network, i.e. a UDE, we can learn parts of an equation as nonlinear functions that then can be translated into mathematical expressions. We implemented this modelling framework as ODINN.jl, a package in the Julia programming language, providing high performance, source-to-source automatic differentiation (AD) and seamless integration with tools and global datasets from the Open Global Glacier Model in Python. We demonstrate this concept for 17 different glaciers around the world, for which we successfully recover a prescribed artificial law describing ice creep variability by solving ∼ 500 000 ordinary differential equations in parallel. Furthermore, we investigate which are the best tools in the scientific machine learning ecosystem in Julia to differentiate and optimize large nonlinear diffusivity UDEs. This study represents a proof of concept for a new modelling framework aiming at discovering empirical laws for large-scale glacier processes, such as the variability in ice creep and basal sliding for ice flow, and new hybrid surface mass balance models.

Inversion methods play an important role in glacier models, both to calibrate and estimate parameters of interest (e.g. Glen's coefficients). However, inversions are usually made for each glacier individually, without using any global information, i.e. without deriving general laws governing the spatiotemporal variability of those parameters. The reason behind this limitation is twofold: the statistical challenge of making constrained inferences with multiple glaciers, and the computational limitation of processing massive glacier datasets. Machine learning powered with differential programming is a tool that can address both limitations.

We introduce a statistical framework for functional inversion of physical processes governing global-scale glacier changes. We apply this framework to invert a prescribed function describing the spatial variability of Glen’s coefficient (A). Instead of estimating a single parameter per glacier, we learn the parameters of a regressor (i.e. a neural network) that encodes information related to each glacier (i.e. long-term air temperature) to the parameter of interest. The inversion is done by embedding a neural network inside the Shallow Ice Approximation PDE - resulting in a Universal Differential Equation - with the goal of minimizing the error on the simulated ice surface velocities. We previously had shown that this hybrid model training is possible thanks to the use of differential programming, enabling differentiation of a PDE, a numerical solver and a neural network simultaneously. In this work we upscale this approach to include larger datasets and with the goal of learning real empirical laws from observations.

This framework is built inside ODINN.jl, an open-source package in the Julia programming language for global glacier evolution modelling using Universal Differential Equations. ODINN exploits the latest generation of ice surface velocities and geodetic mass balance remote sensing products, as well as many preprocessing tools from the Open Global Glacier Model (OGGM).