Glacier evolution models based on temperature-index approaches are commonly used to assess hydrological impacts of glacier changes. However, current model calibration frameworks cannot efficiently transfer information from sparse high-resolution observations across glaciers. This limits their ability to resolve seasonal mass changes on unmonitored glaciers in large-scale applications. Machine learning approaches can potentially address this limitation by learning relationships from sparse data that are transferable in space and time, including to unmonitored glaciers. Here, we present the Mass Balance Machine (MBM), a data-driven mass balance model based on the XGBoost architecture, designed to provide accurate and high spatio-Temporal resolution regional-scale reconstructions of glacier mass balance. We trained and tested MBM using a dataset of approximately 4000 seasonal and annual point mass balance measurements from 32 glaciers across heterogeneous climate settings in mainland Norway, spanning from 1962 to 2021. To assess the advantage of MBM's generalisation capabilities, we compared its predictions on independent test glaciers at various spatio-Temporal scales with those of regional-scale simulations from three glacier evolution models. MBM successfully predicted annual and seasonal point mass balance on the test glaciers (RMSE of 0.59-1.00 m w.e. and bias of-0.01 to 0.04 m w.e.). On seasonal mass balance, MBM outperformed the other models across spatial scales, reducing RMSE by up to 46 % and 25 % on glacier-wide winter and summer mass balance, respectively. Our results demonstrate the capability of machine learning models to generalise across glaciers and climatic settings from relatively sparse mass balance data, highlighting their potential for a wide range of applications.

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).