Universal differential equations for glacier ice flow modelling

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.</p

Data­-driven and learning-­based approaches for the spatio­temporal interpolation of SLA fields from current and future satellite­-derived altimeter data

International audienceThe spatio­temporal interpolation of sea surface tracer fields from satellite­-derived altimeter data generally relies on model­-based schemes, the most popular ones being optimal interpolation schemes which exploit space­time covariance models. The ever increasing availability of in situ, remote sensing and simulation data make more and more appealing data­-driven alternatives, which may learn more complex representations of the underlying dynamics with a view to improving the reconstruction of fine­scale processes. We first review three categories of data-­driven schemes, namely patch­based super-­resolution models [Fablet et al., 2018], analog assimilation models [Lguensat et al., 2017] and neural­-network-­based assimilation models [Fablet et al., 2017]. We give more emphasis to the last two ones, which appear more generic. They are both stated within a Kalman­-based assimilation framework (namely the ensemble Kalman filter and smoother). They differ in the considered data­-driven dynamical model. The analog assimilation models exploit analog forecasting operators (especially locally-­linear analog operators) under the assumption that analog states share similar dynamics. By contrast, neural network (NN) architectures provide explicit representations of the dynamical operator. We focus on residual and convolutional architectures which may be interpreted as numerical integration schemes of differential equations (Fablet et al., 2017). For such data­driven and learning­-based schemes, the representativeness of the training data are critical issues. When dealing with high-­dimensional geophysical dynamics, the curse of dimensionality may make poorly relevant their straightforward application to the entire space­time domain of interest. We then discuss and introduce multiscale patch­-level representation as means to overcome these issues. We present numerical experiments for the spatio­temporal interpolation of SLA fields from satellite­-derived altimeter data using OSSE (Observing System Simulation Experiment) settings. We consider two types of satellite-­derived altimeter data, along-­track nadir data and upcoming wide­-swath SWOT mission. As case­ study region, we consider a region in the western Mediterranean sea with rich mesoscale and submesoscale dynamics. ROMS numerical simulations with a 0.02°x0.02° resolution over 5 years are used to implement the considered OSSE. The first 4 years are used for training. We apply the proposed interpolation schemes to the last one to evaluate their reconstruction performance. Overall, our results support a significant potential improvement for horizontal scales ranging from 20km to 100km with a gain of 42% (12%) in terms of SLA RMSE (correlation) with respect to the optimal Interpolation. Our results also suggest possible additional improvement from the joint assimilation of SWOT and along-­track nadir observations. We further discuss the pros and cons of data­-driven and learning-­based schemes for the reconstruction of SLA fields, especially future research directions to bridge model­-driven and data­-driven reconstruction schemes