Learning parameter-dependent shear viscosity from data, with application to sea and land ice

TL;DR

This work develops a data-driven framework to infer rheologies of non-Newtonian fluids by representing the effective shear viscosity with neural networks while enforcing isotropy and a convex dissipation potential. It formulates two learning paths, one regression-based on stress data and another PDE-constrained approach using velocity data, and demonstrates the methodology on land and sea ice rheologies with external parameter dependencies. The authors show robust recovery of Glen's law for land ice and the viscous-plastic sea ice model, including demonstrations using DEM-generated data, and reveal how external parameters drive shear-thickening and thinning behaviors. The approach provides physically consistent, generalizable rheologies that can reproduce realistic velocity fields and complex floe-scale dynamics, with potential for extension to multiple parameters and convex potential learning.

Abstract

Complex physical systems which exhibit fluid-like behavior are often modeled as non-Newtonian fluids. A crucial element of a non-Newtonian model is the rheology, which relates inner stresses with strain-rates. We propose a framework for inferring rheological models from data that represents the fluid's effective viscosity with a neural network. By writing the rheological law in terms of tensor invariants and tailoring the network's properties, the inferred model satisfies key physical and mathematical properties, such as isotropic frame-indifference and existence of a convex potential of dissipation. Within this framework, we propose two approaches to learning a fluid's rheology: 1) a standard regression that fits the rheological model to stress data and 2) a PDE-constrained optimization method that infers rheological models from velocity data. For the latter approach, we combine finite element and machine learning libraries. We demonstrate the accuracy and robustness of our method on land and sea ice rheologies which also depend on external parameters. For land ice, we infer the temperature-dependent Glen's law and, for sea ice, the concentration-dependent shear component of the viscous-plastic model. For these two models, we explore the effects of large data errors. Finally, we infer an unknown concentration-dependent model that reproduces Lagrangian ice floe simulation data. Our method discovers a rheology that generalizes well outside of the training dataset and exhibits both shear-thickening and thinning behaviors depending on the concentrations.

Figures (8)

• Figure 1: Problem setup for generating a training dataset to infer Glen's law for land ice. An infinitely long slab of ice of uniform thickness $L$ slides down a bedrock inclined at an angle $\alpha$. The training dataset consists of steady solutions to this problem for different angles $\alpha$ and temperatures $T$.
• Figure 2: Land ice problem: Glen's law inferred from noisy stress data (top panels) and velocity data (lower panels). For each model, we show the stress-strain relationship (left) and the velocity profiles which solve \ref{['eq:pstokes_one_dim']} in the training setup (right). In each case, ten models are inferred from ten different noise samples; the results in this figure represent the mean (solid lines) and twice the standard deviation (thickness of colored regions).
• Figure 3: Convergence behavior of loss functions for Glen's law in land ice problem. The top figure shows the penalized velocity loss (solid lines) and the gradient norm (dotted lines) against the number of function evaluations when optimizing with the LBFGS (blue) and ADAM (red) algorithms. The bottom figure shows the analogue results for the penalized stress loss. Both methods use gradients computed using full batch data.
• Figure 4: Land ice problem: Velocity fields computed over a longitudinal section of the Arolla glacier with (a) Glen's law \ref{['eq:glens_law']} and with (b) a rheological model inferred from velocity data with a noise level of $\sigma_v = 0.1\,u_{\max}$. The color map represents the Euclidean norm of the velocity vector $|\boldsymbol{u}|$ (non-dimensional). (c) Temperature variation, (d) difference between the two velocity fields.
• Figure 5: One-dimensional problem setup used for sea ice problems: This setup is used to generate data for both the viscous-plastic model (section \ref{['subsec:hibler']}) and the DEM model (section \ref{['subsec:DEM']}). In this configuration, ice floes on a periodic square patch of ocean of length $L = 100km$ are driven from left to right by a hat-shaped horizontal ocean current $u_o$ (blue arrows).