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arxiv: 2511.10452 · v2 · submitted 2025-11-13 · 🧮 math.NA · cs.NA· math.OC· physics.flu-dyn· physics.geo-ph

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

Gonzalo G. de Diego , Georg Stadler This is my paper

Pith reviewed 2026-05-17 22:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OCphysics.flu-dynphysics.geo-ph
keywords rheology inferenceneural networksice viscositynon-Newtonian fluidsparameter-dependent modelsPDE-constrained optimizationsea iceland ice
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The pith

A neural network can learn temperature- and concentration-dependent ice viscosity from velocity or stress data while preserving isotropy and a convex dissipation potential.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework that represents a fluid's effective viscosity as the output of a neural network whose inputs are the invariants of the strain-rate tensor together with external parameters such as temperature or ice concentration. By construction the network produces models that are frame-indifferent and that derive from a convex potential of dissipation. The method is demonstrated in two settings: recovering the known temperature dependence of Glen's law for land ice and recovering the concentration dependence of the shear term in a viscous-plastic sea-ice model. It is also used to infer an entirely new concentration-dependent rheology from Lagrangian floe simulations. If the approach succeeds, researchers can obtain rheologies that generalize outside the training set without having to guess their functional form in advance.

Core claim

The authors show that a neural network, when written in terms of tensor invariants and external parameters and equipped with architectural constraints that enforce isotropy and convexity of the dissipation potential, can accurately represent the shear viscosity of ice. The same network is trained either by direct regression against stress data or by PDE-constrained optimization against velocity observations. For land ice the network recovers the temperature-dependent Glen's law; for sea ice it recovers a concentration-dependent viscous-plastic shear term; and from floe data it produces a new model that exhibits both shear-thickening and shear-thinning and that continues to predict well at un

What carries the argument

Neural network that maps strain-rate tensor invariants and scalar parameters to effective viscosity, constructed so that the resulting rheology is isotropic and derives from a convex dissipation potential.

If this is right

  • Glen's law temperature dependence for land ice can be recovered from velocity data alone.
  • The shear component of the viscous-plastic sea-ice model can be learned as a function of concentration without assuming a specific algebraic form.
  • The discovered rheology generalizes to parameter values and flow conditions not present in the training data.
  • Both shear-thickening and shear-thinning regimes appear automatically as concentration changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same invariant-based network could be applied to other non-Newtonian materials whose rheology depends on additional scalars such as temperature or particle volume fraction.
  • Embedding the learned viscosity directly into large-scale ice-sheet or sea-ice codes would allow local adaptation of the flow law without manual retuning of parameters.
  • The method supplies a route to quantify how uncertainty in observations propagates into uncertainty in the inferred viscosity function.

Load-bearing premise

The neural network whose architecture is built around tensor invariants can accurately represent the true underlying rheology from noisy or limited velocity or stress data while preserving convexity of the dissipation potential.

What would settle it

If the velocities computed with the learned viscosity disagree systematically with independent measurements of ice motion at temperatures or concentrations lying outside the training range, the inferred rheology does not capture the governing physics.

Figures

Figures reproduced from arXiv: 2511.10452 by Georg Stadler, Gonzalo G. de Diego.

Figure 1
Figure 1. Figure 1: Problem setup for generating a training dataset to infer Glen’s law for land ice. An infinitely long [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Land ice problem: Glen’s law inferred from noisy stress data (top panels) and velocity data (lower [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence behavior of loss functions for Glen’s law in land ice problem. The top figure shows [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Land ice problem: Velocity fields computed over a longitudinal section of the Arolla glacier with (a) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: One-dimensional problem setup used for sea ice problems: This setup is used to generate data for [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Viscous-plastic sea ice problem: Effective shear viscosity inferred from noisy stress data (top panels) [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: DEM sea ice problem: Shown on the left is the shear strain-rate/stress relationship for different [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: DEM sea ice model: Velocity profiles computed with the rheological model inferred from minimizing [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a neural-network framework for inferring parameter-dependent shear viscosity in non-Newtonian fluid models, with applications to land and sea ice. By expressing the rheological law in terms of tensor invariants and tailoring the network architecture, the method enforces isotropic frame-indifference and the existence of a convex dissipation potential. Two inference strategies are presented: direct regression onto stress data and PDE-constrained optimization that fits velocity observations. The approach recovers the temperature-dependent Glen law for land ice and the concentration-dependent viscous-plastic shear term for sea ice, explores robustness to large data errors, and infers an unknown concentration-dependent rheology from Lagrangian floe simulations that is reported to generalize outside the training set and to exhibit both shear-thickening and shear-thinning regimes.

Significance. If the convexity and generalization properties are rigorously verified, the work would be significant for data-driven rheology modeling in glaciology and computational mechanics. It enables inference of complex, parameter-dependent constitutive laws from indirect velocity measurements while preserving key mathematical structure, and the hybrid finite-element/machine-learning implementation for PDE-constrained learning constitutes a practical technical advance.

major comments (2)
  1. [PDE-constrained optimization] PDE-constrained optimization section: the loss is defined on velocity mismatch rather than on the dissipation potential itself. Although the network architecture is designed to guarantee convexity for each fixed parameter value, no post-hoc diagnostic (e.g., eigenvalue analysis of the Hessian of the learned potential evaluated at held-out concentrations) is reported. This verification is load-bearing for the claim that the inferred thickening/thinning behavior is physically meaningful rather than an optimization artifact under noisy velocity data.
  2. [Numerical experiments] Results on the unknown concentration-dependent model (floe-simulation experiment): the generalization statement outside the training concentrations is central to the strongest claim, yet the manuscript provides only qualitative descriptions of the recovered behavior without quantitative error tables or direct comparison against established concentration-dependent baselines on the held-out data.
minor comments (2)
  1. [Abstract] The abstract states that the method 'explores the effects of large data errors' but does not indicate the precise noise levels, error models, or which figures/tables quantify those effects.
  2. [Methods] Notation for the neural-network input (tensor invariants) and output (viscosity) should be introduced with explicit dimension and symmetry properties in the methods section to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the presentation and support for our claims.

read point-by-point responses

  1. Referee: PDE-constrained optimization section: the loss is defined on velocity mismatch rather than on the dissipation potential itself. Although the network architecture is designed to guarantee convexity for each fixed parameter value, no post-hoc diagnostic (e.g., eigenvalue analysis of the Hessian of the learned potential evaluated at held-out concentrations) is reported. This verification is load-bearing for the claim that the inferred thickening/thinning behavior is physically meaningful rather than an optimization artifact under noisy velocity data.

    Authors: We agree that an explicit post-training verification of convexity would provide stronger evidence that the observed shear-thickening and shear-thinning regimes are physically meaningful. Although the architecture enforces convexity of the dissipation potential by construction for each fixed parameter (via the choice of convex activations and invariant-based input), we will add a post-hoc diagnostic in the revised manuscript. Specifically, we will report the minimum eigenvalues of the Hessian of the learned potential evaluated at several held-out concentration values to confirm positive semi-definiteness and rule out optimization artifacts. revision: yes

  2. Referee: Results on the unknown concentration-dependent model (floe-simulation experiment): the generalization statement outside the training concentrations is central to the strongest claim, yet the manuscript provides only qualitative descriptions of the recovered behavior without quantitative error tables or direct comparison against established concentration-dependent baselines on the held-out data.

    Authors: We acknowledge that quantitative support would better substantiate the generalization claim. Because the target rheology is unknown a priori, established baselines do not exist; however, we can still quantify performance via velocity and stress prediction errors on held-out concentrations. In the revision we will add tables reporting these errors for the inferred model on both training and held-out data, together with comparisons against simple parametric baselines (e.g., constant-viscosity and linear-in-concentration models) to demonstrate the advantage of the learned nonlinear dependence. revision: yes

Circularity Check

0 steps flagged

No circularity: data-driven inference with architecture-enforced properties

full rationale

The paper's core contribution is a neural-network representation of parameter-dependent viscosity, constructed explicitly in terms of tensor invariants so that frame-indifference and convexity of the dissipation potential hold by architectural design rather than by any derived claim. Both the direct regression and PDE-constrained routes fit the network weights to external stress or velocity observations; the reported generalization, shear-thickening/thinning behavior, and recovery of Glen's law are therefore empirical outcomes on held-out data or simulation snapshots, not algebraic identities or self-citations. No load-bearing step reduces a prediction to a fitted input, renames a known result, or imports uniqueness from prior author work; the derivation chain remains self-contained against the supplied data.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the true rheology can be expressed as a function of tensor invariants and approximated by a neural network that preserves convexity and frame-indifference; the learned weights are the primary free parameters.

free parameters (1)
  • neural network weights and biases
    These are fitted either to stress data or through PDE-constrained optimization to velocity data.
axioms (2)
  • domain assumption The rheological law can be written in terms of tensor invariants to guarantee isotropic frame-indifference.
    Invoked to ensure the learned model is physically admissible independent of coordinate choice.
  • domain assumption A convex potential of dissipation exists for the inferred viscosity.
    Required for thermodynamic consistency and well-posedness of the flow equations.

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