Non-Newtonian viscous fluid models with learned rheology accurately reproduce Lagrangian sea ice simulations

This paper presents a machine learning framework that infers accurate, concentration-dependent non-Newtonian rheological models from discrete element method simulations, enabling efficient and precise large-scale Lagrangian sea ice modeling that captures complex behaviors like shear-thinning and shear-thickening across varying ice concentrations.

The Gist

Imagine the Arctic Ocean as a giant, frozen puzzle made of millions of individual ice chunks, or "floes." These chunks drift, crash into each other, rub against one another, and sometimes pile up into ridges. For decades, scientists have tried to predict how this giant puzzle moves using computer models.

The Old Way: A Rough Guess
Traditionally, scientists treated the sea ice like a thick, sticky fluid (like honey or paint). They used a 50-year-old recipe to guess how "thick" or "sticky" the ice would be under pressure. This recipe works okay when the ice is packed tight in the center of the ocean, but it falls apart when the ice is thinner or near the edges. It's like trying to predict how a crowd of people moves by assuming everyone is a single, solid block of clay; it ignores the fact that people bump, slide, and push each other individually.

The New Way: Learning from the "Particles"
The authors of this paper wanted a better recipe. They started with a super-detailed computer simulation called a "Discrete Element Method" (DEM). Think of this as a high-end video game where every single ice chunk is a separate character with its own physics. It calculates every collision and friction point. This is incredibly accurate, but it's so computationally heavy that it's impossible to run for the whole world's oceans.

So, the team asked: Can we teach a simpler model to act like this super-detailed game?

The Solution: A "Smart" Fluid
They built a new model that treats the ice as a fluid again, but instead of using a fixed, old recipe for how "thick" it is, they used Artificial Intelligence (AI) to learn the recipe on the fly.

Here is how they did it, using a simple analogy:

1. The Teacher: The super-detailed "video game" (DEM) acts as the teacher. It runs simulations and shows the resulting speed and direction of the ice.
2. The Student: The new, simpler fluid model acts as the student. It has a "brain" (a neural network) that guesses how thick the ice is at any given moment.
3. The Lesson: The student tries to mimic the teacher's results. If the student's prediction of the ice's speed is wrong, the AI brain adjusts its internal settings to get closer to the teacher's answer.
4. The Rulebook: Crucially, they didn't just let the AI guess anything. They forced the AI to follow the laws of physics (like energy conservation and symmetry) so the results make sense in the real world.

What They Discovered
By letting the AI learn from the detailed simulation, they found some surprising things about sea ice:

• It's Not Just Sticky; It's Smart: The ice doesn't behave the same way all the time.
  • When the ice is moderately packed, it acts like shear-thickening fluid (like cornstarch and water). If you push it faster, it gets harder and more resistant, almost like it's turning into solid rock.
  • When the ice is very tightly packed, it acts like shear-thinning fluid (like ketchup). If you push it faster, it actually flows more easily.
• Tiny Changes, Huge Effects: A tiny change in how much of the ocean is covered by ice (just 5% more or less) can change the ice's "thickness" (viscosity) by thousands of times. It's like a light switch that goes from "runny" to "solid" with the slightest tweak.
• It Works Everywhere: Even though they only taught the AI with simple, straight-line wind and water currents, the model could successfully predict how the ice would move in complex, swirling, or changing weather patterns. It even worked when they tested it on a 2D map, not just a straight line.

Why This Matters
The paper concludes that this method is a major step forward. Instead of guessing how ice behaves with old, imperfect formulas, we can now "learn" the rules directly from high-fidelity data. This allows scientists to create models that are both fast enough to run on a global scale and accurate enough to capture the complex, bumpy reality of how ice floes actually interact.

In short, they taught a simple fluid model to "think" like a complex crowd of ice chunks, resulting in a much more accurate way to predict how our frozen oceans will move.

Technical Summary

Technical Summary: Non-Newtonian Viscous Fluid Models with Learned Rheology for Sea Ice

Problem Statement
Accurate prediction of sea ice dynamics is critical for climate modeling and coastal safety, yet existing continuum models face significant limitations. The standard approach, the Hibler viscous-plastic model (based on AIDJEX work from 50 years ago), performs moderately well for packed ice in the central ice pack but loses predictive accuracy in the marginal ice zone. In these regions, lower ice concentrations lead to granular effects—such as collisions, frictional contact, and ridging—that the Hibler model does not capture. Conversely, Discrete Element Methods (DEMs), which resolve individual ice floes and their interactions, offer high fidelity but are computationally prohibitive for large-scale simulations. While data-driven approaches using scientific machine learning (sciML) have emerged to parameterize rheology, many rely on pre-defined functional forms or stress-strain data that may be noisy or unavailable from observations. Furthermore, existing work often targets fluids where standard nonlinear viscosity models are expected to suffice, whereas the rheological structure of complex DEM sea ice data is not clearly defined.

Methodology
The authors propose a framework to infer a continuum non-Newtonian viscous fluid model that reproduces velocity fields computed by a DEM (specifically the SubZero model). The core methodology involves the following steps:

1. Data Generation: The authors generate training data using the SubZero DEM, which simulates irregular polygonal ice floes interacting via collisions and friction. The simulations are driven by ocean drag and wind forces in a unidirectional parallel shear flow configuration. The DEM resolves the motion of thousands of floes, from which horizontal velocities and shear stresses are spatially averaged over grid cells to create Eulerian datasets.
2. Rheological Parameterization: Instead of assuming a specific functional form, the effective shear viscosity $\psi$ is represented as a feedforward neural network (NN). The rheology is constrained by physical principles:
   • Frame-Indifference: The constitutive law is formulated as a scalar function of the principal invariants of the strain-rate tensor (in 1D, $\tau = 2\psi(|\dot{\gamma}|, A)\dot{\gamma}$).
   • Positivity and Monotonicity: The NN architecture enforces non-negative viscosity (using ELU activation) and monotonicity of the stress-strain map to ensure the resulting PDE is well-posed and uniquely solvable.
3. Training Strategy (PDE-Constrained Optimization): Unlike standard approaches that minimize stress-strain misfit, this framework minimizes the misfit between the DEM's velocity data and the solution of the continuum momentum equation. The training objective is $J_v(\theta) = \sum |u_k - \langle u(\theta, A) \rangle_k|^2$, where $u(\theta, A)$ is the solution to the PDE $\rho_i H \partial_t u - \partial_y \tau = \tau_o + \tau_w$.
   • The optimization is performed using a combination of the finite element method (Firedrake) and automatic differentiation (PyTorch).
   • Gradients are computed via the adjoint method, solving the linearized adjoint problem to update the NN weights $\theta$.
   • A two-step optimization is employed: first minimizing a stress-strain misfit ($J_s$) to provide an initial guess, then refining with the velocity misfit ($J_v$).

Key Results
The study demonstrates that a learned, nonlinear rheology can accurately reproduce DEM velocity fields across a wide range of conditions:

• Shear-Thickening and Shear-Thinning: The inferred rheology reveals a transition in behavior dependent on ice concentration ($A$). For lower concentrations (e.g., $A=0.85$), the sea ice exhibits shear-thickening behavior (viscosity increases with strain rate). For higher concentrations (e.g., $A=0.95$), the behavior transitions to shear-thinning.
• Sensitivity to Concentration: The effective shear viscosity is found to increase by several orders of magnitude with small changes in ice concentration (as little as 5%).
• Generalization: The trained model, with fixed weights, successfully generalizes to:
  • Different Forcing: Unsteady problems driven by time-dependent wind and ocean profiles, including "unseen" concentrations (e.g., $A=0.875$) not present in the training set.
  • 2D Configurations: The model accurately reproduces velocity fields in a two-dimensional incompressible setup, preserving symmetries observed in the DEM, though it cannot capture compressible effects like local concentration redistribution.
• Comparison to Stress-Strain Fitting: The authors note that a model closely fitting the noisy stress-strain data from the DEM does not necessarily accurately reproduce the velocity fields. The velocity-based optimization ($J_v$) is shown to be superior for capturing the macroscopic dynamics.

Significance and Claims
The paper claims that this framework represents a "major step" toward developing non-Newtonian models that accurately reproduce observed sea ice dynamics. By inferring rheology directly from velocity data (which is more readily available from satellite imagery than stress data) and embedding the governing PDE into the training loop, the method overcomes the limitations of phenomenological parameterizations and the high cost of DEMs.

The authors emphasize that their approach does not assume a pre-existing rheological form, allowing the discovery of complex behaviors (such as the transition from shear-thickening to shear-thinning) that standard models like Hibler's (which predicts plastic behavior for all concentrations) or simple collisional models might miss. While the current model assumes incompressibility and ignores mechanical deformations like fracturing and ridging, the framework provides a physically sensible, computationally efficient continuum model that captures the essential dynamics of granular sea ice flows. The work suggests that data-driven rheology inference can bridge the gap between high-fidelity discrete simulations and scalable continuum models for Earth System Models.