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

This paper proposes a physics-informed machine learning framework that uses neural networks to infer non-Newtonian rheological models from either stress or velocity data, demonstrating its ability to accurately recover temperature-dependent land ice laws and concentration-dependent sea ice behaviors.

The Gist

Imagine you are trying to figure out the "personality" of a mysterious substance, like a strange new type of slime or a thick syrup, but you aren't allowed to touch it or look at its chemical formula. All you can do is watch how it moves when you push it, or measure how much it resists being squeezed.

This paper describes a high-tech way to play "detective" with fluids—specifically the massive, slow-moving ice sheets on land and the floating ice packs in the ocean.

Here is the breakdown of how they do it, using some everyday analogies.

1. The Problem: The "Secret Recipe" of Ice

Every fluid has a "recipe" called rheology. This recipe tells the fluid how much it should resist flowing when a force is applied.

• Water is simple: if you push it twice as hard, it moves twice as fast.
• Non-Newtonian fluids (like the ice in this paper) are "moody." If you push them harder, they might suddenly turn into a solid (like cornstarch and water) or become much thinner and runnier.

For scientists studying climate change, knowing this "secret recipe" for ice is vital. If we know exactly how "runny" a glacier is, we can predict how fast it will melt into the sea and how much the oceans will rise. But the recipe for ice is incredibly complex because it changes based on temperature (land ice) or how crowded the ice is (sea ice).

2. The Solution: The "Digital Mimic" (Neural Networks)

Instead of guessing a mathematical formula (which is like trying to guess a song by only hearing one note), the researchers use a Neural Network.

Think of the Neural Network as a Digital Mimic. You don't tell the Mimic what the recipe is; instead, you show it videos of the ice moving and tell it, "Hey, try to adjust your internal settings until your movements look exactly like these videos."

The researchers designed this Mimic to be a "polite" student of physics. They didn't just let it guess wildly; they gave it strict rules:

• The "No Magic" Rule (Isotropy): The fluid shouldn't behave differently just because you rotated your perspective.
• The "No Energy from Nothing" Rule (Dissipation): The fluid can't create energy out of thin air; it must obey the laws of thermodynamics.

3. Two Ways to Play Detective

The researchers found two ways to train their Digital Mimic:

• Method A: The "Stress" Test (The Pressure Gauge): This is like putting the fluid in a machine that measures exactly how much pressure it pushes back with. It’s very direct, but in the real world, measuring "internal stress" is incredibly hard.
• Method B: The "Velocity" Test (The Speedometer): This is like watching the fluid flow and measuring its speed. This is much easier to do in real life (we can see glaciers moving!), but it’s a harder math problem because you have to work backward from the speed to find the recipe.

4. The Results: Proving the Detective is Good

To see if their detective work actually worked, they did three tests:

1. The Land Ice Test: They gave the Mimic data about how glaciers move. The Mimic successfully "re-discovered" Glen’s Law—a famous math formula scientists have used for decades. It’s like giving a student a math problem and having them "accidentally" reinvent calculus.
2. The Sea Ice Test: They tested it on the "moody" sea ice. Even when the data was "noisy" (messy and imperfect, like a blurry photo), the Mimic was able to figure out the recipe for how ice concentration changes its flow.
3. The "Unknown" Test (The Ultimate Challenge): They used a super-complex simulation of individual ice chunks bumping into each other (like a crowded mosh pit). They didn't give the Mimic any formula. The Mimic looked at the chaotic movement and successfully figured out a brand-new recipe that described how the ice transitions from thick to thin.

Why does this matter?

In the past, if we didn't know the "recipe" for a certain type of ice, we just had to guess. This paper provides a way to learn the recipe directly from observation.

By using AI that respects the laws of physics, we can take messy, real-world satellite data of moving ice and turn it into precise mathematical models. This helps us build better "crystal balls" to predict the future of our planet's ice and sea levels.

Technical Summary

Technical Summary: Learning Parameter-Dependent Shear Viscosity from Data

Authors: Gonzalo G. de Diego and Georg Stadler (Courant Institute, NYU)

1. Problem Statement

The modeling of complex, non-Newtonian fluids (such as glaciers, sea ice, or polymer solutions) requires a rheological model—a constitutive relation that links internal stresses to strain rates. Traditionally, these models are either phenomenological (simple analytical fits) or derived from first principles (kinetic theory). However, the inherent complexity of many materials makes finding accurate, physically consistent models difficult.

The authors address the challenge of data-driven rheology inference: how to discover the functional form of a fluid's viscosity from observational data (stress or velocity) while ensuring the resulting model obeys fundamental physical laws (e.g., thermodynamics and isotropy).

2. Methodology

The authors propose a framework that replaces traditional analytical functions with Neural Networks (NNs) to represent the effective shear viscosity $\psi$.

• Physical Constraints: To ensure the model is physically meaningful, the authors derive a framework based on:
  • Isotropic Frame-Indifference: The model is formulated using tensor invariants, ensuring the physics remains consistent regardless of the coordinate system.
  • Dissipative Nature: The model must satisfy the second law of thermodynamics (entropy must increase), meaning the viscous stress must be purely dissipative.
  • Existence of a Convex Potential: They assume the existence of a convex potential of dissipation, which guarantees the mathematical well-posedness (existence and uniqueness of solutions) of the underlying Partial Differential Equations (PDEs).
• Neural Network Parameterization: The effective shear viscosity $\psi$ is parameterized as $\psi_\theta(|\text{S}_u|, \lambda)$, where $\lambda$ is an external parameter (like temperature or concentration). The NN architecture uses hyperbolic tangent activations and logarithmic inputs to handle multiple orders of magnitude in strain rates and ensure stability.
• Two Learning Approaches:
  1. Stress-based Regression ($J_s$): A standard regression approach that fits the NN directly to stress-strain data.
  2. PDE-Constrained Optimization ($J_v$): A more sophisticated method that infers the rheology from velocity data. This involves minimizing the misfit between observed and simulated velocities by solving the governing PDEs using the Finite Element Method (FEM). This is implemented by coupling the machine learning library (PyTorch) with a finite element library (Firedrake) using adjoint-based gradients.

3. Key Contributions

• Theoretical Framework: Derivation of a general, isotropic, and dissipative rheological framework that is more flexible than previous monotone or convex-only models.
• Hybrid Computational Pipeline: Integration of automatic differentiation (AD) from machine learning with adjoint-based PDE solvers from classical numerical analysis.
• Robustness to Noise: Demonstration that the framework can recover accurate physical laws even when training data is corrupted by significant noise.
• Discovery of Unknown Laws: The ability to move beyond "fitting" known laws to "discovering" entirely new, complex rheologies from high-fidelity simulation data (DEM).

4. Results

The authors validate the method through three distinct applications:

• Land Ice (Recovering Glen’s Law): The method successfully recovered the temperature-dependent Glen’s law. They found that velocity-based inference is significantly more robust; even with 10% noise in velocity, the inferred velocity profiles remained highly accurate, whereas stress-based inference produced much larger errors in velocity prediction.
• Sea Ice (Recovering the Viscous-Plastic Model): The framework successfully recovered the shear component of the standard Hibler model. Interestingly, they noted a different sensitivity here: unlike land ice, noise in the velocity data for sea ice led to larger errors in the inferred viscosity, likely due to the influence of ocean drag.
• Sea Ice (Discovery from DEM Data): In a "blind" test, the method was used to infer a rheology from Discrete Element Method (DEM) data (which simulates individual ice floe collisions). The NN discovered a complex, non-linear viscosity that transitions from shear-thickening to shear-thinning as ice concentration increases—a behavior that matches the underlying physics of the complex floe interactions.

5. Significance

This work represents a significant step forward in Scientific Machine Learning (SciML). By embedding rigorous physical principles (isotropy, convexity, and dissipation) directly into the neural network architecture, the authors bridge the gap between "black-box" machine learning and classical continuum mechanics. This ensures that the discovered models are not just statistically accurate, but are physically consistent and capable of generalizing to new, unseen physical scenarios. This has profound implications for climate modeling, where accurate ice rheology is critical for predicting sea-level rise and ocean circulation.