Learning constitutive models and rheology from partial flow measurements

This paper presents an end-to-end framework that combines a differentiable non-Newtonian solver with a frame-invariant tensor basis neural network to learn form-agnostic constitutive models from partial flow measurements, enabling the discovery of interpretable, geometry-portable rheological laws through automated Bayesian model selection.

The Gist

Imagine you are trying to figure out how a specific type of honey behaves. You know that some honey is runny, some is thick, and some changes its thickness depending on how fast you stir it. In the world of science, this is called rheology—the study of how materials flow and deform.

For a long time, scientists have had a hard time figuring out the "rules" (constitutive models) that govern how complex fluids like blood, paint, or polymer solutions behave. Here is the problem:

The Old Way: The "Taste Test" in a Vacuum

Traditionally, to understand a fluid, scientists put it in a simple, boring machine (a rheometer) that squeezes or stirs it in a straight line. It's like trying to understand how a car handles by only driving it in a straight, empty parking lot at 5 mph.

• The Flaw: Real life is messy. Fluids flow through crooked pipes, around obstacles, and into tiny blood vessels. The simple "parking lot" tests often fail to predict how the fluid will act in these complex, real-world situations.
• The Result: Scientists often guess the wrong mathematical formula for the fluid, leading to bad predictions when they try to use it in a real engine or a human body.

The New Way: The "Digital Twin" Detective

This paper introduces a clever new method called "Digital Rheometry." Instead of guessing the rules from a simple test, they let the fluid show them the rules by watching it flow through a complex environment.

Here is how they did it, using a few simple analogies:

1. The "Differentiable Solver" (The Super-Teacher)

Imagine you have a video game engine that simulates fluid flow perfectly. Usually, if you change the rules of the game (like making the fluid thicker), you have to restart the whole game to see what happens.
The authors built a special version of this engine where every single step is mathematically reversible.

• The Analogy: Think of a teacher who can not only solve a math problem but also instantly see exactly which number you changed to get the wrong answer. If the simulation predicts the fluid moves too fast, the "teacher" knows exactly which rule to tweak to fix it. This allows them to learn from the data instantly, rather than guessing and checking.

2. The "TBNN" (The Shape-Shifting Translator)

They used a type of AI called a Tensor Basis Neural Network (TBNN).

• The Analogy: Imagine you are trying to describe a complex dance to someone who has never seen it. Instead of memorizing every single step, you teach the AI the principles of the dance (like "if the music speeds up, the dancer spins faster").
• The TBNN learns the relationship between how the fluid is being stretched/squeezed (the dance moves) and how much stress (resistance) it creates. Crucially, it learns the universal rules of the dance, not just the specific moves in one room. This means if you take this learned AI and put it in a different room (a new geometry), it still knows how to dance correctly.

3. The "Distillation" (Turning AI into a Recipe)

AI is great at predicting, but it's often a "black box"—you don't know why it made a decision. Scientists need simple, understandable formulas (like a recipe) to use in engineering.

• The Analogy: The AI is like a master chef who can cook a perfect dish but can't explain the recipe. The authors developed a way to watch the AI cook and then reverse-engineer the recipe.
• They use a statistical tool (Bayesian Information Criterion) to ask: "Is this complex AI recipe actually necessary, or can we explain the flavor with a simple, classic recipe (like the Carreau-Yasuda model)?"
• They found that their AI could perfectly mimic complex fluids, and then they successfully extracted the exact mathematical "recipe" that describes those fluids.

Why This Matters

This framework is a game-changer for three reasons:

1. It works in the real world: You don't need to take the fluid out of its environment. You can learn the rules just by watching it flow through a pipe, a micro-vessel, or an oil well. It's like learning how a car handles by driving it on a mountain road, not just in a parking lot.
2. It's robust: Even if your measurements are blurry, noisy, or low-resolution (like a shaky video), the physics of the simulation acts as a "guardrail," correcting the AI so it doesn't learn nonsense.
3. It finds the truth: It can tell you if a fluid is simple or complex. If the data is too simple to tell the difference between two complex models, the system honestly says, "We can't tell the difference yet," rather than forcing a wrong answer.

The Bottom Line

The authors have built a universal translator for fluids. They can take messy, real-world flow data, use a super-smart, physics-aware AI to learn the hidden rules, and then translate those rules back into simple, human-readable equations. This allows engineers and scientists to predict how complex fluids will behave in any situation, from drug delivery in the body to oil extraction in the ground, without needing perfect, idealized lab tests.

Technical Summary

1. Problem Statement

Predicting the flow of complex fluids (biological and industrial) through real-world geometries requires accurate constitutive laws that relate fluid stress to deformation. Current characterization methods face three major limitations:

• Idealized Measurements: Traditional benchtop rheometers measure bulk, averaged responses in simplified, 1D flows. These often fail to capture physics relevant to complex geometries (e.g., strong resistance to stretching in contractions) where shear and extension coexist.
• Model Ambiguity: Multiple constitutive laws can fit simple flow data equally well but diverge dramatically in complex kinematics. Selecting a model a priori and fitting parameters often leads to overfitting or physically ungrounded solutions.
• Data-Driven Limitations: Existing machine learning approaches (e.g., PINNs, symbolic regression) often suffer from:
  • Geometry Specificity: They are difficult to transfer to unseen environments.
  • Overfitting: They struggle to infer thousands of network weights from sparse, low-dimensional scalar data.
  • Lack of Interpretability: They act as "black boxes" without extracting physical parameters.

2. Methodology

The authors propose a two-part, end-to-end framework that unifies measurement, constitutive discovery, and model selection using automatic differentiation and differentiable simulations.

A. Differentiable Non-Newtonian Solver

• Core Engine: The authors developed a fully differentiable fluid solver built on JAX (using JAX-CFD and Immersed Boundary methods).
• Capability: It solves the incompressible Navier-Stokes equations with spatially varying viscosity. It supports arbitrary geometries and boundary conditions.
• Differentiability: The solver provides exact gradients of any flow observable (e.g., velocity fields) with respect to model parameters or neural network weights, enabling end-to-end optimization without explicit adjoint derivations.

B. Tensor Basis Neural Network (TBNN) for Constitutive Discovery

• Architecture: Instead of learning a black-box stress field, the framework embeds a Tensor Basis Neural Network (TBNN) directly into the solver.
• Physics Invariance: The TBNN maps scalar kinematic invariants (e.g., $tr(D^2)$, $tr(W^2)$) to stress coefficients, enforcing frame invariance (Galilean invariance).
• Function: It learns a geometry-agnostic mapping from local kinematics to stress. By training on local velocity fields from complex flows, it recovers the underlying stress-strain relationship without presupposing a specific mathematical form.

C. Differentiable Model Fitting & Selection

• Distillation: Once the TBNN is trained, its learned closure is interrogated using a "digital rheometer." The TBNN is subjected to controlled deformation histories (e.g., oscillatory shear) to generate stress-time series.
• Symbolic Regression via BIC: A library of classical constitutive models (Newtonian, Carreau-Yasuda, Oldroyd-B, Giesekus, Linear PTT) is fitted to the TBNN-generated data using a differentiable ODE solver.
• Selection: The Bayesian Information Criterion (BIC) is used to select the optimal model. BIC balances predictive power against model complexity, penalizing over-parameterization. This allows the data to determine both the model form and its physical parameters.

3. Key Contributions

1. End-to-End Differentiable Framework: A novel pipeline that leverages automatic differentiation to learn constitutive laws directly from complex flow data, bypassing the need for inverse simulations for every candidate model.
2. Geometry Portability: By learning a local, frame-invariant material response, the method generalizes to unseen geometries and flow conditions, unlike standard PINNs which are often geometry-specific.
3. Interpretability & Physical Parameter Extraction: The framework distills complex neural closures into interpretable symbolic forms (classical equations) with physically meaningful parameters, bridging the gap between data-driven flexibility and physical rigor.
4. Automated Model Selection: The first demonstration of using differentiable simulations and BIC to automatically select among multiple classical constitutive families from rheometry time series, addressing the issue of model ambiguity.

4. Results

• Constitutive Recovery: The TBNN successfully learned the stress-strain relationship of a Carreau-Yasuda (CY) fluid from a complex pressure-driven flow through a constriction-expansion channel.
  • The trained model achieved convergence with a loss reduction of >4 orders of magnitude.
  • The reconstructed velocity fields were visually indistinguishable from ground truth.
• Generalization: The learned TBNN closure was tested in a new, unseen geometry (bidisperse porous medium) with a different pressure gradient. It predicted flow fields with uniformly low relative error across all strain rates, demonstrating true transferability.
• Robustness: The framework remained accurate even with:
  • Coarse Resolution: 10x reduction in spatial resolution (down to 13x13 grids).
  • Noise: Correlated, heteroskedastic noise up to 4% of the flow velocity.
• Model Identification:
  • The framework correctly identified the ground-truth model (e.g., Giesekus) from synthetic noisy rheometer data in nearly 100% of cases for simpler models and ~70% for complex ones.
  • BIC Analysis: When the data did not justify a complex model (e.g., specific parameter regimes where complex and simple models behave similarly), BIC correctly selected the simpler model, highlighting the limits of identifiability based on forcing protocols.
  • Parameter Accuracy: For correctly identified models, the estimated physical parameters (viscosity, relaxation time, etc.) were within 1% of ground truth values.

5. Significance and Impact

• "Digital Rheometry": This work establishes a foundation for characterizing complex fluids directly within their operating environments (e.g., micro-vessels, oil reservoirs, industrial pipes) rather than relying on idealized lab tests.
• Overcoming the Information Bottleneck: By using the governing equations as hard constraints (physical regularization), the method extracts maximum information from sparse or noisy data, overcoming the limitations of bulk rheometry.
• Experimental Design: The differentiable nature of the solver opens the door to gradient-based experimental design, where forcing protocols can be optimized to maximize the information content for distinguishing between competing constitutive hypotheses.
• Future Applications: The approach is applicable to a wide range of engineering and physical sciences problems involving non-Newtonian fluids, viscoelasticity, and multiphase flows, potentially automating the discovery of new material laws where classical models fail.

In summary, the paper presents a paradigm shift from "fitting models to data" to "learning physics from data," providing a rigorous, interpretable, and portable tool for constitutive modeling in complex fluid dynamics.