Learning constitutive models and rheology from partial flow measurements

TL;DR

The paper develops an end-to-end differentiable rheology framework that learns constitutive stress–strain relations directly from arbitrary flow measurements. It embeds a tensor-basis neural network (TBNN) closure inside a differentiable CFD solver to infer the constitutive law from flow fields, enabling transfer across geometries and robustness to noise. It then interrogates the learned closure with differentiable model fitting to identify classical laws (e.g., Carreau–Yasuda, Oldroyd–B, Giesekus) and extract parameters, using a digital rheometer and Bayesian information criteria for model selection. The approach demonstrates a path toward in-line, physics-regularized rheometry that can guide experimental design and enable interpretable, data-driven rheology across complex fluids and flow configurations. $\nabla\cdot\mathbf{u}=0$, $\rho(\partial_t\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u})=-\nabla p+\nabla\cdot\boldsymbol{\sigma}+\mathbf{f}$, $\boldsymbol{\sigma}=2\eta(I_1)\boldsymbol{D}$ with $\eta(I_1)$ learned via TBNN, and the framework can project to classical models through differentiable fitting and BIC-based selection. $Carreau-Yasuda$ and other constitutive forms are benchmarked, demonstrating accurate stress–flow predictions and interpretable parameter recovery. The results highlight the value of differentiable rheometry in turning complex flow data into both predictive models and mechanistic insights for non-Newtonian fluids. $\text{Digital rheometer}$ operations further illustrate how model selection and parameter identification can be performed efficiently from bulk measurements, emphasizing the role of experimental design in achieving identifiability. $\,$

Abstract

Constitutive laws are at the core of fluid mechanics, relating the fluid stress to its deformation rate. Unlike Newtonian fluids, most industrial and biological fluids are non-Newtonian, exhibiting a nonlinear relation. Accurately characterizing this nonlinearity is essential for predicting flow behavior in real-world engineering and translational applications. Yet current methods fall short by relying on bulk rheometer data and simple fits that fail to capture behaviors relevant in complex geometries and flow conditions. Data-driven approaches can capture more complex behaviors, but lack interpretability or consistency. To close this gap, we leverage automatic differentiation to build an end-to-end framework for robust rheological learning. We develop a differentiable non-Newtonian fluid solver with a tensor basis neural network closure that learns stress directly from arbitrary flow measurements, such as velocimetry data. In parallel, we implement differentiable versions of major constitutive relations, enabling Bayesian model parametrization and selection from rheometer data. Our framework predicts flows in unseen geometries and ensures physical consistency and interpretability by matching neural network responses to known constitutive laws. Ultimately, this work lays the groundwork for advanced digital rheometry capable of comprehensively characterizing non-Newtonian and viscoelastic fluids under realistic in-situ or in-line operating conditions.

Figures (12)

• Figure 1: Learning rheological models from flow data.Top: A tensor basis neural network (TBNN) that maps scalar invariants of the kinematics to the stress is embedded in a differentiable flow solver. Training on any type of flow data with JAX gradients yields a TBNN that generalizes across conditions and geometries and enables flow prediction. Bottom: For interpretability and extrapolation, we also implement a differentiable ODE framework that fits the learned TBNN model's to classical constitutive laws (e.g. Giesekus) and selects the best model and parameters using the Bayesian Information Criterion (BIC).
• Figure 2: Learning a tensor basis neural network closure for stress. (a) Ground-truth steady-state $x$-velocity for pressure-driven flow through a constriction (pressure gradient $G=5$). (b) Training loss versus iteration (c) Steady-state $x$-velocity predicted by the simulation with the trained TBNN. (d) $x$-velocity at the constriction throat ($x=4$).
• Figure 3: Flow prediction in an unseen geometry. (a) Steady-state $x$-velocity prediction for pressure-driven flow in a bidisperse porous medium with $G = 7.5$. (b) Relative error compared to ground truth, binned as a function of local strain rate.
• Figure 4: Extracted Carreau--Yasuda parameters from a TBNN. (a) Representative oscillatory forcing: TBNN shear-stress output (points) with the best-fit Newtonian response (dashed) and Carreau--Yasuda (CY) response (solid) fit to the same trace; the CY model follows the waveform closely while the Newtonian fit misses the extrema. (b) Parity plot of the shear-thinning exponent $n$ learned from the TBNN versus ground truth across eight runs; dashed line indicates $y=x$. (c) Parity for the onset timescale $k$ on log--log axes; points fall on the identity over more than an order of magnitude. In these runs $\eta_0$ was fixed, and $a$ and $\eta_\infty$ are weakly constrained; numerical values for all parameters, the best-fit Newtonian viscosity, and model comparison statistics are given in Table \ref{['tab:tbnn-cy-extraction']}. Across all runs the CY model is very strongly favored by BIC ($\Delta\mathrm{BIC}=\mathrm{BIC}_{\mathrm{N}}-\mathrm{BIC}_{\mathrm{CY}}\gg 0$).
• Figure 5: Demonstration of fitting different constitutive models to the same ground-truth data. We simulate rheometer data using the same forcing functions described in the text, taking the strain rate as the controlled quantity and measuring the resulting shear stress in a Giesekus model. We fit several different constitutive models to this data and then examine how they perform when predicting the shear stress response under a new forcing that was not included in the fitting data. This new forcing and ground-truth response are shown in the top panel. In the bottom panel we show the response of the different best-fit models to this new forcing.