Structure-preserving neural networks in data-driven rheological models

TL;DR

This work develops a theory- and data-driven framework for structure-preserving rheology in non-Newtonian Stokes flows by learning the viscous stress through Input-Convex Neural Networks (ICNNs). The core idea is to model the viscosity as $\tau_{\bm{\theta}}(\varepsilon(\mathbf{u})) = \mathrm{ICNN}_{\bm{\theta}}(|\varepsilon(\mathbf{u})|)\varepsilon(\mathbf{u})$, ensuring convexity in the input to guarantee well-posedness, with the sign reflecting shear-thinning/thickening behavior. The paper provides a perturbation-based convergence analysis that links the ICNN approximation error to finite element errors, and validates the approach with numerical tests on Carreau-type laws and real xanthan datasets, demonstrating that ICNNs can capture complex rheological trends and outperform traditional parametric models. The results offer a theoretically grounded, data-driven path for reliable simulations of complex fluids, with open-source code for reproducibility and potential extensions to broader multiphysics settings.

Abstract

In this paper we address the importance and the impact of employing structure preserving neural networks as surrogate of the analytical physics-based models typically employed to describe the rheology of non-Newtonian fluids in Stokes flows. In particular, we propose and test on real-world scenarios a novel strategy to build data-driven rheological models based on the use of Input-Output Convex Neural Networks (ICNNs), a special class of feedforward neural network scalar valued functions that are convex with respect to their inputs. Moreover, we show, through a detailed campaign of numerical experiments, that the use of ICNNs is of paramount importance to guarantee the well-posedness of the associated non-Newtonian Stokes differential problem. Finally, building upon a novel perturbation result for non-Newtonian Stokes problems, we study the impact of our data-driven ICNN based rheological model on the accuracy of the finite element approximation.

Figures (7)

• Figure 1: Approximation of $y(x) = |x| + \sin(x)$: comparison between $\text{ICNN}_{\bm{\theta}}$ and standard $\text{NN}_{\bm{\theta}}$ with same architecture, same training procedure.
• Figure 1: $\text{ICNN}_{\bm{\theta}}$ approximations of the Carreau law \ref{['eqn:carreau']} with parameters: $k_\infty=0,k_0=2,\lambda=2,n=1.2, 1.6, 2, 2.4, 2.8$.
• Figure 2: Approximation of the non-convex function $f(x,y)= |x| + |y| + \sin(x+y)$ (red colour) by $\text{ICNN}_{\bm{\theta}}$ (viridis): two different points of view.
• Figure 2: Velocity error $\|\overline{\mathbf{u}} - \mathbf{u}_{h,ICNN} \|_{[W^{1,r}(\Omega)]^2}$ (left) and pressure error $\|\overline{p} - p_{h,ICNN} \|_{L^{r'}(\Omega)}$for $r \in \{ 1.2,1.6,2, 2.4, 2.8\}$.
• Figure 3: Comparison between $\text{ICNN}_{\bm{\theta}}$ and a standard feed-forward neural network $NN_{\bm{\theta}}$ in the approximation of real rheological measures: viscosity curves $k(t)$ (left) and function $k(t)t$ (right), with $k\in \{\text{ICNN}_{\bm{\theta}}, \text{NN}_{\bm{\theta}}\}$. DATASET: NaCL05_XG

Theorems & Definitions (9)

• Remark 2.1
• Proposition 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Remark 4.1
• Theorem 4.2: Perturbation result
• Remark 4.3
• Proof 1