Finding the Underlying Viscoelastic Constitutive Equation via Universal Differential Equations and Differentiable Physics

TL;DR

This work introduces a physics-informed Universal Differential Equation (UDE) framework to recover missing terms in viscoelastic constitutive equations for four Maxwell-type models (UCM, Johnson-Segalman, Giesekus, ePTT) using differentiable physics and synthetic LAOS data. A Tensor Basis Neural Network (TBNN) embedded into the constitutive update recovers nonlinear terms, producing a dimensionless, frame-indifferent formulation that is trained via continuous adjoint optimization. The results show strong shear-stress predictions across most models, with ePTT posing challenges due to the exponential nonlinearity, and reveal that normal-stress predictions can be accurate even without direct data; a distillation step demonstrates that a surrogate Maxwell model can mimic nonlinear models, offering a pathway for cross-model translation and simplified rheological surrogates. These findings support the potential of UDEs in rheology and suggest avenues for improved tensor bases and differentiable-physics approaches to enable robust, data-efficient constitutive identification and digital-twin applications.

Abstract

This research employs Universal Differential Equations (UDEs) alongside differentiable physics to model viscoelastic fluids, merging conventional differential equations, neural networks and numerical methods to reconstruct missing terms in constitutive models. This study focuses on analyzing four viscoelastic models: Upper Convected Maxwell (UCM), Johnson-Segalman, Giesekus, and Exponential Phan-Thien-Tanner (ePTT), through the use of synthetic datasets. The methodology was tested across different experimental conditions, including oscillatory and startup flows. While the UDE framework effectively predicts shear and normal stresses for most models, it demonstrates some limitations when applied to the ePTT model. The findings underscore the potential of UDEs in fluid mechanics while identifying critical areas for methodological improvement. Also, a model distillation approach was employed to extract simplified models from complex ones, emphasizing the versatility and robustness of UDEs in rheological modeling.

Figures (14)

• Figure 1: Tensor basis neural network architecture (TBNN). The invariants ($\tau_i^*$) of the basis tensor ${T_{ij}^*}^{(n)}$ are the inputs. The output of the multilayer perceptron is a scalar coefficient $g^{(n)}(\tau_i^*;\theta)$. $\mathscr{N}_{ij}$ is the final result produced by the Hadamard product (element-wise) $\odot$ between $g^{(n)}(\tau_i^*;\theta)$ and ${T_{ij}^*}^{(n)}$. The MLP model is composed of four layers: an input and output layer with nine neurons each and two hidden layers with 32 neurons each that utilize the tanh activation function.
• Figure 2: Flowchart of the algorithm to solve UDE. At $k=900$ iterations, a new time series is added in the optimization process until all eight-time series have been trained $( k=7000)$.
• Figure 3: Loss function for each model used as a priori information in the UDE.
• Figure 4: Evaluation for the extrapolation of UDE models for shear stress ($\sigma^{*}_{12},t^*$) with input $\dot\gamma^*(t^*)=$ 3 cos (1.5 $t^*$). The points highlighted in green correspond to training points, $t^*\in[0,20]$. The blue curve illustrates the numerical solution of the differential equation, referred to as the "ground truth" solution. The black curve depicts the UDE model pre-training, while the red curve illustrates the UDE model post-training. The following parameters were used $\xi=0.4$ (Johnson-Segalman), $\alpha=0.2$ (Giesekus) and ($\xi$, $\epsilon$)=(0,0.4) for (ePTT).
• Figure 5: Evaluation for the extrapolation of UDE models for first normal stress difference in shear (N$_1^* = \sigma^*_{11} - \sigma^*_{22}$)(N$_1^*,t^*$) with input $\dot\gamma^*(t^*)=$ 2 cos $t^*$. The blue curve illustrates the numerical solution of the differential equation, referred to as the "ground truth" solution. The black curve depicts the UDE model pre-training, while the red curve illustrates the UDE model post-training. The following parameters were used $\xi=0.4$ (Johnson-Segalman), $\alpha=0.2$ (Giesekus) and ($\xi$, $\epsilon$)=(0,0.4) for (ePTT).