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

arXiv: https://arxiv.org/abs/2501.00556 · HTML: https://mlsanity.com/paper/2501.00556/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 78 references that are in the map.
- First posted on arXiv 31 Dec 2024.

## Builds on (most cited first, 18 of 18)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Knowledge Distillation: A Survey](https://mlsanity.com/paper/2006.05525/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [Unsupervised discovery of interpretable hyperelastic constitutive laws](https://mlsanity.com/paper/2010.13496/) (2020)
- [Bayesian-EUCLID: discovering hyperelastic material laws with uncertainties](https://mlsanity.com/paper/2203.07422/) (2022)
- [Differentiable Programming for Differential Equations: A Review](https://mlsanity.com/paper/2406.09699/) (2024)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations](https://mlsanity.com/paper/2302.03598/) (2023)
- [Towards a theory of model distillation](https://mlsanity.com/paper/2403.09053/) (2024)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
- [Scientific Machine Learning for Modeling and Simulating Complex Fluids](https://mlsanity.com/paper/2210.04431/) (2022)
- [Learning orbital dynamics of binary black hole systems from gravitational wave measurements](https://mlsanity.com/paper/2102.12695/) (2021)
- [Efficient hybrid modeling and sorption model discovery for non-linear advection-diffusion-sorption systems: A systematic scientific machine learning approach](https://mlsanity.com/paper/2303.13555/) (2023)

## Abstract (arXiv)

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.