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

D\'ario A.B. Oliveira; Elias C. Rodrigues; Roberto F. Ausas; Roney L. Thompson

REVIEW 3 major objections 7 minor 79 references

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

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A hybrid neural-differential-equation method can recover the missing nonlinear term in Maxwell-type viscoelastic constitutive equations from shear-stress time series alone, and then predict normal stress differences it was never trained on.

desk verdict Solid, honestly reported benchmark extending RUDEs to four viscoelastic models with shear-stress-only training; identifiability is a real caveat, and the paper needs code, error metrics, and a toned-down QL-LAOS claim. read the letter →

arxiv 2501.00556 v2 pith:KTDHW7TC submitted 2024-12-31 physics.flu-dyn cs.LG

Elias C. Rodrigues , Roney L. Thompson , Dário A.B. Oliveira , Roberto F. Ausas This is my paper · ORCID

classification physics.flu-dyncs.LG

keywords universaldifferentialequationsviscoelasticconstitutivemodelstensorbasisneuralnetworkframeindifferencelargeamplitudeoscillatoryshearmodeldistillationnormalstressdifferencesphysics-informedmachinelearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to show that a Universal Differential Equation (UDE), a mechanistic constitutive equation with a neural network embedded inside it, can reconstruct the missing nonlinear term in Maxwell-type viscoelastic models from data. Using only time series of the shear stress $\sigma^*_{12}$ from synthetic oscillatory shear flows, the method recovers the correct tensor-valued function for the Upper Convected Maxwell, Johnson–Segalman, and Giesekus models, with learned coefficients matching the generating values ($g \approx 1$, $0.2$, and $0.2$). The reason to care is that the trained model then predicts the first and second normal stress differences, quantities it was never shown, and also forecasts startup-flow viscosity. The exponential Phan-Thien–Tanner model is recovered less faithfully because its exponential term is harder to express in the tensor basis. The same machinery also serves as a model-distillation tool, fitting a simpler Maxwell model that absorbs part of the nonlinear behavior of Giesekus and ePTT.

What carries the argument

The load-bearing object is the tensor-basis neural network with eight tensor basis elements $T^{(n)}_{ij}$ and eight invariants $\tau^*_i$, an architecture that guarantees Euclidean invariance of the learned term $N_{ij} = \sum_{n=1}^8 g^{(n)}(\{\tau^*_i\}_{i=1}^8;\theta)\,T^{(n)}_{ij}$. A multilayer perceptron maps the eight invariants to scalar coefficients $g^{(n)}$, and the Hadamard product with the basis tensors produces a tensor-valued output; because the basis spans symmetric isotropic tensor functions of $\sigma^*$ and $\dot\gamma^*$, the ansatz can represent any frame-indifferent missing term up to the capacity of the network. The machinery does two jobs: it inserts the missing physics directly into the differential equation as a hard constraint, and its invariant structure lets the network generalize from shear-stress training to normal-stress predictions.

What would settle it

Train the same UDE on shear-stress-only data from a constitutive model whose missing term is a linear combination of two basis tensors, for instance $\alpha\,\sigma\cdot\sigma + \frac{\xi}{2}(\sigma\cdot\dot\gamma + \dot\gamma\cdot\sigma)$, and check whether the recovered coefficients match the generating $\alpha$ and $\xi$. If two different coefficient combinations produce indistinguishable $\sigma^*_{12}$ over the training window but different normal stresses, then shear-only data do not identify the constitutive term, and the central claim would fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a frame-indifferent tensor-basis neural network, placed inside the dimensionless Maxwell-type equation $\frac{d\sigma^*}{dt^*} = \dot\gamma^* + (\nabla v)^*{}^T\cdot\sigma^* + \sigma^*\cdot(\nabla v)^* - N(\sigma^*,\dot\gamma^*)$, can learn the missing term $N$ from shear-stress observations alone. The network is built from eight basis tensors and eight invariants through the representation theorem for isotropic functions of two symmetric tensors, so the learned tensor function is Euclidean invariant. Across eight oscillatory experiments covering Deborah numbers $0.25$ to $1$ and Weissenberg numbers $1$ to $2$, training on $\sigma^*_{12}$ alone yields coefficients $g^{(2)} \approx 0.99$, $g^{(6)} \approx 0.199$, and $g^{(4)} \approx 0.199$ for UCM, Johnson–Segalman, and Giesekus, respectively, recovering $\sigma^*$, $\frac{\xi}{2}(\sigma^*\cdot\dot\gamma^* + \dot\gamma^*\cdot\sigma^*)$, and $\alpha\,\sigma^*\cdot\sigma^*$. The resulting model extrapolates to untrained shear-rate amplitudes, to normal-stress differences $N^*_1$ and $N^*_2$, and to startup shear, with the ePTT case showing amplitude discrepancies because its missing term is exponential in $\operatorname{tr}(\sigma^*)$. The paper also reports a quasi-linear large-amplitude oscillatory shear regime for Johnson–Segalman and ePTT at higher Deborah numbers, and a distillation procedure in which an optimized UCM surrogate reproduces Giesekus and ePTT responses with renormalized viscosity and relaxation time.

Load-bearing premise

The argument depends on the assumption that shear-stress time series from the chosen oscillatory flows, together with the UCM baseline and the tensor-basis ansatz, uniquely determine the tensor-valued function $N$ over the region the training visits; if two different $N$ produced the same $\sigma^*_{12}$ history, the accurate normal-stress predictions would be luck rather than identification.

Editorial extensions

If this is right

If correct, the method identifies the missing tensor term in Maxwell-type constitutive equations directly from rheometric shear-stress data, without needing normal-stress measurements.
The recovered coefficients are interpretable as model parameters: $g^{(2)}=0.99$, $g^{(6)}=0.199$, and $g^{(4)}=0.199$ match the generating values $1$, $\xi/2$, and $\alpha$ for the respective models.
The trained UDE generalizes across flow type: it predicts the first and second normal stress differences and startup-flow viscosity from oscillatory shear training.
For models with exponential stress dependence, like ePTT, the current tensor basis is insufficient and accuracy degrades as the extensibility parameter increases, indicating a needed architectural modification.
Model distillation shows that a reparametrized UCM can absorb much of the nonlinear behavior of Giesekus and ePTT, with effective viscosity and relaxation time reduced according to the recovered coefficients.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension left implicit by the paper is applying the shear-only pipeline to experimental LAOS data, turning rheometry into a constitutive-equation discovery tool, once the identifiability question is settled.
The near-exact recovery of coefficients for polynomial-type nonlinearities suggests that adding $\operatorname{tr}(\sigma^*)$-dependent features or an exponential activation to the tensor-basis network would improve recovery for exponential models like ePTT.
The distillation result implies that comparisons between linear and nonlinear constitutive models should re-optimize the linear model's parameters rather than simply zeroing the nonlinear term, which would change how model discrepancy is quantified in rheology.
A testable consequence is that training on data from a superposition of Giesekus and Johnson–Segalman nonlinearities should yield additive recovered coefficients if the identifiability assumption holds.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The paper does what it says: it takes Lennon et al.'s RUDE/TBNN setup, trims the basis to Smith's eight tensors, and tests recovery of missing terms for UCM, Johnson-Segalman, Giesekus, and ePTT from synthetic shear-stress-only oscillatory data. The main new results are the coefficient recovery (g≈1, 0.2, 0.2 for the three clean models), the extrapolation to normal stress differences and startup flows, and the distillation of a surrogate Maxwell from Giesekus/ePTT data. The ePTT failure is reported frankly, which I appreciate.

Where it's soft: the identifiability issue is real. Training on σ12 alone, with a six-component stress state, means the map from N to the observed trajectory could have multiple preimages. The upper-convected coupling does carry normal-stress information into σ12, which is why the extrapolation works here, but the paper provides no uniqueness argument, no repeated-initialization study, and no error bars on the recovered coefficients. For a benchmark on noise-free data with known generative models, that's a caveat, not a fatal flaw; the empirical evidence is strong for three models. But a reader asking 'would this work on real rheometer data with noise?' has no answer.

Second, no code or data is provided, and there are no quantitative error metrics—just figures and the recovered coefficients. That's a reproducibility gap the referees should close.

Third, the QL-LAOS 'important finding' (elliptical Lissajous curves at higher De for JS and ePTT) is overclaimed relative to the evidence. It's an interesting observation, but it's presented as a discovery about viscoelastic structure without the analysis to back it up. I'd ask them to soften that language or support it properly.

Overall, this is a useful extension, not a breakthrough. It deserves serious refereeing, and I'd send it out. Conditional on the authors providing code/data, error metrics, and toning down the QL-LAOS claim, I'd be happy to see it published. The identifiability analysis could be asked for as a major revision item, though I wouldn't block on it if the empirical demonstration holds.

Referee Report

3 major / 7 minor

Summary. The manuscript proposes a Universal Differential Equation (UDE) framework for recovering missing terms in Maxwell-type viscoelastic constitutive equations. The mechanistic prior is a dimensionless UCM-like evolution for the extra stress (Eqs. (9)-(10)); the missing term N is represented by a frame-invariant Tensor Basis Neural Network (TBNN) built on eight Smith-type basis tensors and invariants (Eqs. (11)-(13)). Training data are synthetic shear-stress time series sigma*12 from eight large-amplitude oscillatory shear protocols for UCM, Johnson-Segalman, Giesekus, and ePTT. The authors report coefficient recovery (g(2) about 0.99 for UCM, g(6) about 0.199 for Johnson-Segalman, g(4) about 0.199 for Giesekus), extrapolation to first and second normal stress differences and to startup flow, and a 'viscoelastic model distillation' procedure that fits an effective UCM surrogate to Giesekus/ePTT data.

Significance. The paper addresses a timely problem: data-driven discovery of objective constitutive equations from rheometric time series. Its main strengths are the principled TBNN construction, the deliberately sparse measurement setting (only sigma*12 used for training), and the honest documentation of the ePTT failure mode. The synthetic-data setup is a standard self-consistency benchmark rather than circular reasoning, and the normal-stress predictions are genuine holdouts that provide meaningful evidence. If the identifiability and robustness concerns below are resolved, the result would be a useful proof-of-concept for the rheology and scientific-machine-learning communities.

major comments (3)

[Section 3.3 and Section 4.1; Eqs. (9)-(11), (14); Figs. 5-6, 9-10] The central extrapolation claim, that training on sigma*12 alone determines the full tensor-valued function N, requires an identifiability analysis that the paper does not provide. The ODE state is the full six-component stress tensor and the TBNN outputs all six components N_ij, while the loss in Eq. (14) sees only sigma*12. The upper-convected terms (gamma* sigma11, gamma* sigma12) and the shared scalar coefficients g(n) in Eq. (11) make sigma*12 informative about normal stresses, so the claim is plausible, but the paper does not rule out coefficient combinations that match every sigma*12 training history yet differ in N11 and N22. For UCM, for example, the sigma*12 equation involves sigma11 but does not contain sigma22 directly, so whether N22 is pinned down depends on the shared-coefficient ansatz and is not analyzed. L1 regularization selects a sparse solution but does not by itself establish that the selected solution is the true constitutive term. I recommend adding an observability/identifiability study for the TBNN class, or at minimum a numerical experiment that fits the sigma*12 data with deliberately restricted basis subsets and shows whether the resulting N1 and N2 predictions change.
[Section 4.1 and Eq. (14)] The quantitative evidence for the method rests on a small number of recovered coefficients (g(2)=0.99, g(6)=0.199, g(4)=0.199) and on visual agreement in Figs. 4-10. No error bars, no multiple initializations, and no sensitivity study with respect to the L1 penalty kappa or the neural-network width are reported. Since the optimization landscape for UDEs is nonconvex, a single run is not enough to support the recovery claim. Please report statistics over random seeds and a kappa sweep, and give quantitative errors (for example relative L2 norms for sigma12, N1, N2 in the extrapolation tests).
[Section 5; Section 4.1, Figs. 6 and 11] Section 5 states that 'Regardless of the synthetic data generation model utilized, the UDE is capable of capturing patterns beyond the training data for normal stress differences.' This overstates the ePTT result: Fig. 6 shows a non-negligible spurious N2 and Fig. 11 shows that accuracy degrades as epsilon increases. Because the ePTT case is the main negative result, the conclusion should be sharpened to say that the method currently recovers the polynomial/linear corrections (Johnson-Segalman and Giesekus) and the UCM baseline reliably, while the exponential ePTT correction is only approximated in the tested range.

minor comments (7)

[Eq. (14)] The notation sigma*12(data)(ti; theta) makes the data appear to depend on the network parameters; the data term should be written sigma*12(data)(ti).
[Section 3.2] The text says the MLP output layer has nine neurons, but Eq. (11) uses eight tensor coefficients g(n); please reconcile this with the eight-term basis.
[Section 3.3 and Fig. 2] The flowchart caption states k=7000 while the text states 7200 total iterations; please align the numbers. Also, 'EDOSolver' in Eq. (14) should be 'ODESolver'.
[Section 4.1] The (Wi,De)=(2,1) case is called an extrapolation experiment, but this driving protocol is in the training set; the extrapolation is to unobserved stress components, so the wording should be clarified.
[Section 3.2, Eqs. (12)-(13)] The claim that Smith's representation theorem sets the number of basis tensors at eight needs a precise citation or derivation, since the standard integrity basis for two symmetric tensors contains ten invariants; the completeness of the eight-term basis in Eq. (13) is not demonstrated.
[Section 4.1, Fig. 8] The QL-LAOS claim is stronger than the presented evidence: a quantitative criterion (e.g., harmonic content or ratio of higher harmonics) is needed to conclude that a quasi-linear regime is reached.
[Reproducibility] No code or data availability statement is provided; adding one would improve reproducibility. There are also minor typos, e.g., 'tinny deviation' in Section 4.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: normal-stress and startup predictions are genuine holdouts, and the TBNN basis is a representation ansatz rather than an input-output identity.

full rationale

The derivation chain is self-contained. Section 3.3 states that "training data exclusively consists of the time series of the shear stress component (σ∗12)", and the loss in Eq. (14) involves only σ∗12; the normal-stress comparisons in Figs. 5, 6, 9, and 10, and the startup test in Fig. 12, use stress components and kinematics never present in the training loss. These are therefore genuine extrapolation tests rather than fitted inputs renamed as predictions. The TBNN in Eq. (11) uses Smith's representation theorem to span all isotropic tensor functions of σ∗ and γ̇∗, and the target terms (σ∗, σ∗·γ̇∗+γ̇∗·σ∗, and σ∗·σ∗) are among the basis tensors; recovering scalar coefficients by optimization is standard function fitting within a complete basis, not circularity. The synthetic data being generated from the same Maxwell-type family that supplies the mechanistic prior is a self-consistency benchmark, a common and legitimate validation design. The paper's self-citations (e.g., [70]-[73] for the QL-LAOS interpretation and [78,79] for dimensionless interpretation of distillation) support side discussions, not the central reconstruction claim. The lack of an identifiability analysis for recovering the full tensor function from shear-stress-only data is a genuine correctness risk, but the absence of a uniqueness proof is not evidence that the result reduces to its input by construction. No load-bearing step in this paper exhibits the required reduction, so the honest finding is no significant circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

All parameters are either hand-chosen hyperparameters or coefficients fitted in the distillation step. No new physical entities are postulated. The main axioms are the homogeneous-flow restriction, the applicability of Smith's representation theorem to the missing stress term, and the accuracy of the synthetic ground truth.

free parameters (4)

L1 regularization coefficient kappa = 1e-2
Hand-chosen; affects coefficient sparsity and could bias the recovered g(n) values. Introduced in Eq. (14).
Neural network architecture = 2 hidden layers x 32 neurons, tanh, learning rate 1e-3
Hand-chosen; no ablation study on architecture sensitivity.
Surrogate coefficients g(2), g(3) for Maxwell-Giesekus = 1.408, 0.944
Fitted in distillation, used to define effective viscosity and relaxation time via Eqs. (18) and (19).
Surrogate coefficients g(2), g(3) for Maxwell-ePTT = 1.413, 0.997
Fitted in distillation, used in Eqs. (21) and (22).

assumptions (4)

domain assumption Homogeneous flow, so grad sigma = 0 and the advective term vanishes.
Stated in Section 3.1: 'Given that rheometric experiments are generally carried out in small gaps, we investigate homogeneous flow (nabla sigma = 0) in all scenarios.' This restricts applicability to flows without stress gradients.
domain assumption The missing term N(sigma*, gamma-dot*) is an isotropic tensor function representable by Smith's eight basis tensors with coefficients depending on the eight invariants.
Section 3.2 uses Smith's representation theorem to set the basis. The ePTT exponential term is not exactly in this span, which the authors acknowledge as a limitation, so this assumption is load-bearing.
standard math The numerical ground truth from Tsit5 is an accurate reference.
Section 3.3: synthetic data is generated by Tsit5; the paper does not report tolerance or convergence checks, so any solver error is inherited by the benchmark.
domain assumption The neural network can approximate the required function given the training data and L1 regularization.
This is the standard universal-approximation assumption; the paper provides no approximation error bounds.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Finding the Underlying Viscoelastic Constitutive Equation via Universal Differential Equations and Differentiable Physics." pith.science (2026). https://pith.science/paper/KTDHW7TC

@misc{pith2026250100556,
  author       = {Pith},
  title        = {Pith review of: Finding the Underlying Viscoelastic Constitutive Equation via Universal Differential Equations and Differentiable Physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTDHW7TC}},
  note         = {Machine review of arXiv:2501.00556}
}

read the original 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

Figures reproduced from arXiv: 2501.00556 by the authors.

**Figure 2.** Flowchart of the algorithm to solve UDE. At [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Loss function for each model used as a priori information in the UDE. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Evaluation for the extrapolation of UDE models for shear stress ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Evaluation for the extrapolation of UDE models for first normal stress difference in shear (N [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Evaluation for the extrapolation of UDE models for second normal stress difference (N [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: Lissajous-Bowditch curves for the extrapolation of UDE models for shear stress ( [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Lissajous-Bowditch curve for different Deborah numbers with input [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: Lissajous-Bowditch curves for the extrapolation of UDE models for the normal stress difference ( [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: Lissajous-Bowditch curves for the extrapolation of UDE models for normal stress difference ( [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: Comparison between UDE prediction for Linear and Exponential PTT models for different extensibility [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: Test for predicting UDE models in a startup experiment with [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: Surrogate Maxwell for Giesekus model. The red curve illustrates the surrogate model post-training. The [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: Surrogate Maxwell for ePTT model. The red curve illustrates the surrogate model post-training. The black [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

79 extracted references · 77 canonical work pages

[1]

Nonlinear rheology of colloidal dispersions

J M Brader. Nonlinear rheology of colloidal dispersions. Journal of Physics: Condensed Matter, 22(36):363101, aug 2010

work page 2010
[2]

M. I. Jordan and T. M. Mitchell. Machine learning: Trends, perspectives, and prospects. Science, 349(6245): 255–260, jul 2015

work page 2015
[3]

Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang

George Em Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics- informed machine learning. Nature Reviews Physics, 3(6):422–440, may 2021

work page 2021
[4]

What is SciML?, 2024

SciML Research Group - Brown University. What is SciML?, 2024. URL https://sites.brown.edu/ bergen-lab/research/what-is-sciml/ . Accessed: 08-05-2024

work page 2024
[5]

Scientific Machine Learning, 2024

Joost Iwema. Scientific Machine Learning, 2024. URL https://sciml.wur.nl/reviews/sciml/sciml. html. Accessed: 08-05-2024

work page 2024
[6]

Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review

Sohrab Zendehboudi, Nima Rezaei, and Ali Lohi. Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review. Applied Energy, 228:2539–2566, oct 2018

work page 2018
[7]

Tartakovsky, Jeff Burghardt, and Eric Darve

Kailai Xu, Alexandre M. Tartakovsky, Jeff Burghardt, and Eric Darve. Learning viscoelasticity models from indirect data using deep neural networks. Computer Methods in Applied Mechanics and Engineering, 387:114124, December 2021. ISSN 0045-7825

work page 2021
[8]

Rausch, Francisco Sahli Costabal, and Adrian Buganza Tepole

Vahidullah Taç, Manuel K. Rausch, Francisco Sahli Costabal, and Adrian Buganza Tepole. Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations. Computer Methods in Applied Mechanics and Engineering, 411:116046, June 2023. ISSN 0045-7825

work page 2023

Show all 79 references

[9]

A mechanics-informed deep learning framework for data-driven nonlinear viscoelasticity

Faisal As’ad and Charbel Farhat. A mechanics-informed deep learning framework for data-driven nonlinear viscoelasticity. Computer Methods in Applied Mechanics and Engineering, 417:116463, December 2023. ISSN 0045-7825

2023
[10]

Unsupervised discovery of interpretable hyperelastic constitutive laws

Moritz Flaschel, Siddhant Kumar, and Laura De Lorenzis. Unsupervised discovery of interpretable hyperelastic constitutive laws. Computer Methods in Applied Mechanics and Engineering, 381:113852, August 2021. ISSN 0045-7825

2021
[11]

Bayesian-euclid: Discovering hyperelastic material laws with uncertainties

Akshay Joshi, Prakash Thakolkaran, Yiwen Zheng, Maxime Escande, Moritz Flaschel, Laura De Lorenzis, and Siddhant Kumar. Bayesian-euclid: Discovering hyperelastic material laws with uncertainties. Computer Methods in Applied Mechanics and Engineering, 398:115225, August 2022. I...

2022
[12]

Constitutive model characterization and discovery using physics-informed deep learning

Ehsan Haghighat, Sahar Abouali, and Reza Vaziri. Constitutive model characterization and discovery using physics-informed deep learning. Engineering Applications of Artificial Intelligence, 120:105828, April 2023. ISSN 0952-1976

2023
[13]

Roy and Suman Guha

Arunabha M. Roy and Suman Guha. A data-driven physics-constrained deep learning computational framework for solving von mises plasticity. Engineering Applications of Artificial Intelligence, 122:106049, June 2023. ISSN 0952-1976

2023
[14]

Physics-informed few-shot deep learning for elastoplastic constitutive relationships

Chen Wang, You-quan He, Hong-ming Lu, Jian-guo Nie, and Jian-sheng Fan. Physics-informed few-shot deep learning for elastoplastic constitutive relationships. Engineering Applications of Artificial Intelligence , 126: 106907, November 2023. ISSN 0952-1976

2023
[15]

Raissi, P

M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, feb 2019

2019
[16]

Scientific machine learning through physics–informed neural networks: Where we are and what’s next

Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing, 92(3), jul 2022

2022
[17]

Hartt, George Em Karniadakis, and Safa Jamali

Mohammadamin Mahmoudabadbozchelou, Marco Caggioni, Setareh Shahsavari, William H. Hartt, George Em Karniadakis, and Safa Jamali. Data-driven physics-informed constitutive metamodeling of complex fluids: A multifidelity neural network (MFNN) framework. Journal of Rheology, 65(2...

2021
[18]

Rheology-informed neural networks (RhINNs) for forward and inverse metamodelling of complex fluids

Mohammadamin Mahmoudabadbozchelou and Safa Jamali. Rheology-informed neural networks (RhINNs) for forward and inverse metamodelling of complex fluids. Scientific Reports, 11(1), jun 2021

2021
[19]

Kamani, Simon A

Mohammadamin Mahmoudabadbozchelou, Krutarth M. Kamani, Simon A. Rogers, and Safa Jamali. Digital rheometer twins: Learning the hidden rheology of complex fluids through rheology-informed graph neural networks. Proceedings of the National Academy of Sciences, 119(20), may 2022....

2022
[20]

Data-driven selection of constitutive models via rheology-informed neural networks (rhinns)

Milad Saadat, Mohammadamin Mahmoudabadbozchelou, and Safa Jamali. Data-driven selection of constitutive models via rheology-informed neural networks (rhinns). Rheologica Acta, 61(10):721–732, aug 2022. ISSN 1435-1528

2022
[21]

Fractional rheology-informed neural networks for data-driven identification of viscoelastic constitutive models

Donya Dabiri, Milad Saadat, Deepak Mangal, and Safa Jamali. Fractional rheology-informed neural networks for data-driven identification of viscoelastic constitutive models. Rheologica Acta, 62(10):557–568, August 2023. ISSN 1435-1528

2023
[22]

Nagrani, Ritwik V

Pranay P. Nagrani, Ritwik V . Kulkarni, Parth U. Kelkar, Ria D. Corder, Kendra A. Erk, Amy M. Marconnet, and Ivan C. Christov. Data-driven rheological characterization of stress buildup and relaxation in thermal greases. Journal of Rheology, 67(6):1129–1140, oct 2023. ISSN 1520-8516

2023
[23]

Ardekani

Sukirt Thakur, Maziar Raissi, and Arezoo M. Ardekani. Viscoelasticnet: A physics informed neural network framework for stress discovery and model selection. Journal of Non-Newtonian Fluid Mechanics, 330:105265, August 2024. ISSN 0377-0257

2024
[24]

Understanding and mitigating gradient pathologies in physics- informed neural networks

Sifan Wang, Yujun Teng, and Paris Perdikaris. Understanding and mitigating gradient pathologies in physics- informed neural networks. Preprinted at arXiv, 2020

2020
[25]

When and why PINNs fail to train: A neural tangent kernel perspective

Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449:110768, jan 2022

2022
[26]

Rohrhofer, Stefan Posch, Clemens Gobnitzer, and Bernhard C

Franz M. Rohrhofer, Stefan Posch, Clemens Gobnitzer, and Bernhard C. Geiger. Understanding the difficulty of training physics-informed neural networks on dynamical systems. Preprinted at arXiv, 2022

2022
[27]

Park, and Kyung Hyun Ahn

Howon Jin, Sangwoong Yoon, Frank C. Park, and Kyung Hyun Ahn. Data-driven constitutive model of complex fluids using recurrent neural networks. Rheologica Acta, 62(10):569–586, 2023

2023
[28]

Young, Patrick T

Charles D. Young, Patrick T. Corona, Anukta Datta, Matthew E. Helgeson, and Michael D. Graham. Scattering- informed microstructure prediction during lagrangian evolution (simple)—a data-driven framework for modeling complex fluids in flow. Rheologica Acta, 62(10):587–604, Sept...

2023
[29]

Brunton, Joshua L

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences , 113(15): 3932–3937, mar 2016

2016
[30]

Kamani, Simon A

Mohammadamin Mahmoudabadbozchelou, Krutarth M. Kamani, Simon A. Rogers, and Safa Jamali. Unbiased construction of constitutive relations for soft materials from experiments via rheology-informed neural networks. Proceedings of the National Academy of Sciences, 121(2), January ...

2024
[31]

Naozuka, Heber L

Gustavo T. Naozuka, Heber L. Rocha, Renato S. Silva, and Regina C. Almeida. Sindy-sa framework: enhancing nonlinear system identification with sensitivity analysis. Nonlinear Dynamics, 110(3):2589–2609, August 2022. ISSN 1573-269X

2022
[32]

Universal differential equations for scientific machine learning

Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, and Ali Jasim Ramadhan. Universal differential equations for scientific machine learning. Preprinted at ArXiv, 2021

2021
[33]

Differentiable programming for differential equations: A review

Facundo Sapienza, Jordi Bolibar, Frank Schäfer, Brian Groenke, Avik Pal, Victor Boussange, Patrick Heimbach, Giles Hooker, Fernando Pérez, Per-Olof Persson, and Christopher Rackauckas. Differentiable programming for differential equations: A review. Preprinted at ArXiv, 2024

2024
[34]

Junbang Liang and Ming C. Lin. Differentiable physics simulation. In ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations, pages 1–5, 2019

2020
[35]

Differentiable physics: A position piece

Venkatasubramanian Viswanathan Bharath Ramsundar, Dilip Krishnamurthy. Differentiable physics: A position piece. Preprinted at arXiv, 2024

2024
[36]

Physics-based Deep Learning

Nils Thuerey, Philipp Holl, Maximilian Mueller, Patrick Schnell, Felix Trost, and Kiwon Um. Physics-based Deep Learning. 2021. URL https://physicsbaseddeeplearning.org

2021
[37]

Structural identification with physics-informed neural ordinary differential equations

Zhilu Lai, Charilaos Mylonas, Satish Nagarajaiah, and Eleni Chatzi. Structural identification with physics-informed neural ordinary differential equations. Journal of Sound and Vibration, 508:116196, sep 2021

2021
[38]

Alvarez, and Daniel Bedoya-Ruíz

Sebastián Delgado-Trujillo, Diego A. Alvarez, and Daniel Bedoya-Ruíz. Hysteresis modeling of structural systems using physics-guided universal ordinary differential equations. Computers and Structures, 280:106988, May 2023. ISSN 0045-7949

2023
[39]

J. Koch. Data-driven surrogates of rotating detonation engine physics with neural ordinary differential equations and high-speed camera footage. Physics of Fluids, 33(9), September 2021. ISSN 1089-7666. 23 A PREPRINT - MAY 26, 2025

2021
[40]

Neural network aided approximation and parameter inference of non-markovian models of gene expression

Qingchao Jiang, Xiaoming Fu, Shifu Yan, Runlai Li, Wenli Du, Zhixing Cao, Feng Qian, and Ramon Grima. Neural network aided approximation and parameter inference of non-markovian models of gene expression. Nature Communications, 12(1), May 2021. ISSN 2041-1723

2021
[41]

Brendan Keith, Akshay Khadse, and Scott E. Field. Learning orbital dynamics of binary black hole systems from gravitational wave measurements. Physical Review Research, 3(4):043101, November 2021. ISSN 2643-1564

2021
[42]

Idelfonso B. R. Nogueira, Vinicius V . Santana, Ana M. Ribeiro, and Alírio E. Rodrigues. Using scientific machine learning to develop universal differential equation for multicomponent adsorption separation systems. The Canadian Journal of Chemical Engineering, 100(9):2279–229...

2022
[43]

Santana, Erbet Costa, Carine M

Vinicius V . Santana, Erbet Costa, Carine M. Rebello, Ana Mafalda Ribeiro, Christopher Rackauckas, and Idelfonso B.R. Nogueira. Efficient hybrid modeling and sorption model discovery for non-linear advection- diffusion-sorption systems: A systematic scientific machine learning...

2023
[44]

Capturing missing physics in climate model parameterizations using neural differential equations

Ali Ramadhan, John Marshall, Andre Souza, Xin Kai Lee, Ulyana Piterbarg, Adeline Hillier, Gregory LeClaire Wagner, Christopher Rackauckas, Chris Hill, Jean-Michel Campin, and Raffaele Ferrari. Capturing missing physics in climate model parameterizations using neural differenti...

2020
[45]

Universal differential equations for glacier ice flow modelling

Jordi Bolibar, Facundo Sapienza, Fabien Maussion, Redouane Lguensat, Bert Wouters, and Fernando Pérez. Universal differential equations for glacier ice flow modelling. Geoscientific Model Devolopment, 2023

2023
[46]

Lennon, Gareth H

Kyle R. Lennon, Gareth H. McKinley, and James W. Swan. Scientific machine learning for modeling and simulating complex fluids. Proceedings of the National Academy of Sciences, 120(27), jun 2023

2023
[47]

Beris, Jeffrey S

Antony N. Beris, Jeffrey S. Horner, Soham Jariwala, Matthew J. Armstrong, and Norman J. Wagner. Recent advances in blood rheology: a review. Soft Matter, 17(47):10591–10613, 2021

2021
[48]

Ardekani

Gaojin Li, Eric Lauga, and Arezoo M. Ardekani. Microswimming in viscoelastic fluids.Journal of Non-Newtonian Fluid Mechanics, 297:104655, November 2021

2021
[49]

Rheological properties of synovial fluids

H Fam, JT Bryant, and M Kontopoulou. Rheological properties of synovial fluids. Biorheology, 44(2):59–74, 2007

2007
[50]

Rheology and processing of polymeric materials: Volume 1: Polymer Rheology , volume 1

Chang Dae Han. Rheology and processing of polymeric materials: Volume 1: Polymer Rheology , volume 1. Oxford University Press, 2007

2007
[51]

Polymers and rheology: A tale of give and take

Leire Sangroniz, Mercedes Fernández, and Antxon Santamaria. Polymers and rheology: A tale of give and take. Polymer, 271:125811, April 2023. ISSN 0032-3861

2023
[52]

Clerk Maxwell

J. Clerk Maxwell. On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London, 157:49–88, 1867
[53]

Ch. W. Macosko. Rheology-Principles, Measurements, and Applications (Advances in Interfacial Engineering). Wiley-VCH, 1994

1994
[54]

On the formulation of rheological equations of state

James Gardner Oldroyd. On the formulation of rheological equations of state. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 200(1063):523–541, feb 1950

1950
[55]

Continuum Mechanics

I.-Shih Liu. Continuum Mechanics. Springer Berlin Heidelberg, 2002. ISBN 9783642077029

2002
[56]

The role of the gordon–schowalter derivative term in the constitutive models—improved flexibility of the modified xpp model

Radek Pivokonsky, Petr Filip, and Jana Zelenkova. The role of the gordon–schowalter derivative term in the constitutive models—improved flexibility of the modified xpp model. Colloid and Polymer Science, 293(4): 1227–1236, January 2015. ISSN 1435-1536

2015
[57]

Ronald G. Larson. Constitutive equations for polymer melts and solutions. Butterworths, 1988. ISBN 0409901199

1988
[58]

Giesekus

H. Giesekus. A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. Journal of Non-Newtonian Fluid Mechanics, 11(1–2):69–109, January 1982. ISSN 0377-0257

1982
[59]

Phan-Thien

N. Phan-Thien. A nonlinear network viscoelastic model. Journal of Rheology, 22(3):259–283, June 1978. ISSN 1520-8516

1978
[60]

Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. Neural ordinary differential equations, 2018

2018
[61]

Reynolds averaged turbulence modelling using deep neural networks with embedded invariance

Julia Ling, Andrew Kurzawski, and Jeremy Templeton. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. Journal of Fluid Mechanics, 807:155–166, oct 2016

2016
[62]

G.F. Smith. On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. International Journal of Engineering Science, 9(10):899–916, October 1971. ISSN 0020-7225. 24 A PREPRINT - MAY 26, 2025

1971
[63]

van de Ven, Tinne Tuytelaars, and Andreas S

Gido M. van de Ven, Tinne Tuytelaars, and Andreas S. Tolias. Three types of incremental learning. Nature Machine Intelligence, 4:1185–1197, 2022

2022
[64]

Runge–kutta pairs of order 5 (4) satisfying only the first column simplifying assumption

Ch Tsitouras. Runge–kutta pairs of order 5 (4) satisfying only the first column simplifying assumption. Computers & Mathematics with Applications, 62(2):770–775, 2011

2011
[65]

Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia

Christopher Rackauckas and Qing Nie. Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia. Journal of Open Research Software, 5(1):15, 2017

2017
[66]

Kingma and Jimmy Ba

Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. The International Conference on Learning Representations (ICLR), 2015

2015
[67]

Optimization.jl: A unified optimization package

Vaibhav Kumar Dixit and Christopher Rackauckas. Optimization.jl: A unified optimization package. Zenodo, 2023

2023
[68]

Maklad and R.J

O. Maklad and R.J. Poole. A review of the second normal-stress difference; its importance in various flows, measurement techniques, results for various complex fluids and theoretical predictions. Journal of Non-Newtonian Fluid Mechanics, 292:104522, June 2021

2021
[69]

A. A. Collyer. Rheological Measurement. Springer Netherlands, 1998

1998
[70]

de Souza Mendes and Roney L

Paulo R. de Souza Mendes and Roney L. Thompson. A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids. Rheologica Acta, 52(7):673–694, apr 2013

2013
[71]

Thompson, Alexandra A

Roney L. Thompson, Alexandra A. Alicke, and Paulo R. de Souza Mendes. Model-based material functions for SAOS and LAOS analyses. Journal of Non-Newtonian Fluid Mechanics, 215:19–30, 2015

2015
[72]

de Souza Mendes, Roney L

Paulo R. de Souza Mendes, Roney L. Thompson, Alexandra A. Alicke, and Ricardo T. Leite. The quasilinear large-amplitude viscoelastic regime and its significance in the rheological characterization of soft matter. Journal of Rheology, 58:537–561, 2014

2014
[73]

Constructing a thixotropy model from rheological experiments

Paulo R de Souza Mendes, Behbood Abedi, and Roney L Thompson. Constructing a thixotropy model from rheological experiments. Journal of Non-Newtonian Fluid Mechanics, 261:1–8, 2018

2018
[74]

Distilling the knowledge in a neural network

Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. Preprinted at ArXiv, 2015

2015
[75]

Maybank, and Dacheng Tao

Jianping Gou, Baosheng Yu, Stephen J. Maybank, and Dacheng Tao. Knowledge distillation: A survey. Interna- tional Journal of Computer Vision, 129(6):1789–1819, 2021

2021
[76]

Towards a theory of model distillation

Enric Boix-Adsera. Towards a theory of model distillation. Preprinted at ArXiv, 2024

2024
[77]

Model compression

Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, page 535–541, 2006

2006
[78]

Reynolds and Weissenberg numbers in viscoelastic flows

Roney L Thompson and Cassio M Oishi. Reynolds and Weissenberg numbers in viscoelastic flows. Journal of Non-Newtonian Fluid Mechanics, 292(104550), 2021

2021
[79]

On more insightful dimension- less numbers for computational viscoelastic rheology

Rafael A Figueiredo, Cassio M Oishi, Fernando T Pinho, and Roney L Thompson. On more insightful dimension- less numbers for computational viscoelastic rheology. Journal of Non-Newtonian Fluid Mechanics, 331(105282), 2024. 25

2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Nonlinear rheology of colloidal dispersions

J M Brader. Nonlinear rheology of colloidal dispersions. Journal of Physics: Condensed Matter, 22(36):363101, aug 2010

work page 2010

[2] [2]

M. I. Jordan and T. M. Mitchell. Machine learning: Trends, perspectives, and prospects. Science, 349(6245): 255–260, jul 2015

work page 2015

[3] [3]

Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang

George Em Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics- informed machine learning. Nature Reviews Physics, 3(6):422–440, may 2021

work page 2021

[4] [4]

What is SciML?, 2024

SciML Research Group - Brown University. What is SciML?, 2024. URL https://sites.brown.edu/ bergen-lab/research/what-is-sciml/ . Accessed: 08-05-2024

work page 2024

[5] [5]

Scientific Machine Learning, 2024

Joost Iwema. Scientific Machine Learning, 2024. URL https://sciml.wur.nl/reviews/sciml/sciml. html. Accessed: 08-05-2024

work page 2024

[6] [6]

Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review

Sohrab Zendehboudi, Nima Rezaei, and Ali Lohi. Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review. Applied Energy, 228:2539–2566, oct 2018

work page 2018

[7] [7]

Tartakovsky, Jeff Burghardt, and Eric Darve

Kailai Xu, Alexandre M. Tartakovsky, Jeff Burghardt, and Eric Darve. Learning viscoelasticity models from indirect data using deep neural networks. Computer Methods in Applied Mechanics and Engineering, 387:114124, December 2021. ISSN 0045-7825

work page 2021

[8] [8]

Rausch, Francisco Sahli Costabal, and Adrian Buganza Tepole

Vahidullah Taç, Manuel K. Rausch, Francisco Sahli Costabal, and Adrian Buganza Tepole. Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations. Computer Methods in Applied Mechanics and Engineering, 411:116046, June 2023. ISSN 0045-7825

work page 2023