Inferring viscoplastic models from velocity fields: a physics-informed neural network approach

Martin Lardy; Sham Tlili; Simon Gsell

REVIEW 3 major objections 4 minor 43 references

Inferring viscoplastic models from velocity fields: a physics-informed neural network approach

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A physics-informed neural network can recover viscoplastic rheological parameters from a single noisy two-dimensional velocity snapshot without knowing boundary conditions.

desk verdict A solid PINN study that recovers dimensionless flow-curve shape from velocity snapshots, but the absolute viscosity scale is unidentifiable from Stokes data and the paper should say so. read the letter →

arxiv 2506.17681 v1 pith:BLP2M5CX submitted 2025-06-21 physics.flu-dyn

Martin Lardy , Sham Tlili , Simon Gsell This is my paper · ORCID

classification physics.flu-dyn

keywords rheologyinferencephysics-informedneuralnetworksviscoplasticfluidsHerschel–BulkleymodelCarreauPapanastasiouvelocityfielddataconstitutiveselection

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 claims that the full rheological behavior of a viscoplastic fluid can be read off from a single two-dimensional velocity field, without knowing the boundary conditions. The authors build a physics-informed neural network that represents the velocity through a stream function, uses automatic differentiation to form the Stokes momentum residual with a parameterized viscosity law, and trains the network and the rheological parameters together against noisy velocity data. On synthetic flows generated from Herschel–Bulkley, Carreau, and Papanastasiou models, the inferred parameters and reconstructed viscosity curves match the simulations, and a model-selection variant picks the correct constitutive law among the three candidates. If this holds beyond the simulated benchmark cases, it would give an in situ alternative to laboratory rheometry for materials such as biological tissues, foams, emulsions, and geophysical fluids that are hard to load into a rheometer.

What carries the argument

The load-bearing object is a physics-informed neural network whose output is not the velocity directly but a stream function $\psi_\theta$, so the incompressibility condition $\nabla\cdot U^* = 0$ is satisfied by construction. Automatic differentiation of the network produces velocity gradients and pressure gradients needed to evaluate the non-dimensional Stokes residual $L_{\mathrm{phy}} = \|-\kappa\nabla P^*_\theta + \nabla\cdot(2\eta^*_\alpha S^*_\theta)\|$, where $\eta^*_\alpha$ is one of the three parameterized constitutive laws; minimizing this residual jointly with the data loss is what turns raw velocity samples into rheological parameters. For model selection, the machinery is the AIC-style total physical loss $L_{\mathrm{phy,tot}} = \sum_i \beta_i(L_{\mathrm{phy},i} + \|\eta_i\|_0)$ with learned activation weights $\beta_i$ and a sparsity loss $L_\beta = 2\max_i(n_i)|1-\sum_i \beta_i|$, which forces convergence to a single candidate model without always selecting the most complex one.

What would settle it

Take a two-dimensional creeping microchannel flow of a well-characterized viscoplastic gel, measure the velocity field by particle image velocimetry, and run the algorithm with the three candidate models; compare the inferred yield stress and power-law index with independent rheometer measurements. If the selected model is wrong or the inferred parameters fall outside the rheometer confidence intervals, the claim that rheology can be inferred from velocity fields alone outside synthetic benchmarks is disproved.

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Extended reading notes

Core claim

Working in the creeping Stokes regime and assuming incompressibility, the paper treats rheology inference as an inverse problem: find a generalized-Newtonian viscosity function $\eta(\dot\gamma)$ and a pressure field $P$ such that the non-dimensional momentum balance $\nabla P^* = \nabla\cdot(2\eta^* S^*)$ is satisfied by the observed velocity samples. A fully connected network outputs a stream function $\psi_\theta$ and pressure; the velocity is recovered as $u_\theta = \partial\psi_\theta/\partial y^*$ and $v_\theta = -\partial\psi_\theta/\partial x^*$, which enforces incompressibility exactly. The total loss combines a data-fidelity term on the measured velocities, a physical residual built from the Stokes equation using automatic differentiation, and a positivity constraint on the inferred rheological parameters. The paper demonstrates on lattice-Boltzmann synthetic velocity fields that the parameters of Herschel–Bulkley, Carreau, and Papanastasiou models can be recovered from a few hundred sample points, that inference tolerates Gaussian or uniform noise up to roughly 10–20% amplitude, and that a shear-rate-biased sampling strategy improves accuracy by widening the interquartile range of probed shear rates. A model-selection extension introduces per-model weights $\beta_i$, an AIC-like complexity penalty $\|\eta_i\|_0$, and a sparsity loss; in 30-run tests it selects the generating model for each of the three candidates, with occasional confusion only between Herschel–Bulkley and its regularized Papanastasiou limit.

Load-bearing premise

The benchmark velocity fields are exact solutions of the same two-dimensional steady Stokes equations and generalized-Newtonian viscosity laws that are encoded in the network's physical loss, so the tests never check the method against a flow whose true physics differs from the candidate models; an experimental flow with inertia, three-dimensionality, wall slip, or a different constitutive behavior could break the inference.

Editorial extensions

If this is right

Given a correct model family, rheological parameters can be recovered from a single two-dimensional velocity snapshot, with the pressure field reconstructed as a by-product and no boundary conditions supplied.
The method tolerates significant measurement noise: with up to 10–20% artificial Gaussian or uniform noise on velocity samples, inferred parameters stay close to their true values and the network output is a denoised velocity field.
Sampling strategy matters: biased sampling toward high-shear regions increases the shear-rate interquartile range and consistently lowers the viscosity inference error across uniform-channel, wavy-channel, and obstacle geometries.
Model selection among Herschel–Bulkley, Carreau, and Papanastasiou candidates picks the correct law in most runs, with the only systematic confusion occurring between Herschel–Bulkley and its regularized Papanastasiou limit.
Because boundary conditions are not needed, the approach is positioned for experimental velocimetry data in microfluidic and geophysical contexts where stress boundary conditions are unknown.

Reading between the lines

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

A natural next test beyond the paper is to apply the same pipeline to experimental particle-image-velocimetry fields of a well-characterized viscoplastic gel and compare the inferred yield stress and power-law index with rheometer measurements; the present benchmark cannot say whether the method survives real measurement noise and boundary slip.
The AIC-style selection loss suggests a Bayesian reading: the weights $\beta_i$ could be interpreted as posterior model probabilities, so a practical extension would be to report uncertainty in model choice from repeated runs rather than a single winning model.
The shear-rate-distribution result implies a concrete experimental-design guideline: choose channel geometries and obstacle placements that maximize the range of shear rates sampled, for example by combining a wide channel and a narrow constriction in the same field of view.
If the method generalizes to three dimensions and inertial corrections, it could become a non-contact rheometer that maps viscosity heterogeneities spatially, which the paper hints at but does not demonstrate.

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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

The stress-test note is correct and it matters. The Stokes momentum balance is invariant under simultaneous rescaling of pressure and viscosity, and because the pressure is a free network output, only dimensionless ratios are recoverable from velocity alone. The authors' own normalization shows this, but they never state it. The abstract and title promise inferring viscoplastic models from velocity fields alone, which reads as recovering dimensional constitutive parameters. In fact, what is recovered is the shape of the flow curve in dimensionless form, e.g. the Bingham number and power-law index for Herschel-Bulkley, not the consistency k or the yield stress magnitude. An independent pressure-drop or force measurement would close the scale. This is a mathematical property of Stokes flow, not a numerical artifact, and the paper needs a prominent identifiability discussion before any experimental claims. That said, the paper does something concrete and useful. It demonstrates that a simple fully connected PINN can infer Herschel-Bulkley, Carreau, and Papanastasiou parameters from a single 2D velocity snapshot, with robustness to added noise. The shear-rate-biased sampling analysis is a nice practical contribution, and the AIC-style model-selection loss (Eq. 18) is a reasonable way to choose among candidate laws, with the sparsity loss preventing degenerate solutions. The code is released. Prior PINN work covered power-law and viscoelastic fluids, but this specific application to these three viscoplastic families is new, and the systematic study of how sampling distribution affects inference is a genuine plus. The weak spots beyond identifiability are two. First, the pressure rescaling factor kappa for Herschel-Bulkley is set to gamma_dot^(n-1), which depends on the unknown power-law index n we are trying to infer. That looks like a leak. It may be just a training convenience, but the authors owe readers an explanation of why the inference is not trivialized by an n-dependent loss. Second, all validation uses synthetic fields generated by the same equation family used in the physical loss. The model-selection test is also constrained by forcing Papanastasiou parameters above 1, which dodges the most interesting overlap between models. These are common limitations in PINN papers, but they should be stated plainly. This paper deserves a serious referee. The central idea is sound, the numerical work is reproducible, and the pitfalls are addressable. I would send it out, with the request that the authors add an identifiability section and address the kappa concern. For readers working on rheology inference or microfluidics, it is worth a look; I would not cite it as evidence for recovering absolute viscosities, only for dimensionless curve shape.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a physics-informed neural network (PINN) framework for inferring generalized-Newtonian/viscoplastic constitutive laws from planar velocity field data. The network represents the velocity through a streamfunction and also outputs pressure; the loss combines a data term (Eq. 11) with the residual of the dimensionless Stokes momentum balance (Eq. 12), while the viscosity is parameterized by one of three candidate models: Herschel-Bulkley (Eq. 5), Carreau (Eq. 7), or Papanastasiou (Eq. 9). The rheological parameters and the network weights are optimized jointly. The authors validate the method on lattice-Boltzmann synthetic velocity fields in several confined geometries, study the effect of collocation point number and noise, propose a shear-rate-biased sampling strategy to improve inference, and introduce an AIC-like model-selection loss (Eq. 18) to choose among candidate constitutive laws. They report accurate recovery of the dimensionless rheological parameters, robustness to artificial noise, and correct model selection in most runs.

Significance. If the claims hold, the paper offers a practical, code-available tool for estimating dimensionless constitutive parameters from a single noisy velocity snapshot without explicit boundary-condition knowledge, and its analysis of shear-rate sampling is a useful practical contribution. The model-selection mechanism with an explicit complexity penalty is a meaningful step beyond simple parameter fitting. The principal caveats are that the absolute viscosity scale is not identifiable from Stokes velocity data, so only dimensionless ratios such as σ0/(kγdot_c^n), η∞/η0, and τ0/(μ0γdot_c) are recovered, and that the validation is entirely in-distribution, generated by the same PDE family used in the physical loss. These caveats do not invalidate the method but materially bound what the title and abstract can claim, and they should be stated explicitly.

major comments (3)

[§2.1–2.2, Table 1] The inference target is the set of dimensionless ratios α_i, not dimensional material constants. Equation (1b) is invariant under the simultaneous rescaling (U, P, η) → (U, cP, cη), so neither the consistency k in the Herschel-Bulkley model, nor η0 in the Carreau model, nor μ0 in the Papanastasiou model can be identified from velocity data alone; only α0 = σ0/(kγdot_c^n), η∞/η0, and τ0/(μ0γdot_c) are identifiable. The manuscript never states this limitation explicitly, and the title and abstract can be read as promising dimensional constitutive parameters. This is a mathematical property of the inverse problem, not a training or noise artifact, so the authors should add an explicit identifiability statement and qualify the claims accordingly.
[§2.3, Eq. (12)] The pressure rescaling for the Herschel-Bulkley model is κ = γdot_c^{n−1}, and n is precisely the inferred parameter α1. If κ is evaluated with the true n, the Herschel-Bulkley inference experiments in Section 3 use the answer in the loss; if κ is evaluated with the current estimate of n, the physical loss is not the non-dimensional Stokes residual (3b), and after minimization over the unconstrained pressure network the residual carries an n-dependent prefactor that can bias the inference of n independently of the constitutive fit. For γdot_c ≠ 1 the formula is also not dimensionless. Please specify whether κ uses the true or current n, justify the choice, or set κ to a constant independent of the inferred parameters (e.g., κ = 1).
[§2.4 and §6] The validation is entirely synthetic and in-distribution: the lattice-Boltzmann benchmark solves the same steady Stokes equations and the same three constitutive laws used in the physical loss (12), so the experiments never probe robustness to model-form error such as inertia, three-dimensionality, boundary slip, or a rheology outside the candidate set. The Section 6 claim that the method is 'particularly promising for analyzing experimental data' therefore goes beyond what is demonstrated. The AIC-like penalty in Eq. (18) is an appropriate external criterion and mitigates circularity for the model-selection part, but the parameter-inference experiments would benefit from at least one out-of-distribution test, for example data from a mildly inertial or 3D flow, a slip boundary, or a constitutive law not in the candidate set, or an explicit statement that experimental applicability is untested.

minor comments (4)

[Figure 3] The pressure comparison uses an ad hoc scaling factor a chosen to minimize the absolute difference; since the pressure level and scale are arbitrary in this inverse problem, the reported pressure error should be defined after this optimal rescaling and the normalization should be stated in the caption.
[Eq. (18)] The notation ∥η_i∥0 for the number of parameters of model i is misleading because η_i is a function, not a vector; please rename this quantity n_i or |α_i|.
[Global] There are several typos and placeholder fields: 'Panastasiou' in the abstract, 'Taken togher' in Section 5, 'IRQ' for IQR in the Figure 7 caption, and PACS/MSC codes '0000, 1111' in the header.
[Section 4] The biased-sampling discussion would benefit from clarifying that the inter-quartile range is computed on the sampled shear-rate values after resampling, not on the full shear-rate field; the current text mentions binning the field but the distinction is important for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inference pipeline is a genuine inversion benchmark, and the main caveats are identifiability and validation limitations, not circular reductions.

full rationale

The derivation chain is self-contained rather than circular. The velocity data are produced by an independent lattice-Boltzmann forward solver ([36]) using Herschel-Bulkley, Carreau, and Papanastasiou models, while the PINN minimizes a data term (11) and the nondimensional Stokes residual (12) with respect to parameters α that are not defined from the target snapshots. This is a standard forward-model/inverse-problem pairing, not a prediction equivalent to its input by construction. The κ prefactor in Eq. (12) is a pure gauge: because pressure is an unconstrained network output, replacing P* by κP* leaves the physical loss unchanged for any constant κ, so the choice κ=γdot_c^{n-1} cannot smuggle the true n into the inference. The model-selection penalty ∥η_i∥0 in Eq. (18) is an external AIC-like regularizer, not a fitted quantity. The self-citations ([30], [31], [36]) concern experimental platforms and a numerical method; none is invoked as authority for the inference claim. The genuine caveats are that benchmarks use synthetic data generated from the same constitutive family as the loss, and that Stokes flow has an absolute-scale degeneracy (η→cη, P→cP with U fixed), so only dimensionless parameter ratios are identifiable; the paper itself defers experimental application to future work and calls for deeper identifiability studies in Section 6.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the correctness of the Stokes-flow model, the fidelity of the synthetic benchmark generator, the adequacy of the chosen network to represent pressure and velocity, and the hand-tuned training hyperparameters. No new physical entity is introduced.

free parameters (5)

Pressure rescaling factor κ (Herschel-Bulkley) = γ̇c^(n-1) (text ambiguous; n is the true power-law index)
Chosen in Section 2.3 to rescale the pressure residual for Herschel-Bulkley flows. If n is the ground-truth value, the optimizer is partially informed of the answer it is supposed to recover.
Loss-balance weight W = 10^w-1 with w = floor(log10(Ldata/Lphy))+1 at initialization
Ad hoc schedule in Section 2.3; the final inferred parameters depend on this balance.
Shear-bias exponent c = 0, 1, 4 in Section 4
User-defined parameter in Eq. (16) controlling sampling; inference accuracy improves as c grows, so the reported accuracy is partly tuned by this choice.
Model-selection slope r = 10
Set in Section 5 for the activation β_i (Eq. 19); no sensitivity or robustness analysis is given.
Network architecture and learning rate = 4 hidden layers of 20 tanh neurons; learning rate 0.0025
Fixed across all experiments (Section 2.3); no study of how architecture affects the inference.

assumptions (5)

domain assumption Flows are steady, incompressible, two-dimensional creeping flows obeying ∇·U=0 and ∇P=∇·(2ηS(U)) (Eq. 1).
The physical loss is built from Eq. (3); the inference is only valid if the true flow satisfies this model.
domain assumption The true viscosity belongs to one of the three candidate laws (Herschel-Bulkley, Carreau, Papanastasiou).
The inverse problem assumes the model class; Section 5 further constrains Papanastasiou parameters to keep it distinct from Herschel-Bulkley, so unconstrained real materials may be misidentified.
domain assumption Lattice-Boltzmann simulations (Ref [36]) faithfully solve Stokes flow with the prescribed viscosity.
All ground-truth values are taken from these simulations; solver errors near yield surfaces or boundaries propagate into the error metric (15) and the inferred parameters.
domain assumption A 4x20 tanh network represents velocity and pressure accurately enough that automatic-differentiation gradients are reliable, including in unyielded plug regions.
No approximation error or convergence analysis is provided; the paper's central mechanism depends on these gradients being accurate.
domain assumption Artificial noise is additive and zero-mean Gaussian or uniform (Section 3.2).
Real experimental noise may be correlated or biased; the authors note a Newtonian bias under strong noise but do not study other noise structures.

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Pith. "Pith review of Inferring viscoplastic models from velocity fields: a physics-informed neural network approach." pith.science (2026). https://pith.science/paper/BLP2M5CX

@misc{pith2026250617681,
  author       = {Pith},
  title        = {Pith review of: Inferring viscoplastic models from velocity fields: a physics-informed neural network approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLP2M5CX}},
  note         = {Machine review of arXiv:2506.17681}
}

read the original abstract

Fluid-like materials are ubiquitous, spanning from living biological tissues to geological formations, and across scales ranging from micrometers to kilometers. Inferring their rheological properties remains a major challenge, particularly when traditional rheometry fails to capture their complex, three-dimensional, and often heterogeneous behavior. This difficulty is exacerbated by system size, boundary conditions, and other material-specific physical, chemical, or thermal constraints. In this work, we explore whether rheological laws can be inferred directly from flow observations. We propose a physics-informed neural network (PINN) framework designed to learn constitutive viscoplastic laws from velocity field data alone. Our method uses a neural network to interpolate the velocity field, enabling the computation of velocity gradients via automatic differentiation. These gradients are used to estimate the residuals of the governing conservation laws, which implicitly depend on the unknown rheology. We jointly optimize both the constitutive model and the velocity field representation by minimizing the physical residuals and discrepancies from observed data. We validate our approach on synthetic velocity fields generated from numerical simulations using Herschel-Bulkley, Carreau and Panastasiou models under various flow conditions. The algorithm reliably infers rheological parameters, even in the presence of significant noise. We analyze the dependence of inference performance on flow geometry and sampling, highlighting the importance of shear rate distribution in the dataset. Finally, we explore preliminary strategies for model-agnostic inference via embedded model selection, demonstrating the potential of PINNs for identifying the most suitable rheological law from candidate models.

Figures

Figures reproduced from arXiv: 2506.17681 by the authors.

**Figure 2.** Overview of the inference process by infering Herschel-Bulkley parameters in different configura [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Velocity and pressure fields at the output of the neural network, for the three examples shown [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Impact of the number of colocation points on the inference accuracy. A series of 20 inferences [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Rheology inference on noisy velocity data. (a,b) Mean values of the infered Herschel-Bulkley [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Illustration of the shear-biased sampling in the case of the flow past an obstacle. (a-c) Sampling [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Effect of the inter-quartile range (IRQ) of the shear-rate distribution on the inference accuracy, [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Rheology model selection. (a) Overview of the various viscosity functions [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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