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arxiv: 2512.05600 · v2 · pith:BS44RAUBnew · submitted 2025-12-05 · ⚛️ physics.flu-dyn

Development of Rheological Constitutive Modeling Method Using a Sparse Identification Algorithm: A Case Study for Extensional Flows

Takeshi Sato , Souta Miyamoto , Shota Kato This is my paper

Pith reviewed 2026-05-21 18:05 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords constitutive modelingsparse identificationRheo-SINDyextensional flowGiesekus modelFENE dumbbelldata-driven rheologypolymer rheology
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The pith

Rheo-SINDy recovers the exact Giesekus model from extensional flow data and yields a simple approximate model for FENE dumbbells.

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

The paper applies the Rheo-SINDy sparse identification method to data generated under extensional flow. It first verifies that the framework exactly reconstructs the known Giesekus constitutive equation when fed data from that same model. For data from the more complex FENE dumbbell model, it produces a compact approximate constitutive model after the library of candidate terms is manually shaped with rheological insight. The resulting model reproduces key extensional properties and continues to work reasonably in regions outside the original data set.

Core claim

The Rheo-SINDy framework can identify the exact expression of the Giesekus model under extensional flow. For the FENE dumbbell model it discovers a relatively simple approximate constitutive model by manually designing the library matrix based on rheological knowledge; this identified model reasonably predicts extensional rheological properties, including an extrapolation region.

What carries the argument

Rheo-SINDy, the sparse regression procedure that selects a minimal set of terms from a user-supplied library of candidate functions to form a constitutive equation for the stress or conformation tensor.

If this is right

  • Rheo-SINDy can be used to build constitutive models directly from extensional rheometry data without first writing a full analytical theory.
  • The identified approximate model remains useful for predicting behavior in untested strain-rate regimes.
  • Manual incorporation of rheological knowledge into the candidate library is essential for obtaining compact and interpretable equations.
  • The same workflow can be tested on other flow kinematics once the library is adjusted accordingly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the library-design step can be partially automated, the method could scale to a wider range of complex fluids.
  • The approach may serve as a quick way to generate reduced-order models for use in large-scale flow simulations.
  • Extending the framework to mixed shear-plus-extension flows would test its robustness for realistic processing conditions.

Load-bearing premise

The library of candidate functions must be manually designed to contain the relevant physical terms.

What would settle it

Generate new extensional flow data for the FENE dumbbell model at strain rates outside the training range and check whether the identified approximate constitutive model deviates markedly from the full FENE simulation results.

Figures

Figures reproduced from arXiv: 2512.05600 by Shota Kato, Souta Miyamoto, Takeshi Sato.

Figure 1
Figure 1. Figure 1: shows the framework of our data-driven strategy. Assuming that the upper convected derivative expresses the derivative terms in CMs, we can write the general expression for CMs as ▽ τ = f(τ ,κ), where ▽ τ (= dτ (t)/dt−τ ·κ T−κ·τ ) is the upper convected derivative of the stress tensor τ , κ is the velocity gradient tensor, and κ T is the transpose of κ. Using the general expression for CMs, we obtain the e… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hyperparameter ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Hyperparameter ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The identified coefficient values for the FENE dumbbell model obtained by a-Lasso with [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The test simulation results for the oscillatory (a) uniaxial, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The test simulation results for the uniaxial extensional flows [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The identified coefficient values of the equation for [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Deriving constitutive models (CMs) from numerical data has been an attractive approach as a systematic CM building method. One recent study is Rheo-SINDy, which extended the sparse identification of nonlinear dynamics (SINDy) method to rheology. Although the Rheo-SINDy framework discovered an approximate CM from numerical data under shear flow, its versatility has not been investigated. To clarify its applicability to other types of flows, this study applied Rheo-SINDy to numerically generated data under extensional flow conditions. As baseline tests for extensional flow, we considered two problems: (i) whether the Rheo-SINDy framework can reproduce the famous Giesekus model from data generated by that model, and (ii) whether it can derive an approximate CM from data generated by a dumbbell model with a finite extensible nonlinear elastic (FENE) spring. For problem (i), we confirmed that Rheo-SINDy can identify the exact expression of the Giesekus model under extensional flow. For problem (ii), the Rheo-SINDy framework discovered a relatively simple expression of the approximate CM by manually designing the library matrix based on rheological knowledge. The identified approximate CM can reasonably predict extensional rheological properties of the FENE dumbbell model, including an extrapolation region. These findings demonstrate the fundamental validity of using Rheo-SINDy under extensional flow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript applies the Rheo-SINDy sparse identification method to constitutive modeling under extensional flows. It demonstrates exact recovery of the Giesekus model from data generated by that model under extensional conditions. For the FENE dumbbell model, it obtains an approximate constitutive model after manually designing the candidate function library using rheological knowledge, and reports that the resulting expression reasonably predicts extensional rheological properties including in an extrapolation region.

Significance. If the results hold, this extends the Rheo-SINDy framework beyond shear flows to extensional conditions, which is relevant for modeling stretching-dominated processes in rheology and fluid dynamics. The exact recovery of the Giesekus model from independent extensional data provides clear validation of the method's core identification capability. The FENE approximation illustrates potential for practical constitutive modeling but depends on informed library curation. The work supplies a concrete case study that could support further data-driven rheology research.

major comments (1)
  1. [Problem (ii)] Problem (ii) section: The claim that Rheo-SINDy 'discovered' the approximate constitutive model is qualified by the manual design of the library matrix based on rheological knowledge. This means the sparse regression step selects among a pre-chosen set of functional forms rather than performing blind identification from a general library. Because the reported predictive accuracy, including extrapolation, is shown only under this curated library, the interpretation of the method's versatility for unknown functional forms requires additional discussion or tests with a less informed library.
minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly separate the exact recovery result for the Giesekus model from the library-dependent approximation for the FENE case to avoid conflating the two outcomes.
  2. [Methods] Clarify the precise composition of the candidate function library used in each problem (e.g., list the basis functions or reference the supplementary material) so readers can assess the degree of prior rheological input.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. The comment on Problem (ii) raises an important point about the role of library design, which we address below by clarifying the method's scope and committing to revisions that strengthen the discussion of its applicability.

read point-by-point responses

  1. Referee: Problem (ii) section: The claim that Rheo-SINDy 'discovered' the approximate constitutive model is qualified by the manual design of the library matrix based on rheological knowledge. This means the sparse regression step selects among a pre-chosen set of functional forms rather than performing blind identification from a general library. Because the reported predictive accuracy, including extrapolation, is shown only under this curated library, the interpretation of the method's versatility for unknown functional forms requires additional discussion or tests with a less informed library.

    Authors: We agree that the library was manually designed using rheological knowledge, as already stated in the manuscript. This design choice is deliberate: an exhaustive general library without domain guidance often leads to prohibitive computational costs, ill-conditioned regression, and difficulty in interpreting results for high-dimensional rheological problems. The sparse regression step nonetheless performs genuine selection among the candidate terms, identifying which functional forms are active. The manuscript does not claim fully blind discovery for the FENE case; rather, it demonstrates that Rheo-SINDy can recover a compact, predictive model when reasonable rheological priors are incorporated into the library. To address the referee's concern about versatility for unknown functional forms, we will expand the discussion in the revised manuscript to explicitly distinguish between informed-library and broader-library scenarios, and we will add results from a test with a less curated (more general) library to illustrate the trade-offs in accuracy and sparsity. revision: yes

Circularity Check

0 steps flagged

Rheo-SINDy extensional flow tests are independent validations with no circular reduction to inputs

full rationale

The paper generates independent numerical data from the Giesekus model and FENE dumbbell model under extensional flows, then applies the Rheo-SINDy sparse regression procedure. For problem (i) it recovers the exact known Giesekus form as a verification test; for problem (ii) it selects an approximate constitutive model from a manually curated library and checks that the resulting expression reproduces target rheological properties including in an extrapolation region. These steps constitute standard model identification and out-of-sample validation rather than any self-definitional loop, fitted-input prediction, or load-bearing self-citation that collapses the claimed result back onto its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach inherits the core SINDy assumption that the governing equation is a sparse linear combination of library terms. For the FENE case an additional manual selection step is required.

free parameters (1)
  • SINDy sparsity threshold or regularization strength
    Standard hyperparameter in sparse regression methods that controls which terms are retained; value not specified in abstract.
axioms (1)
  • domain assumption The true constitutive relation lies within the span of the chosen library of candidate functions.
    Invoked when the library is constructed and when claiming the identified expression is a valid approximation.

pith-pipeline@v0.9.0 · 5796 in / 1259 out tokens · 55165 ms · 2026-05-21T18:05:43.656905+00:00 · methodology

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