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arxiv: 2606.13336 · v1 · pith:WD2WXJJ7new · submitted 2026-06-11 · ⚛️ physics.flu-dyn

Data-Driven Equation Discovery for Nonlinear Liquid Film Flows

Sebastian T. Dooley , Albert P. Bart\'ok , James E. Sprittles , Radu Cimpeanu This is my paper

Pith reviewed 2026-06-27 05:48 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords equation discoverydata-driven modelingliquid film flowsnonlinear PDEssparse regressionmulti-collinearityfluid dynamicsthin film equations
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The pith

Careful data curation with expert knowledge allows data-driven methods to recover the governing equations for nonlinear liquid film flows despite identifiability issues.

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

The paper tries to establish that data-driven equation discovery can identify the underlying partial differential equations for nonlinear liquid film flows when simulation data is carefully selected using expert knowledge of the problem. A sympathetic reader would care because liquid films have known asymptotic models, creating a controlled test case for seeing how far these methods reach into systems that resist traditional derivation. The authors show that multi-collinearity from monomial basis functions in the multi-scale setting and early-time transients with high residuals create real identifiability problems. Pinpointing these limits helps set boundaries for current techniques and points toward more stable algorithms.

Core claim

By leveraging expert knowledge and the ability to carefully curate data, we establish a best-case scenario for identifying the underlying governing equations. Even here, multi-collinearity stemming from the choice of monomial basis functions in our multi-scale flow configuration introduces complex identifiability issues. Early-time transients compound this further, as the most dynamically rich behaviour carries the largest residuals in training data.

What carries the argument

Sparse regression over a monomial basis applied to curated time-series data from liquid film flow simulations, to recover the terms of the governing partial differential equations.

If this is right

  • Data-driven discovery can succeed on PDE systems such as liquid films when data selection incorporates domain knowledge.
  • Multi-collinearity from monomial bases must be handled explicitly to achieve reliable term identification in multi-scale flows.
  • Early-time transients require special treatment because they produce the largest residuals while containing the richest dynamics.
  • Mapping these vulnerabilities guides the creation of more numerically stable discovery algorithms for partial differential equations.

Where Pith is reading between the lines

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

  • Alternative basis choices or explicit regularization could reduce the multi-collinearity problem in similar multi-scale settings.
  • The curation strategy might be adapted to experimental measurements, where noise would add a further test of robustness.
  • The same approach could be applied to other nonlinear film or free-surface flows to check whether the identified limits are general.

Load-bearing premise

That a monomial basis remains usable for discovery even though multi-collinearity arises from the choice of basis functions in the multi-scale flow.

What would settle it

Applying the discovery procedure to the curated dataset and finding that it fails to recover the known thin-film equations or returns inconsistent models across data subsets would falsify the claim of successful identification in this best-case scenario.

Figures

Figures reproduced from arXiv: 2606.13336 by Albert P. Bart\'ok, James E. Sprittles, Radu Cimpeanu, Sebastian T. Dooley.

Figure 1
Figure 1. Figure 1: Schematic of a falling liquid film down an inclined plane. The inclination angle is denoted by 𝜃 and a rotated Cartesian coordinate system aligning with the flat, smooth substrate is used. The gravity vector g, fluid properties, spatio-temporally varying interfacial height ℎ(𝑥, 𝑡) and mean film height ℎ ∗ 0 are also illustrated. the fluid is gravity, which acts at an angle 𝜃 to the 𝑦-axis. Surface tension … view at source ↗
Figure 2
Figure 2. Figure 2: The non-dimensional system (4.3) evolved from the initial condition (4.5) over spatio-temporal mesh (𝑥, 𝜏 ˜ ) ∈ [0, 1]× [0, 𝑡∗ 𝐹 ]. The evolution shows rapid decay of high-frequency waves and slower decay of lower frequency waves. The initial perturbation amplitude is large and all values satisfy ℎ(𝑥, 𝑡) > 0. A range of alternative periodic initial conditions can be used to specify the IVP and produce data… view at source ↗
Figure 3
Figure 3. Figure 3: An example of a feature selection tolerance line search. For each term in the function library, an importance score is calculated over the range of tolerance values, relating to whether it was frequently retained from the sequentially thresholded feature selection process. The tolerance values are rescaled with respect to the critical tolerance. Applying an importance cut-off heuristically set to 0.8, we o… view at source ↗
Figure 4
Figure 4. Figure 4: An array of model evaluation metrics. The overall minimum corrected AIC score (best) is highlighted in red and a line connects the minimum value for each model complexity. Additional points show scores of alternative models that were outperformed by an equally complex model. The top row uses residuals from the fitted linear system for error calculation. The bottom row panels involve the solution of discove… view at source ↗
Figure 5
Figure 5. Figure 5: a) The dynamics of the discovered equation. b) The evolution of the governing equation and is the testing data used. We compare the two surfaces in panel c), displaying relative percentage error. Errors include: RMSE=0.00413, MAE= 0.00340 and ||𝑒||∞ = 0.0116. With the thin film equation recovered using clean data, we can briefly consider the robustness of this result to artificial noise, which can be found… view at source ↗
Figure 6
Figure 6. Figure 6: Varying temporal resolution in the training data, as the number of temporal samples remains constant. We begin by noting a large region of success for ever-decreasing proportions of the time span used. This indicates that sampling from the initial transient region with highly nonlinear dynamics is fruitful for [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fixing the temporal resolution in the training data and varying the number of temporal samples to account for this. Successful recovery continues to the maximum tested value of 10.0𝜏 ∗ 𝐹 [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: a) Characteristic time, 𝑇 (Φ), b) Characteristic length, 𝐿(Φ). We see that the constant solution ℎ = 1 is a steady state of the system. Performing a linear stability analysis, one recovers the dispersion relation 𝜔(𝑘) = −3𝑖𝑘 − Φ𝑘 2 − 𝑘 4 , (5.3) with ℜ(𝜎(𝑘)) < 0, ∀𝜃 ∈ [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Minimum 𝐿2 error normalised with respect to the null error for corresponding Φ. (b) Where valid, the normalised 𝐿2 error on the testing dataset associated with the correct model. (c) The success-failure classification with near-successes also coloured. For the lowest value case of Φ ≈ 0.001, we see models ℎ𝑡 =  −1.01ℎ 3 ℎ𝑥 𝑥 𝑥  𝑥 and ℎ𝑡 =  −0.998ℎ 3 − 0.998ℎ 3 ℎ𝑥 𝑥 𝑥  𝑥 . The first such model shows… view at source ↗
Figure 10
Figure 10. Figure 10: Heat map of low 𝐿2 error models selected from the Pareto frontier with coefficients pro￾portional to colour. Grey squares indicate an unselected term with a corresponding coefficient value of 0, as indicated by the provided colour bar. Model selection is performed using a marginal gain crite￾rion. One-sided gradients of error with respect to model complexity are evaluated. Then, starting from the lowest c… view at source ↗
Figure 11
Figure 11. Figure 11: Panel (𝑎) − (𝑖) correspond to the models in table 2. This shows a green-pink binary clas￾sification of regions in the parameter space in which the corresponding model structure was recovered. Green corresponds to a successful recovery. Limitations were also identified for the gravity-enriched model. Disparities in the magnitudes of coefficients appeared to obstruct recovery of the governing equation and i… view at source ↗
Figure 12
Figure 12. Figure 12: (a) shows the solution data for a validation initial condition solved using finite differences. (b) shows the evolution of the same initial condition, using the equation discovered from clean data using our pipeline. (c) shows the evolution of the same initial condition, using the equation discovered from noisy data using our pipeline without filtering/smoothing. The evolution (d) relates to the equation … view at source ↗
read the original abstract

Over the past decade data-driven equation discovery emerged as a powerful alternative to first principles-based methodologies traditionally used in mathematical modelling cycles. The approach provides a promising path towards deep, physical insight into systems that have previously evaded rigorous mathematical derivation procedures, often due to intractable complexity. The strengths of such techniques have been successfully established for many problem classes described by systems of ordinary differential equations and continue to be extended, with their reach into partial differential equation systems gaining momentum, though comparatively nascent. Herein we tackle such a frontier: elucidating the dynamics of liquid film flows, a problem space providing a rich backdrop in terms of asymptotic analytical building blocks. By leveraging expert knowledge and the ability to carefully curate data, we establish a best-case scenario for identifying the underlying governing equations. Even here, multi-collinearity, stemming from the choice of monomial basis functions in our multi-scale flow configuration, introduces complex identifiability issues. Early-time transients compound this further, as the most dynamically rich behaviour carries the largest residuals in training data. Pinpointing such vulnerabilities allows us to better define the boundaries of current discovery techniques and paves the way for the next generation of more resilient, numerically stable algorithms.

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

2 major / 0 minor

Summary. The manuscript applies data-driven sparse regression to discover governing PDEs for nonlinear liquid film flows. Leveraging expert knowledge for data curation, the authors construct what they describe as a best-case scenario for equation recovery. They report that monomial basis functions induce multi-collinearity in the multi-scale setting and that early-time transients produce the largest residuals, creating identifiability difficulties that are used to delineate the practical boundaries of current discovery methods.

Significance. If the manuscript supplied quantitative demonstrations (recovered equations, coefficient errors, condition numbers, or ablation studies on data weighting), the work could usefully bound the applicability of monomial-based sparse regression for multi-scale PDE systems in fluid mechanics. As presented, the contribution remains prospective rather than substantiated.

major comments (2)
  1. [Abstract] Abstract: the central claim that expert curation 'establishes a best-case scenario for identifying the underlying governing equations' is load-bearing yet unsupported; the text only enumerates identifiability problems without reporting any recovered equations, regression residuals, condition numbers of the library matrix, or verification that the chosen terms match known asymptotic models for film flows.
  2. [Abstract] Abstract: the assertion that multi-collinearity 'introduces complex identifiability issues' is presented as a finding, but no quantitative evidence (e.g., singular-value spectrum of the regression matrix, sensitivity of selected terms to noise or weighting, or comparison of fits with/without curated data) is supplied to show that these issues actually prevent reliable recovery under the claimed best-case conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive report. Our manuscript aims to delineate practical limitations of monomial-based sparse regression for multi-scale fluid systems by showing that even expert-curated data does not yield clean recovery. We agree the abstract requires clarification and quantitative support, and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that expert curation 'establishes a best-case scenario for identifying the underlying governing equations' is load-bearing yet unsupported; the text only enumerates identifiability problems without reporting any recovered equations, regression residuals, condition numbers of the library matrix, or verification that the chosen terms match known asymptotic models for film flows.

    Authors: We acknowledge the abstract phrasing risks implying successful recovery. The manuscript's core contribution is demonstrating that identifiability issues persist even under expert-curated conditions; no clean governing equations are recovered. To address this, we will revise the abstract to state explicitly that the work identifies boundaries and limitations rather than successful discovery. We will also add quantitative metrics (condition numbers, residual distributions, and comparison to known film-flow asymptotics) in the main text or appendix of the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that multi-collinearity 'introduces complex identifiability issues' is presented as a finding, but no quantitative evidence (e.g., singular-value spectrum of the regression matrix, sensitivity of selected terms to noise or weighting, or comparison of fits with/without curated data) is supplied to show that these issues actually prevent reliable recovery under the claimed best-case conditions.

    Authors: We agree that explicit quantitative support for the multi-collinearity claim would strengthen the paper. The current text describes the source of the problem (monomial bases in a multi-scale setting) but does not include singular-value spectra or ablation studies. We will incorporate these quantitative demonstrations, including condition numbers of the library matrix and sensitivity to data weighting, in the revised manuscript to substantiate the identifiability difficulties. revision: yes

Circularity Check

0 steps flagged

No circularity: data-driven discovery uses external curated data and known physics without self-referential reduction

full rationale

The paper applies standard sparse regression (likely SINDy-style) to curated simulation data from liquid film flows, leveraging known asymptotic building blocks only for data selection and basis choice. No derivation step equates a claimed prediction to its own fitted inputs by construction, nor does any load-bearing premise reduce to a self-citation chain. Multi-collinearity and identifiability issues are explicitly flagged as limitations rather than hidden. The central claim (best-case recovery is possible with curation) rests on external data generation and domain knowledge, remaining falsifiable against held-out simulations or analytic limits. This is the expected non-finding for a well-posed data-driven study.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the monomial basis choice is noted as a source of multi-collinearity but not quantified.

pith-pipeline@v0.9.1-grok · 5751 in / 1067 out tokens · 18993 ms · 2026-06-27T05:48:31.457151+00:00 · methodology

discussion (0)

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