Data-driven discovery of the equations of turbulent convection

Alasdair T. Roy; Christopher J. Wareing; Matthew Golden; Roman O. Grigoriev; Steven M. Tobias

REVIEW 3 major objections 5 minor 39 references

Data-driven discovery of the equations of turbulent convection

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read SPIDER, a symmetry-adapted weak-form sparse-regression framework, recovers from DNS data the governing equations, the incompressibility constraint, and the boundary conditions of turbulent convection over a wider range of Rayleigh numbers…

desk verdict A useful and mostly honest benchmark of SPIDER vs pySINDy on convection DNS, with a practical subdomain-sizing heuristic and a real DNS-validation idea, but the headline comparison is not yet fair enough to take at face value. read the letter →

arxiv 2505.10109 v1 pith:YJOGWI5F submitted 2025-05-15 astro-ph.SR physics.flu-dyn

Christopher J. Wareing , Alasdair T. Roy , Matthew Golden , Roman O. Grigoriev , Steven M. Tobias This is my paper · ORCID

classification astro-ph.SRphysics.flu-dyn

keywords boundaryconditionsNavier-Stokesequationsmachinelearningdata-driventechniquessparseregressionRayleigh-Bénardconvectionturbulentweakformulation

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 compares two data-driven equation-discovery methods on simulated turbulent convection: the standard weak-form sparse regression approach in the pySINDy package, and the SPIDER framework, which builds its candidate term libraries from the symmetries of the problem. The authors find that SPIDER recovers the governing momentum and temperature equations over a wider range of Rayleigh numbers, in two and three dimensions, and also recovers the incompressibility constraint and the boundary conditions, things pySINDy did not produce in this study. The practical payoff is that equation discovery need not begin from a huge brute-force menu of terms; a small symmetry-selected library plus a weak-form regression can do the job with less user effort and less memory. The paper also reads failures at the highest Rayleigh numbers as evidence that the underlying simulation is under-resolved, so the method can double as a solver-validation tool.

What carries the argument

The load-bearing mechanism is weak-form sparse regression on symmetry-adapted libraries. Each candidate term is multiplied by a smooth weight function and integrated over rectangular spatiotemporal subdomains; integration by parts shifts derivatives from the data, where numerical differentiation is noisy, onto the weight function. SPIDER constructs its candidate terms by combining the physical fields $\{u,p,T\}$ with differential operators $\partial_t$, $\nabla$, and the symmetry-breaking unit vectors, giving a 26-term scalar library and a 27-term vector library; projections onto irreducible representations under rotations about the boundary normal separate normal and tangential boundary relations. This small, physically structured search space is what allows SPIDER to recover vector equations, the divergence constraint, and boundary conditions without the memory-heavy brute-force libraries and coefficient interlinking that the pySINDy application required.

What would settle it

Run pySINDy with a small symmetry-adapted library comparable in size to SPIDER's (about 26 scalar and 27 vector terms, without additional coefficient interlinking) on the same 2D DNS data; if it then recovers the Navier-Stokes and heat equations at $R=10^{10}$ or $R=10^{12}$, the claim that SPIDER is inherently easier and more capable would be refuted, because the difference would trace to library construction rather than to the algorithm.

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Extended reading notes

Core claim

The central claim is that for Rayleigh-Bénard convection and plane convective Couette flow, the SPIDER framework recovers the full set of statements that govern the data—the momentum (Navier-Stokes) equation, the heat advection-diffusion equation, the incompressibility constraint $\nabla\cdot u=0$, and the stress-free or no-slip boundary conditions—whereas the weak-form SINDy implementation, applied through a large brute-force library, recovers the 2D equations only for a narrower range of Rayleigh numbers, with the diffusive terms (coefficients proportional to $1/\sqrt{R}$) the first to be lost as turbulence strengthens. SPIDER works from a 26-term scalar and 27-term vector library built from the fields $\{u,p,T\}$ and differential operators, using no problem-specific knowledge beyond symmetry and the choice of variables. At $R=10^{12}$, SPIDER recovers the Euler equation but not the diffusive term; the paper shows via horizontal power spectra above the boundary layer that this failure coincides with under-resolved small scales in the DNS, and concludes the failure is a property of the data rather than of the method. In 3D convection and in the Couette flow, SPIDER recovers the governing equations and boundary conditions throughout the range studied.

Load-bearing premise

The paper's central comparison assumes that the way pySINDy is configured here—a 104-term brute-force library pruned and interlinked using knowledge of the exact equations—is a fair and representative way to use that method; if that configuration is not good SINDy practice, the reported advantage of SPIDER could reflect the implementation rather than the algorithms.

Editorial extensions

If this is right

Diffusive terms are the first to disappear as the Rayleigh number rises: both methods recover the Euler (inviscid) equations before the $1/\sqrt{R}$-scaled Laplacian terms, so anyone applying these methods at high $R$ should expect viscosity to be the hardest physics to retrieve.
SPIDER recovered equations, the incompressibility constraint, and boundary conditions in three-dimensional convection ($R=10^4$ to $10^7$) and in two-dimensional convective Couette flow ($R=10^8$), cases in which the pySINDy implementation could not be applied because of memory limits in this study.
The spatial correlation length and correlation time of the flow give a sensible default subdomain size; the optimal pySINDy subdomains and the SPIDER integration boxes both fall near those scales, so future users can start there instead of sweeping hyperparameters blindly.
Because SPIDER recovers the Euler rather than the Navier-Stokes equation exactly where the horizontal power spectra show unresolved boundary-layer scales, equation-discovery residuals can serve as an independent check on whether a DNS is adequately resolved.

Reading between the lines

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

If the symmetry-adapted library is the real source of SPIDER's advantage, then a pySINDy user who hand-builds a comparably small library instead of pruning a 104-term brute-force library should recover much of the gap; this is a direct test the paper does not run.
The correlation-scale heuristic should transfer to other weak-form discovery problems: compute the autocorrelation length and time of the data before choosing integration subdomains, rather than sweeping sizes; the paper demonstrates the idea only on 2D Rayleigh-Bénard convection.
SPIDER's selection of the Euler equation at $R=10^{12}$ could be repurposed as a cheap resolution diagnostic for boundary-layer-resolving simulations; the paper suggests machine-learning validation, and this is the specific failure mode to look for.
For the intended next step of finding subgrid-scale closure equations, the same machinery would need to be run on filtered fields and should first recover the known unfiltered equations as a sanity check before any new terms are trusted.

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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

Two things to know. First, this is a serious, reproducible benchmark: the authors run two equation-discovery methods on their own DNS of Rayleigh–Bénard convection and convective plane Couette flow, and they publish Dedalus, pySINDy, and SPIDER scripts plus data. Second, the central comparison is not yet a fair one. The paper claims SPIDER recovers governing equations, constraints, and boundary conditions more easily and for a wider range of Rayleigh numbers, but pySINDy was handicapped by an artificially constrained library built using the known equations (Section 4.1.2), while SPIDER used symmetry-adapted libraries. So the claim of intrinsic superiority is plausible but not proven.

What is actually new: this is the first SPIDER application to buoyancy-driven convection and convective plane Couette flow, including recovery of boundary conditions. The correlation-time/scale heuristic for choosing subdomain sizes is a practical contribution that should save users trial and error. The resolution analysis—showing that failure at R=10^12 coincides with under-resolved boundary-layer spectra—is a genuine use of equation discovery as a diagnostic. Coefficient recovery at R=10^10 is well documented. The citation pattern is fine; the SPIDER self-citations are to the papers that introduced the method.

Soft spots: the pySINDy library was built by brute force (104 terms) and then constrained using exact knowledge: incompressibility imposed, diffusion reduced to Laplacian, coefficients interlinked. The authors are transparent about this, calling it ‘justified only by comparison to the known correct model.’ That transparency is good, but it means the headline ‘SPIDER is easier and more capable’ is read off an asymmetric test. Also, pySINDy was only applied in 2D due to RAM limits, so the wider-R claim is partly a memory limitation, not purely algorithmic. These issues are fixable: a cost-matched benchmark with the same prior knowledge, derivative evaluation, and compute budget, ideally by someone not involved with SPIDER, plus a resolution study that varies grid points at fixed R.

For whom: practitioners deciding whether to reach for SPIDER or pySINDy on fluid problems, and people working on equation discovery. The guidance is useful, the scripts are reusable, and the under-resolution diagnostic is worth copying. But read the comparison as provisional. I would accept this for peer review and ask for a cost-matched follow-up.

Referee Report

3 major / 5 minor

Summary. This manuscript benchmarks two sparse-regression equation-discovery frameworks, pySINDy (weak-form SINDy with MIOSR) and SPIDER, on noiseless spectral DNS data of 2D and 3D Rayleigh-Bénard convection and 2D convective plane Couette flow. The authors show that with a constrained library pySINDy recovers the 2D governing equations up to R = 10^8 (and only the Euler equation at R = 10^12), while SPIDER recovers the momentum and temperature equations, the incompressibility constraint, and the boundary conditions over a wider range, including the 3D cases. They further propose that flow correlation length scales and timescales should inform the initial choice of weak-form subdomain sizes, and they argue that failures at the highest Rayleigh numbers are caused by under-resolved DNS rather than by the discovery methods.

Significance. If the comparison is accepted, the paper is a practically useful demonstration that symmetry-adapted libraries and weak-form regression can reduce user burden in equation discovery, that constraints and boundary conditions can be recovered from data, and that equation recovery can serve as a diagnostic of numerical resolution. The manuscript is unusually transparent: it documents hyperparameter sweeps, residual tables, recovered coefficients, and power spectra, and it provides simulation and analysis scripts in a data repository. Those strengths do not remove the main weakness: the headline advantage of SPIDER over pySINDy rests on a comparison that is not cost-matched in library construction, prior knowledge, or computational effort, so the central claim is only conditional.

major comments (3)

[Sections 4.1.2 and 4.2] The central 'smaller library and more easily recovered' claim is not established because the two methods are given different prior information. The pySINDy search starts from a brute-force 104-term library in 2D and is then artificially pruned and interlinked using exact knowledge of the solution: incompressibility is imposed via Eq. (9), diffusion is restricted to the Laplacian form, coefficients are interlinked across vector components, and individual terms are switched off; the text states these choices are 'justified only by comparison to the known correct model.' SPIDER instead starts from the 26-term and 27-term symmetry libraries of Eqs. (11)-(12), with no term-level knowledge of the exact equations. The comparison therefore conflates the algorithm with the library-construction protocol. I would ask for either a cost-matched comparison (for instance, SPIDER with the same brute-force library, or pySINDy with a library built from the same symmetry principles) or an explicit reporting of user effort, memory, and wall-clock time for a shared discovery task. Without this, the abstract's 'more easily' is an artifact of the setup rather than an established algorithmic property.
[Sections 5.1 and 5.2.2] The claim that SPIDER 'goes beyond pySINDy' by recovering constraints and boundary conditions is not supported as a comparison, because pySINDy is only applied to bulk 2D governing equations. No pySINDy experiment is reported for recovering ∇·u = 0 or for recovering boundary conditions, so the paper demonstrates that these tasks were not attempted with pySINDy, not that pySINDy cannot perform them. This claim should be either tested in the same framework (for example, by a weak-form regression on boundary data) or explicitly restated as a capability demonstration for SPIDER only.
[Sections 5.2.1 and 6.2] The interpretation that failures at R = 10^12 are due to under-resolution, while plausible and supported by the power spectra in Fig. 8, is not proven because no counterfactual is provided. A higher-resolution run at the same Rayleigh number (for example, with 2048 or 4096 points in the horizontal direction) is needed to show that SPIDER then recovers the diffusive term and the full Navier-Stokes equation. Without such a test, the failure could also be attributed to hyperparameter selection, weak-form quadrature error, or the near-machine-precision magnitude of the diffusion coefficient (~10^-6). This point is load-bearing because the 'wider range of R' and the proposed validation/verification use both rest on attributing the high-R failure to the data rather than to the methods.

minor comments (5)

[Figure 3 and Figure 4 captions] The captions spell 'Rayleigh' as 'Rayeigh'; please correct this typo.
[Section 3] The text states Dedalus version 3.0.11, but the accompanying URL footnote points to v3.0.1; please reconcile this version discrepancy.
[Section 5.1.1] The fourth pySINDy test is described as 'inconclusive,' but no quantitative criterion for success or failure is given; please specify what would have qualified as a conclusive recovery.
[Section 6.1 and Table 4] The correlation scales are computed only for the 2D Rayleigh-Bénard simulations, yet the guidance to use them as a default is presented generally for pySINDy and SPIDER; please flag this as an extrapolation to the 3D and Couette cases.
[Equation (12)] The term u·(∇u) in the vector library L1 is not defined; please clarify whether it denotes (u·∇)u, a contraction with the velocity-gradient tensor, or another expression.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: this is a benchmark against data generated from known equations; the disclosed known-model-constrained pySINDy tuning is a fairness issue, not a circular derivation.

full rationale

Walks the claimed derivation chain: there is no equation-level circularity. The paper does not claim to derive the Boussinesq/Navier-Stokes equations from first principles; it benchmarks two sparse-regression frameworks on DNS generated from those equations, so recovering the generating equations is the intended test, not a circular derivation. The SPIDER libraries (Eqs. 11-12) are generic symmetry-adapted libraries; their terms include the target equation terms, but the incompressibility constraint, momentum equation, heat equation, and boundary conditions are selected by regression and residual analysis rather than imposed term-by-term. The pySINDy setup does encode known-model information: Section 4.1.2 states that the 104-term library 'includes all the terms that we know appear in the governing equations' and that the library is then artificially constrained using incompressibility, Laplacian-only diffusion, and interlinked vector coefficients 'justified only by comparison to the known correct model.' This is a genuine benchmark-fairness concern, and the constrained-library recovery is partly an in-sample parameter-estimation exercise. However, the paper transparently uses the constrained fits only to guide hyperparameter selection and then performs unconstrained regression in Section 5.1.3. That unconstrained regression is not forced by construction to return the target equations: the tuned quantities are integration-domain hyperparameters, not the equation terms themselves. The under-resolution interpretation at high R is separately checked against boundary-layer power spectra (Fig. 8), so it is not a circular self-confirmation of the SPIDER failure. The paper itself concedes in Section 7 that a harder comparison would include higher derivatives in both libraries and that 'it is little surprise that a method performs better when more knowledge is applied in the generation of the library'; that is an admission about benchmark design, not circular reasoning. Self-citations to Gurevich et al. (2024) and Golden et al. (2023) supply the SPIDER machinery and prior demonstrations, but the present paper independently applies that machinery to new DNS data, so the self-citations are not the load-bearing evidence for the headline claims. Overall: no significant circularity; the score of 2 reflects the disclosed known-model-constrained pySINDy tuning and the developer self-citation, which are validity and fairness concerns rather than circular derivation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The result is an empirical benchmark, so the load-bearing inputs are the DNS dataset, the candidate libraries, and tuned hyperparameters rather than new physics. The most consequential choices are the library compositions and subdomain sizes; the paper shows that varying them changes whether diffusion is recovered. The use of known equations to constrain the pySINDy library is disclosed but makes the constrained-library comparison weaker than the unconstrained one.

free parameters (6)

pySINDy spatiotemporal domain divisions (xdiv, zdiv, tdiv) = 12, 10, 8 at R=1e6
Selected by hyperparameter sweep to minimize residual error; the same values were then used at higher R.
SPIDER weight exponent beta = 4, 8, or 12 depending on case
Tuned per Rayleigh number and flow; varied from the default of 8 to improve recovery.
SPIDER subdomain size and subdomain count = e.g., Omega_i=64 and K=1024 for R=1e10 in 2D
Chosen per case; larger domains needed for laminar flow, smaller domains for turbulent flow.
MIOSR sparsity targets = 15 total (5,6,0,4) for 2D Rayleigh-Benard
Set from the known term counts in the true equations, which presupposes the answer in the constrained tests.
pySINDy interpolating polynomial order = 6
Held constant to keep the sweep tractable; the authors note this can affect weak-form accuracy.
Correlation-scale threshold = Autocorrelation drop to 0.5
Defines the recommended subdomain-size heuristic in Section 6.1.

assumptions (5)

domain assumption The Boussinesq approximation and the stated nondimensionalization correctly describe the simulated convection.
Section 2.1 defines the model; discovery is judged against these equations.
domain assumption The DNS solutions accurately represent the continuous PDE solutions at the stated resolutions.
Section 3 describes the resolutions; Section 6.2 tests whether high-R cases are under-resolved.
domain assumption The candidate libraries contain the true operators that appear in the governing equations.
Sections 4.1 and 4.2 construct libraries; missing terms cannot be discovered.
standard math Weak-form integration, integration by parts, and trapezoidal quadrature are accurate enough for the regression.
Both methods rely on this numerical machinery, described in Sections 4.1 and 4.2.
domain assumption The true equations are sparse linear combinations of library terms.
This is the core sparsity assumption of SINDy and SPIDER, stated in Section 4.

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Cite this review

Pith. "Pith review of Data-driven discovery of the equations of turbulent convection." pith.science (2026). https://pith.science/paper/YJOGWI5F

@misc{pith2026250510109,
  author       = {Pith},
  title        = {Pith review of: Data-driven discovery of the equations of turbulent convection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJOGWI5F}},
  note         = {Machine review of arXiv:2505.10109}
}

read the original abstract

We compare the efficiency and ease-of-use of the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm and Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) framework in recovering the relevant governing equations and boundary conditions from data generated by direct numerical simulations (DNS) of turbulent convective flows. In the former case, a weak-form implementation pySINDy is used. Time-dependent data for two- (2D) and three-dimensional (3D) DNS simulation of Rayleigh-Benard convection and convective plane Couette flow is generated using the Dedalus PDE framework for spectrally solving differential equations. Using pySINDy we are able to recover the governing equations of 2D models of Rayleigh-Benard convection at Rayleigh numbers, R, from laminar, through transitional to moderately turbulent flow conditions, albeit with increasing difficulty with larger Rayleigh number, especially in recovery of the diffusive terms (with coefficient magnitude proportional to 1/R^0.5). SPIDER requires a much smaller library of terms and we are able to recover more easily the governing equations for a wider range of R in 2D and 3D convection and plane flow models and go on to recover constraints (the incompressibility condition) and boundary conditions, demonstrating the benefits and capabilities of SPIDER to go beyond pySINDy for these fluid problems governed by second-order PDEs. [We] demonstrat[e] the potential of machine-learning methods to validate numerical solvers and solutions for such flow problems. We also find that properties of the flow, specifically the correlation time and spatial scales, should inform the initial selection of spatiotemporal subdomain sizes [abbreviated]

Figures

Figures reproduced from arXiv: 2505.10109 by the authors.

**Figure 2.** Results of the first pySINDy test at R = 106 for 2D Rayleigh-B´enard convection. Sampling spatiotemporal domains from a region of the bulk flow, the figures show the effect of varying the size of the spatiotemporal domains on the resulting residual error (in weak form, over K subdomains) of the governing equations recovered by the pySINDy implementation. The error shown is the logarithm of the residual sum of square… view at source ↗

**Figure 3.** Results of constrained-library sparse regression using the pySINDy implementation applied to simulation of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Results of unconstrained-library sparse regression using the pySINDy implementation applied to simulation [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Dependence of the residual on the number of terms [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Dependence of the residual on the number of terms [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Dependence of the residual on the number of terms [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]

**Figure 8.** Figure 8: 1D power spectra along profiles in the x direction (kx only) at a height of 10 Chebyshev nodes above the boundary layer for the 2D simulations of Rayleigh-B´enard convection, so as to sample the power spectra of the smallscale dissipative structures in the boundary la…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter chaptertitle englishtitle edition editor eid howpublished institution journal key month note number organization pages publisher location school series seriesnumber title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.sta...

work page

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION word.pl...

work page

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work page 2024