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REVIEW 4 major objections 5 minor 72 references

Drag modelling for flows through assemblies of spherical particles with machine learning: A comparison of approaches

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read One GNN, trained across Reynolds numbers from 0.1 to 300, learns pairwise drag interactions that genetic programming can rewrite as compact symbolic expressions; the formulas explain the same features but lose a few points of $R^2$.

desk verdict Solid GNN results and an honest regime extension, but the symbolic-vs-GNN comparison is undermined by a target mismatch in the reported R^2. read the letter →

arxiv 2507.05983 v1 pith:G3BLRTCA submitted 2025-07-08 physics.comp-ph physics.flu-dyn

Julia Reuter , Hani Elmestikawy , Sanaz Mostaghim , Berend van Wachem This is my paper
classification physics.comp-phphysics.flu-dyn
keywords dragmodellingparticle-ladenflowsgraphneuralnetworkssymbolicregressiongeneticprogrammingpairwiseinteractiondeviationparticle-resolvedDNS
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

To turn black-box drag predictions into interpretable equations, this paper trains a graph neural network (GNN) on particle-resolved simulation data and then uses genetic programming (GP) to replace the GNN's pairwise interaction rule with compact symbolic expressions. The target is the deviation of each particle's drag from the mean drag in a random assembly, across particle Reynolds numbers from about 0.1 to 300 and volume fractions from 0.1 to 0.6. The central claim is that the symbolic expressions, while slightly less accurate than the GNN, capture the same dominant structure: the normalized neighbour distance and the polar angle matter, the azimuthal angle does not, and the local particle arrangement outweighs global flow conditions. A sympathetic reading is that this is a proof-of-concept that interpretable drag-deviation closures can be derived in finite-Reynolds regimes, not yet a claim that the formulas replace the neural model. If the claim holds, point-particle simulations could carry drag-deviation corrections as cheap closed-form terms instead of evaluating a network.

What carries the argument

The load-bearing object is the pairwise superposition ansatz, Eq. (8), which represents each particle's drag deviation as the sum of a single shared pairwise interaction function over its neighbours. The GNN's shared edge model learns that function from PR-DNS data; the GP algorithm then fits symbolic expressions to the extracted pairwise input\textendash output samples, with a feature-permutation analysis identifying which inputs the GNN relies on. The unit-aware GP framework with tuneable-constant 'joker' units, complexity control, and a Pareto archive supplies the search machinery that keeps the resulting formulas compact and dimensionally meaningful.

What would settle it

Train the genetic programming stage directly on pairwise residual targets formed from PR-DNS data (for instance, the difference between the pairwise-interaction-extended point-particle model and the full simulation) instead of on GNN outputs, and compare the resulting symbolic $R^2$ on the same test set. If the directly fitted formulas do not beat or match the GNN-derived ones, the symbolic expressions are largely reproducing the surrogate's view; if they do, the GNN transmission was the bottleneck. A second check is to test the final equations at intermediate Reynolds numbers (e.g., $\mathrm{Re}_p=20$ and $\mathrm{Re}_p=150$) that were not in the training set.

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Extended reading notes

Core claim

The paper's central claim is that a single GNN trained on PR-DNS data can learn, for all considered Reynolds numbers and volume fractions at once, the pairwise interaction function $\tilde{f}_x(\bar{r}_j,\theta_j,\varphi_j,\varepsilon_p,\mathrm{Re}_p)$ whose sum over the 30 nearest neighbours reproduces the per-particle drag deviation $\Delta F_{x,i}/\langle F_x\rangle$. On the test set the GNN attains $R^2$ values of 0.64\textendash 0.80 across $\mathrm{Re}_p$ and 0.63\textendash 0.79 across $\varepsilon_p$. Feeding the GNN's extracted pairwise outputs to a multi-objective, unit-aware GP search yields symbolic expressions per $\mathrm{Re}_p$ and per $\varepsilon_p$ with test $R^2$ of 0.46\textendash 0.64. The authors take this as evidence that the GP pipeline can find relatively simple symbolic models whose structure mirrors the GNN's learned importance pattern\textemdash distance and polar angle dominate, azimuthal angle never appears, and volume fraction appears less often than expected\textemdash while acknowledging that the symbolic models fall slightly behind the GNN in accuracy.

Load-bearing premise

The whole pipeline assumes the pairwise superposition ansatz is faithful and, more immediately, that the GNN's extracted pairwise outputs are an accurate representation of the true physics; if the GNN errs, the symbolic formulas inherit and propagate that error rather than correcting it.

Editorial extensions

If this is right

  • If the claim is right, drag-deviation corrections for point-particle simulations can be written as small algebraic expressions per flow regime rather than as neural-network evaluations, which is cheaper and inspectable.
  • The symbolic expressions' consistent exclusion of the azimuthal angle supports the pairwise-superposition view that drag variation around the mean is rotationally symmetric about the flow direction.
  • Recurring patterns and sub-expressions in the GP results could be used as building blocks toward a single generalized symbolic drag-deviation model covering multiple $(\mathrm{Re}_p,\varepsilon_p)$ pairs.
  • The accuracy gap between symbolic models and the GNN quantifies the interpretability cost and defines the target for future constraint-based GP refinements.
  • The low feature importance of $\mathrm{Re}_p$ and $\varepsilon_p$ in the GNN suggests that local geometry, not global flow conditions, controls the deviation, so future closures should prioritize local arrangement features.

Reading between the lines

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

  • Editorial inference: because the GP equations are fit to one selected GNN's pairwise outputs, their reported $R^2$ is an upper bound set by the GNN; the true quality relative to PR-DNS could be lower, and the equations may not capture physics the GNN missed.
  • Editorial inference: the suggested multi-view approach\textemdash fitting equation constants separately for each $(\mathrm{Re}_p,\varepsilon_p)$ combination while sharing the functional form\textemdash could close much of the accuracy gap while preserving interpretability, and would be a cheap next experiment.
  • Editorial inference: imposing physically motivated shape constraints, such as requiring the pairwise contribution to vanish as $\bar{r}\to\infty$ and to respect fore-aft symmetry at low $\mathrm{Re}_p$, could make the symbolic models extrapolate more safely than the GNN.
  • Editorial inference: the framework is not specific to drag; the same GNN-to-GP distillation could be applied to lift and torque deviations, or to bidisperse assemblies, wherever a pairwise superposition ansatz is plausible.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper presents a two-step machine-learning pipeline for modelling the variation of drag forces on spherical particles in random assemblies. A graph neural network (GNN) is first trained on particle-resolved direct numerical simulation (PR-DNS) data to predict the deviation of each particle's drag from the mean, using a pairwise superposition ansatz in which the particle-level deviation is the sum of per-neighbour edge-model outputs. In the second step, genetic programming (GP) is applied to the pairwise outputs extracted from the trained GNN edge model, producing compact symbolic expressions for the pairwise contribution. The GNN is trained once over all Reynolds numbers and volume fractions, and its test R^2 values range from about 0.64 to 0.80. The GP symbolic expressions, developed separately at fixed Rep or fixed εp, are reported with R^2 values between about 0.46 and 0.64. The paper concludes that GP can find relatively simple symbolic models whose accuracy 'slightly falls behind' the GNN, and presents a feature-importance analysis identifying inter-particle distance and polar angle as the most influential features.

Significance. If the central comparison were established, the contribution would be a useful, interpretable alternative to black-box GNN drag closures, extending earlier Stokes-regime work to finite Reynolds numbers. The paper has several genuine strengths: the GNN is characterized over 11 repeated runs with a held-out PR-DNS test set; the data are openly available; the GNN fidelity prerequisite is explicitly acknowledged in Section 3.1; and the feature-importance analysis is a reasonable way to link the GNN to the later symbolic expressions. However, the headline comparison between GNN and GP accuracy is not currently supported because the two reported R^2 values are computed on different targets (particle-level PR-DNS deviations versus pair-level GNN outputs). The symbolic expressions are not directly evaluated on PR-DNS data, so the claimed 'slightly fall behind' conclusion and the attribution of the discovered formulas to physical structure are not yet established.

major comments (4)
  1. [Section 4.2, Tables 5-7] The R^2 values in Tables 6 and 7 are computed on the pairwise interaction outputs extracted from the GNN (see Section 3.3 and Table 2), whereas the GNN R^2 values in Table 5 are computed on particle-level drag deviations against PR-DNS using Eq. (9). Comparing these two sets of numbers in Section 4.2 to conclude that the symbolic models 'slightly fall behind' the GNN is not a controlled comparison: a pair-level fit can be numerically different from its particle-level aggregate. No step is described in which the symbolic pairwise expressions are summed over the 30 neighbours, as in Eq. (8), and evaluated against PR-DNS labels. The manuscript should either perform this aggregate-level evaluation and report the resulting R^2 values, or explicitly reframe the claim as measuring how well GP reproduces the GNN's edge model, without implying comparable drag-prediction accuracy.
  2. [Section 3.1, Sections 3.3 and 4.2] The GP symbolic models are fitted exclusively to pairwise outputs of a single selected GNN, never to PR-DNS drag labels. Section 3.1 acknowledges that errors in the GNN will be propagated to the symbolic expressions. Consequently, the reported GP R^2 values bound the fidelity of the symbolic expressions to the GNN's learned pairwise decomposition, not to the physical drag data. To make the central claim that the discovered expressions capture physical structure, the paper should include an evaluation of the full symbolic models (summed over neighbours) against the held-out PR-DNS test set, and ideally quantify the propagation of GNN uncertainty by repeating the GP step on edge outputs from more than one of the 11 trained GNNs.
  3. [Section 4.1, model selection] The selection of the GNN used for GP target extraction is made by computing, for each of the 11 runs, the maximum train-test R^2 difference over Rep, and choosing the model with the lowest gap. This procedure uses the test labels to choose a model, and the same model's test R^2 is then reported in Table 5 and used as the GNN reference in the comparison with GP. This is a test-informed selection and can bias the reported GNN performance. The authors should either select the model using a validation partition only, or report the spread of the selection criterion and acknowledge the potential optimism in the reported GNN numbers.
  4. [Tables 6 and 7] The symbolic expressions in Tables 6 and 7 replace all numerical coefficients with placeholder variables C_k and the actual fitted values are never reported. As presented, these are templates rather than concrete symbolic models: the R^2 values cannot be independently reproduced, and the expressions cannot be used in any downstream simulation. The fitted constants should be reported for each displayed equation, or the code and fitted parameter sets should be made available, so that the symbolic models are actually usable and verifiable.
minor comments (5)
  1. [Section 4.1] In the text near Figure 6, the acronym 'PD-DNS' appears; it should be 'PR-DNS'.
  2. [Section 4.1, feature importance] The paper states that the azimuthal angle φ should have no influence due to rotational symmetry, yet the permutation analysis shows a small increase in MSE when φ is permuted. This discrepancy should be commented on — for example, as a small symmetry violation in the GNN — rather than left as an unexplained observation.
  3. [Section 4.2] The GP runs were repeated 11 times, but Tables 6 and 7 report only a single R^2 value for each displayed expression, with no indication of variance across runs. Reporting the range or standard deviation (or at least stating that the displayed expression is representative) would make the comparison much more informative.
  4. [Section 3.3] The paper says the GP target is the pairwise contribution f~x and the training features exclude either Rep or εp depending on the experiment. The captions of Tables 6 and 7 should state explicitly that the R^2 values are computed on the pair-level GP test set, to avoid the ambiguity discussed in the major comments.
  5. [Section 5] The conclusion says 'patters' instead of 'patterns'; the manuscript would benefit from a careful proofread for this and similar typographical errors.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the symbolic expressions are explicitly distilled from GNN pairwise outputs, and the central GNN evaluation is independently grounded in held-out PR-DNS data.

full rationale

The claimed pipeline is a supervised distillation chain, not a circular derivation. PR-DNS supplies particle-level drag deviations; the GNN is trained on those and decomposes them into pairwise edge outputs; the GP then fits symbolic edge expressions to those GNN-extracted pairwise outputs (Sections 3.1 and 3.3, Table 2). No equation is defined in terms of the quantity it is claimed to predict, and no fitted constant is relabeled as a discovery. The paper explicitly frames the symbolic models as approximating the GNN, stating that the GP 'searches for symbolic expressions that fit the input data' and aims to 'capture the underlying patterns learnt by the GNN,' while acknowledging that errors in the GNN 'will be further propagated to the resulting symbolic expressions.' The GNN R^2 values in Table 5 are evaluated against held-out PR-DNS via Eq. (9), which is independent of the GP fit. The main caveat, that the symbolic R^2 values in Tables 6/7 appear to be computed on the GNN-derived pair-level target rather than on particle-level PR-DNS, is a comparison-validity or correctness concern rather than a circularity of the derivation. Self-citations to the authors' prior Stokes-flow GP work provide context and design rationale but do not carry the empirical weight of the present results, which are benchmarked against the cited PR-DNS data. Therefore no step reduces to its own input by construction.

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

The symbolic models rest on a large set of fitted constants, on a GNN surrogate trained on the authors' open DNS data, and on the pairwise-superposition ansatz. No new physical entities are introduced; the two most load-bearing premises are the fidelity of the GNN pairwise decomposition and the validity of pairwise superposition in dense assemblies.

free parameters (5)
  • Symbolic expression constants C_k = not reported
    Every equation in Tables 6 and 7 contains fitted constants C_0 to C_5; values are omitted, so the printed formulas cannot be evaluated or reproduced.
  • GNN edge-model trainable weights = not disclosed (30 neurons per hidden layer)
    The GNN weights are fitted to PR-DNS data and determine the pairwise interaction targets from which GP learns; they form a fitted surrogate at the core of the pipeline.
  • Neighbour count N_n = 30
    Chosen from prior studies as a compromise; the aggregation in Eq. (8) uses exactly 30 pairwise terms per particle.
  • GP complexity cap and population settings = population 500, generations 200, max complexity 30
    Hand-chosen search parameters that shape which symbolic expressions are reachable and selected.
  • log10(Rep) preprocessing transform = applied
    Preliminary experiments indicated improved accuracy after logarithmic rescaling of Rep; this changes the feature representation for the GNN and GP.
assumptions (5)
  • domain assumption Drag deviation from the mean is a sum of independent pairwise interaction functions (Eq. 8).
    This ansatz is built into the GNN architecture and carried into GP; the paper acknowledges it loses validity in dense assemblies near eps_p = 0.5 to 0.6.
  • domain assumption A perfect mean drag model exists, so only the deviation Delta F_x / <F_x> needs modelling.
    Used in Eq. (9) and throughout; the mean deviation is treated as approximately zero.
  • domain assumption The PR-DNS dataset of van Wachem et al. [66,67] is accurate and representative for all Rep and eps_p pairs.
    All GNN training and test labels come from this openly available dataset generated by the authors' group; there is no independent experimental validation.
  • ad hoc to paper The selected GNN's pairwise interaction outputs are a faithful target for symbolic regression.
    Section 3.1 says the GNN must be accurate or the GP expressions inherit its errors; GP never sees raw PR-DNS labels.
  • domain assumption Rotational symmetry about the streamwise axis makes the azimuthal angle phi nearly irrelevant.
    The authors use this to interpret the absence of phi in symbolic expressions, while the GNN permutation test still shows a small phi influence (Fig. 8).

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Pith. "Pith review of Drag modelling for flows through assemblies of spherical particles with machine learning: A comparison of approaches." pith.science (2026). https://pith.science/paper/G3BLRTCA

@misc{pith2026250705983,
  author       = {Pith},
  title        = {Pith review of: Drag modelling for flows through assemblies of spherical particles with machine learning: A comparison of approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3BLRTCA}},
  note         = {Machine review of arXiv:2507.05983}
}
read the original abstract

Drag forces on particles in random assemblies can be accurately estimated through particle-resolved direct numerical simulations (PR-DNS). Despite its limited applicability to relatively small assemblies, data obtained from PR-DNS has been the driving force for the development of drag closures for much more affordable simulation frameworks, such as Eulerian-Lagrangian point particle methods. Recently, more effort has been invested in the development of deterministic drag models that account for the effect of the structure of the particle assembly. Current successful deterministic models are mainly black-box neural networks which: 1) Assume pairwise superposition of the neighbours' effect on the drag, and 2) Are trained on PR-DNS data for a wide range of particle concentrations and flow regimes. To alleviate the black-box nature of neural networks, we use genetic programming (GP) to develop interpretable models. In our previous research, this has been proven successful in the Stokes regime. In the current contribution, we extend the application of GP to higher particle Reynolds number regimes. This is done by training a graph neural network (GNN) on the PR-DNS data to learn the pairwise interactions among the particles that constitute the drag variation. The significance of the input features of the GNN is assessed via a feature permutation approach. Then, the estimated pairwise interactions as extracted from the GNN are fed to a GP algorithm, which searches for symbolic expressions that fit the input data. A comparison between the trained GNN model and the resulting symbolic expressions is presented, to assess whether the symbolic expression can capture the underlying patterns learnt by the GNN. The comparison demonstrates the potential of GP in finding relatively simple symbolic models. At the same time, the accuracy of the symbolic models slightly fall behind the GNN.

Figures

Figures reproduced from arXiv: 2507.05983 by the authors.

Figure 1
Figure 1. Sample flow field visualizations: 3D (left) and 2D (right), at Rep = 5 and 𝜀p = 0.2. The flow is driven in the positive 𝑥 direction [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Representation of a symbolic expression as a syntax tree. functions and features from the provided sets. Various techniques have been presented in the past to approach SR problems. A popular method for physics and fluids is sparse identification of nonlinear dynamical systems (SINDy), which exploits sparse regression to describe underlying principles of nonlinear dynamical systems [13, 21]. AI Feynman [65, 64] aims … view at source ↗
Figure 3
Figure 3. Examples of crossover and mutation operation, where crossover points are marked as red arrows and mutation points as red nodes. Similar to the area of PIML, the integration of domain-specific knowledge into GP algorithms has become increasingly important in recent research. By incorporating prior knowledge, expressions can be constrained to follow expected behaviours or patterns, while reducing the search space of p… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Symbolic models are generated from simulation data using a GNN as a surrogate model. Physical particles in the simulation translate to nodes in the GNN [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Distribution of 𝑅2 over 11 repeated runs of the GNN, disaggregated by Rep (left) and 𝜀p (right). The markers indicate the 𝑅2 values of the selected model that was used to generate the training data for the GP algorithm [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Correlation plots of the predicted deviation from the mean drag as predicted by the GNN (x-axis) and ground truth from the particle-resolved simulation (y-axis) for different 𝜀p . Training and test data are shown in blue and red markers, respectively. The diagonal line…
Figure 7
Figure 7. Figure 7: Correlation plots of the predicted deviation from the mean drag as predicted by the GNN (x-axis) and ground truth from the particle-resolved simulation (y-axis) for different Rep . Training and test data are shown in blue and red markers, respectively. The diagonal lin…
Figure 8
Figure 8. Figure 8: Feature importance analysis of the selected GNN model: the y-axis indicates how much the MSE increases, ΔMSE, when the feature on the x-axis is permuted. We report ΔMSE over ten permutations per feature to eliminate randomness in the shuffling. a time. This expectation…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.