Data-driven modeling of a settling sphere in a quiescent medium

C. Ricardo Constante-Amores; Haoyu Wang; Isaac J. G. Lewis; Leonardo P. Chamorro; Soohyeon Kang; Yuechao Wang

REVIEW 3 major objections 5 minor 63 references

Data-driven modeling of a settling sphere in a quiescent medium

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Trained on positions alone, neural differential equations forecast a chaotic settling sphere's ensemble statistics—terminal velocity, settling time, lateral motion, and acceleration—without simulating the fluid; settling-time means match…

desk verdict The NODE half is a solid, citable contribution on a genuinely new PTV dataset; the NSDE half is in-sample reconstruction dressed up as forecasting, and the authors' own conclusion admits as much. read the letter →

arxiv 2507.12551 v1 pith:BNQYMFVA submitted 2025-07-16 physics.flu-dyn

Haoyu Wang , Isaac J. G. Lewis , Soohyeon Kang , Yuechao Wang , Leonardo P. Chamorro , C. Ricardo Constante-Amores This is my paper

classification physics.flu-dyn PACS 47.55.-t47.55.Kf47.55.pb

keywords neuralordinarydifferentialequationsstochasticsettlingsphereparticletrackingvelocimetrychaotictrajectoriesdata-drivendynamicsterminalvelocityGalileonumber

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 sets out to show that the ensemble behavior of a freely settling sphere in a chaotic flow regime can be forecast directly from experimental trajectory data, with no simulation of the surrounding fluid. Using roughly 200 particle-tracking runs of a 6 mm sphere in water (Galileo number 1428), the authors train two neural differential equations—a deterministic NODE and a stochastic NSDE—with the particle's position as the only state variable. They report that both models reproduce the ensemble-averaged streamwise and lateral velocity profiles, the settling-time distribution (mean within 0.7 percent for the NODE and 0.4 percent for the NSDE), the probability density of final horizontal positions, and auto- and cross-correlation functions, and that second-order acceleration statistics also match at long times. The practical payoff would be a cheap, purely data-driven surrogate for particle settling in environmental and industrial settings where the governing equations are unknown or too expensive to resolve.

What carries the argument

The machinery is the neural differential equation built on a position-only state. For the NODE, the velocity field $g(x)$ in $dx/dt = g(x)$ is a small multilayer perceptron trained by minimizing the squared difference between predicted and observed one-step-ahead positions, with gradients obtained by automatic differentiation. For the NSDE, the trajectory is a stochastic process $dX_t = f_\theta(X_t,t)\,dt + g_\theta(X_t,t)\,dB_t$ solved with the Euler–Maruyama scheme under Stratonovich integration, where $f_\theta$ is the drift network, $g_\theta$ the diffusion network, and a control term $\bar{X}_t = \xi(t, X_t, \chi(t); \theta_\xi)$ driven by the cubic-spline interpolation of the observed path guarantees a unique strong solution and stabilizes training. This control-based formulation, taken from the NSDE literature, is what allows the stochastic model to be trained with a plain mean-squared-error loss rather than a separate discriminator network.

What would settle it

Release the sphere from rest at two different depths in the same tank, so that trajectories pass the same depth with different velocity histories, and check whether the trained position-only NODE predicts the same velocity at that depth; because the model asserts $dx/dt = g(x)$, any measurable history-dependence of velocity at a fixed position would falsify the Markov assumption.

Watch

Extended reading notes

Core claim

The paper's central claim is that the chaotic three-dimensional settling of a sphere can be learned, at the level of ensemble statistics, by a neural network that maps position to velocity. The deterministic NODE integrates $dx/dt = g(x)$ with $g$ a small multilayer perceptron and accurately reconstructs individual trajectory statistics, generalizing to unseen initial conditions drawn from the same distribution; the stochastic NSDE augments the drift network with a learned diffusion term and a cubic-spline control path, and better reproduces the long-time probability distributions of final horizontal positions. Both models capture the ensemble-averaged streamwise and lateral velocity as functions of depth, the settling time to $z/d = 50$ ($\bar{\tau}_z = 56.2$ for NODE and $55.6$ for NSDE versus $55.8$ for experiment), the decay of auto- and cross-correlation functions, and the ensemble-averaged acceleration toward zero at long times. The authors conclude that 'both models can accurately forecast the statistical characteristics of the settling sphere system up to the second-order dynamics with minor error,' while the short-time transient associated with release and free-surface entry remains poorly captured.

Load-bearing premise

The load-bearing assumption is that a settling sphere's velocity is a unique function of its current position, even though the real sphere carries momentum, added-mass forces, and wake memory from its past motion, which is precisely why the start-up transient and free-surface entry are the paper's reported weak spot.

Editorial extensions

If this is right

A position-only neural ODE trained on roughly 200 experimental trajectories reproduces the ensemble-averaged terminal velocity, settling time (mean within 0.7 percent), and lateral-displacement statistics of a chaotic settling sphere.
The stochastic NSDE, despite missing the short-time transient, matches the long-time probability density of final horizontal positions better than the deterministic NODE.
Both models capture second-order dynamics: ensemble-averaged streamwise and lateral accelerations approach zero correctly at long times, although variances are underestimated.
Model accuracy degrades with dataset size in different ways: NODE trained on 50 percent of the data still captures the general statistical trend, while both models trained on 10 percent fail.
Because the models operate on positions alone and require no fluid solver, they can serve as statistical surrogates for settling in flows whose governing equations are not explicitly known.

Reading between the lines

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

We infer that the position-only Markov assumption, if it holds, implies the deterministic model cannot distinguish two trajectories that pass the same depth with different velocity histories; testing this would sharpen the method's boundary.
We read the authors' contrast between smooth NODE trajectories and noisy experimental lateral velocity as evidence that the effective noise is intrinsic to this flow regime; comparing the NSDE's learned diffusion amplitude with experimental repeatability would test this.
We expect the framework to transfer to other particle shapes and fluid types only if a position-only state remains adequate; adding velocity or a memory term to the state would be the natural extension whenever short-time transients contribute to the statistics of interest.

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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

Quick take: read it for the dataset and the NODE result, not for the NSDE. The NODE is a genuine contribution — an 80/20 train/test split with ensemble statistics reproduced on held-out initial conditions and settling time within 0.7 percent. The NSDE is a different story: its control term feeds the target trajectory's cubic spline coefficients into the model, so every NSDE trajectory in Figs. 2-5 is a reconstruction of training data. The 55.6 versus 55.8 settling-time agreement is interpolation, not prediction. The Conclusion half-admits this — the NSDE 'cannot generalize the short-time dynamics of unseen initial conditions given the exclusive dependence between each trajectory and its own cubic spline coefficients' — but the abstract and the 'both models can accurately forecast' sentence should have been scoped the same way.

What is actually new is the experiment: roughly 189 full three-dimensional chaotic settling trajectories for a sphere at Ga = 1428, released from rest above the free surface and tracked with dual-camera PTV. That is a useful resource on its own. Using NODE versus NSDE to compare deterministic and stochastic descriptions of the lateral wandering is a fair way to ask whether that wandering is intrinsically stochastic, and the dataset-size degradation study (100/50/10 percent) is a sensible robustness check.

Soft spots, in decreasing order. First, the NSDE evaluation gap above. The fix is straightforward: train on a subset of trajectories and test on held-out trajectories, or drop the control term and generate forward trajectories from the learned SDE. Second, the SDE is underspecified: the text says Stratonovich (Eq. 5) but the solver is Euler-Maruyama, an Ito scheme, and the control term defined in Eq. 10 never appears in the solved equation, Eq. 9. With state-dependent diffusion the convention matters, so the NSDE as written is ambiguous. Third, the position-only Markov assumption in Eq. 1 cannot represent the acceleration phase. The authors flag the short-time transient failures themselves, so this is a scoping limitation rather than a hidden flaw; it matters mostly because it is the reason the NSDE jumps unrealistically at early times.

The stress-test note holds up. I'd send this to review: a serious referee can push for held-out NSDE evaluation, a cleaned-up SDE specification, and ideally a public dataset and code release. I would take the NODE result and the dataset, and I would not cite the NSDE numbers as predictive evidence. The right outcome is conditional accept, not rejection.

Referee Report

3 major / 5 minor

Summary. The paper develops deterministic (NODE) and stochastic (NSDE) data-driven models for the three-dimensional trajectories of a settling sphere in a quiescent fluid, using experimental PTV data. The NODE is trained on an 80/20 train/test split and evaluated on both seen-distribution and held-out initial conditions (Appendix A1), while the NSDE uses a control term built from each trajectory's cubic spline coefficients and is trained on the full dataset. The models are compared against experiment for ensemble-averaged velocities, lateral displacement, settling-time distributions, correlation functions, and acceleration statistics. The central claim is that both models accurately forecast the time evolution and statistical characteristics of the system up to second-order dynamics.

Significance. If the forecasting claim were fully supported, this would be a valuable demonstration of data-driven surrogates for a chaotic particle-laden flow, with a rare high-resolution experimental dataset. The NODE validation on a genuinely held-out 20% test set in Appendix A1 is a solid element, as is the systematic comparison of ensemble statistics and the dataset-size sensitivity study. The NSDE, however, is evaluated only on the trajectories used for training, so the evidence for its forecasting capability is currently missing. The paper's own conclusion concedes that the NSDE 'cannot generalize the short-time dynamics of unseen initial conditions' because of the spline-control dependence. Thus the significance of the present results is real but narrower than the stated 'both models' claim.

major comments (3)

[Section III.A, Figs. 2–5, and Conclusions] The NSDE evaluation in Section III.A uses the same 189 experimentally gathered trajectories that were used for training: the control term in Eq. (10) requires the cubic spline coefficients of each target trajectory, and the paper states (Section III.A) that 'it was necessary to use only experimentally gathered data for the NSDE case.' Consequently, Figs. 2(c,f), 3(a,b), 4, and 5 for the NSDE demonstrate reconstruction/interpolation of training data, not forecasting. The Conclusion's statement that 'both models can accurately forecast the statistical characteristics of the settling sphere system' is therefore not supported for the NSDE by the reported evidence. The authors should either validate the NSDE on held-out trajectories (e.g., train on a subset and evaluate on unseen trajectories) or explicitly reframe the NSDE results as a reconstruction/statistical-generation check rather than a forecast.
[Section II.B.2, Eqs. (9) and (10)] The NSDE method is underspecified. The solved equation shown as Eq. (9) is dX_t = f_theta dt + g_theta dB_t, but the control term \bar{X}_t introduced in Eq. (10) is omitted from Eq. (9); it is therefore unclear whether the trained model includes the control term in the drift, as implied by the surrounding text, or not. In addition, the text declares Stratonovich integration in Eq. (5) and discusses midpoint evaluation, yet the numerical solver is stated to be Euler-Maruyama, which is an Ito-type scheme. This discrepancy matters for the interpretation of the diffusion term and for reproducibility. Please specify the exact form of the solved SDE (including the control term) and the integration convention actually used.
[Section II.B.1, Eq. (1)] The modeling assumption dx/dt = g(x) with x the particle position alone restricts the state to a first-order Markovian ODE in position. A settling sphere is a second-order, history-dependent system (added mass, drag, wake memory, Basset history force), and this structural assumption cannot represent the initial acceleration from rest or the transient after free-surface entry. The paper's own observations of poor short-time transient capture (Figs. 2c,f and 3a) are consistent with this limitation, but the text tends to attribute them solely to data or stochastic effects. This assumption should be explicitly acknowledged as a modeling choice with known consequences, and the short-time limitations should be framed accordingly.

minor comments (5)

[Section II.B.2] The solver is repeatedly called 'Euler-Murayama'; the standard spelling is 'Euler-Maruyama'.
[Section III.A, first paragraph] The text says 'Fig. 2a-f illustrates the temporal evolution...' but the panels are grouped as (a-c) for lateral displacement and (d-f) for lateral velocity; please align the panel references with the actual figure layout.
[Appendix A1] Appendix A1 states that the 189 ground-truth trajectories are split 80/20, but the main text says the experiment was repeated 'approximately 200 times' and that NSDE analysis used 'exactly 189 trajectories.' The relationship between these numbers should be clarified for consistency.
[Section III.A, paragraph on NODE deviations] The phrase 'the model trained on more data from the later stages' is not supported by the training description; please rephrase to describe the actual effect of the training data distribution on early-stage accuracy.
[Section II.B.2, Eq. (7)] The statement that 'the only difference in evaluation of the Ito and Stratonovich integrals is in the choice of t*_j' is an oversimplification; the distinction also affects the chain rule and the resulting stochastic calculus. A more careful wording would avoid confusing readers.

Circularity Check

2 steps flagged · score 6.0 of 10

NSDE 'forecast' reduces to reconstruction: Eq. (10) feeds each trajectory's own cubic spline into the model, and all NSDE statistics come from the same 189 training trajectories; the NODE branch is independently tested.

self definitional [Sec. II.B.2, Eq. (10) and following paragraph]
"Specifically, this control term is introduced as ¯Xt = ξ(t, Xt, χ(t); θξ), (10) ... By defining our network ξ with respect to the cubic spline coefficients, the continuously defined paths only need to be pre-initialized while still giving the advantage of a unique strong solution throughout training of the SDE."

Eq. (10) makes the NSDE output a function of the controlled path χ(t), and χ(t) is the cubic spline interpolation of the experimental trajectory being modeled. The predicted trajectory is therefore not generated from initial conditions alone; its own target appears as an input. Statistical agreement of the NSDE output with that target is built into the construction, so comparing them is a reconstruction check rather than an independent forecast.
fitted input called prediction [Sec. III.A (Reconstruction by NODE & NSDE), paragraph before Fig. 2]
"However, because the control term in the NSDE architecture requires prior generation of cubic spline coefficients, it was necessary to use only experimentally gathered data for the NSDE case. As a result, analysis done by the NSDEs consisted of exactly 189 trajectories as opposed to the 400 randomly generated initial conditions gathered for the NODE case described above."

All 189 trajectories used for NSDE analysis are the trajectories on which the 100%-data NSDE was trained ('the NSDE model was trained on the full dataset for the 100% data case'). No held-out trajectories are used for the NSDE statistics in Figs. 2-5. The reported long-time agreement (mean settling time 55.6 vs 55.8, displacement PDFs, correlation functions, acceleration) is therefore an in-sample reconstruction, not a forecast. The conclusion concedes the dependence ('cannot generalize ... exclusive dependence between each trajectory and its own cubic spline coefficients') but still presents both models as forecasting the statistics.

full rationale

The NODE branch is not circular: it is trained on an 80% split, has a genuine held-out test in Appendix A1, and its ensemble check samples new initial conditions from a fitted Gaussian, so its statistical agreement is an out-of-sample statement within the same initial-condition distribution. The NSDE branch, however, is circular in the specific sense that the quantity being 'forecast' is fed into the model. The control term in Eq. (10) uses the cubic spline coefficients of the experimental trajectory, and the 100%-data NSDE was trained and evaluated on the same 189 trajectories; no NSDE evaluation is performed on held-out trajectories. The paper itself concedes the exclusive dependence between each trajectory and its own spline coefficients, yet still summarizes the NSDE results as forecasting the statistical characteristics. This is partial circularity: one of the two models' central claims reduces by construction to a reconstruction check, while the NODE claim retains independent content. No load-bearing self-citation chain is present; prior work by the authors is cited incidentally.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced; the models are purely data-driven. The load-bearing assumptions are the position-only Markovian state and the representativeness of the experimental ensemble. The main fitted quantities are the neural network weights and the initial-condition Gaussian used for NODE validation.

free parameters (4)

NODE network weights theta_f = Trained over 48,000 iterations, batch size 256, AdamW
Learned from 80 percent of the trajectory data to minimize MSE of position predictions.
NSDE drift and diffusion network weights theta = Trained over 30,000 iterations, batch size 16, Adam
Learned from the full dataset to minimize MSE of reconstructed trajectories.
Initial-condition Gaussian (mean, variance) = Mean and variance of experimental initial positions
Used to sample 400 synthetic initial conditions for NODE ensemble validation; fitted to the same experimental data.
Hyperparameters (hidden dimension, batch time, learning rate schedule) = Hidden dimension 64, learning rate 1e-3 to 1e-4, etc.
Chosen by hand; small networks were found necessary to avoid overfitting.

assumptions (5)

ad hoc to paper Position-only Markovian state: dx/dt = g(x) with x the particle position
Assumed in Eq. 1; reduces a second-order, history-dependent fluid-particle system to a first-order ODE in position, which cannot represent the initial transient from rest.
domain assumption The 189-200 PTV trajectories form a representative statistical ensemble
Needed for ensemble averages to be meaningful; the dataset is limited and the paper itself shows sensitivity to dataset size.
domain assumption Finite-difference velocities and accelerations from PTV positions with re-projection error 0.07d are accurate
Positions are differentiated with central differences, and the amplification of tracking noise by finite differences is not quantified.
domain assumption Goossens terminal velocity formula holds for this sphere
Used to normalize time and velocity; the value 363 mm/s closely matches the experimentally measured 360 mm/s, so this is a mild assumption.
standard math Ito and Stratonovich stochastic calculus background, including existence and uniqueness of SDE solutions
Standard mathematical background, but the paper states Stratonovich integration while using the Euler-Maruyama (Ito) solver, creating an inconsistency.

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Cite this review

Pith. "Pith review of Data-driven modeling of a settling sphere in a quiescent medium." pith.science (2026). https://pith.science/paper/BNQYMFVA

@misc{pith2026250712551,
  author       = {Pith},
  title        = {Pith review of: Data-driven modeling of a settling sphere in a quiescent medium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNQYMFVA}},
  note         = {Machine review of arXiv:2507.12551}
}

read the original abstract

We develop data-driven models to predict the dynamics of a freely settling sphere in a quiescent Newtonian fluid using experimentally obtained trajectories. Particle tracking velocimetry was used to obtain a comprehensive dataset of settling motions, which we use to train neural networks that model the spatial evolution of a spherical particle without explicitly resolving the surrounding fluid dynamics. We employ deterministic neural ordinary differential equations (NODEs) and stochastic neural stochastic differential equations (NSDEs) to reconstruct the sphere's trajectory and capture key statistical features of the settling process. The models are evaluated based on short- and long-time dynamics, including ensemble-averaged velocity evolution, settling time distributions, and probability density functions of the final settling positions. We also examine the correlation between lateral displacement and streamwise velocity and assess the impact of dataset size on predictive accuracy. While NODEs excel in trajectory reconstruction and generalization across different initial conditions, NSDEs effectively capture statistical trends in the long-time behavior but are more sensitive to data availability. Acceleration profiles computed via second-order finite difference schemes confirm that both approaches accurately capture long-time dynamics, though short-time transients pose challenges.

Figures

Figures reproduced from arXiv: 2507.12551 by the authors.

**Figure 2.** FIG. 2. (a-c) Dimensionless lateral displacement [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) Experimental and reconstructed ensemble-averaged streamwise, [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. (a) Probability density distribution of the dimensionless settling time [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: FIG. 5. Ensemble-averaged acceleration components for the NODE and NSDE models, and ex [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Ensemble-averaged streamwise and lateral velocities, (b) ensemble-averaged lateral [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

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Pith tools

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

[1] [1]

Department of Mechanical Science and Engineering, University of Illinois, Urbana, Illinois 61801, USA

work page

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Department of Aerospace Engineering, University of Illinois, Urbana, Illinois 61801, USA

work page

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Department of Civil and Environmental Engineering, University of Illinois, Urbana, Illinois 61801, USA

work page

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Department of Earth Science and Environmental Change, University of Illinois, Urbana, Illinois 61801, USA. † 1 arXiv:2507.12551v1 [physics.flu-dyn] 16 Jul 2025 Abstract We develop data-driven models to predict the dynamics of a freely settling sphere in a quiescent Newtonian fluid using experimentally obtained trajectories. Particle tracking velocimetry w...

work page Pith review arXiv 2025

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A common approach in data-driven modeling is to approximate the time evolution of x using a learned function g as follows dx dt = g(x)

NODE: Deterministic modeling Let’s assume a deterministic, Markovian dynamics, and x ∈ RdN represent the state, which is defined as the position of the particle center in the ( x, y, z) coordinate system. A common approach in data-driven modeling is to approximate the time evolution of x using a learned function g as follows dx dt = g(x). (1) here, g is t...

work page

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We use automatic differentiation to determine the derivatives of g with respect to θf Chen et al

(3) where B is the batch size. We use automatic differentiation to determine the derivatives of g with respect to θf Chen et al. [29]. To optimize the loss function described in equation 3, we use an AdamW optimizer in PyTorch Paszke. [46]. Table 1 summarizes the network architectures used in this study for the NODE. Finally, while stabilization terms hav...

work page

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Neural network architectures for the NODE and NSDE

NSDE: Stochastic modeling Due to the nature of the experimental data gathered and the underlying physical pro- cesses, a promising modification to classical deterministic data-driven models is to represent the system by a stochastic differential equation (e.g., due to potential noise in experimental 7 TABLE I. Neural network architectures for the NODE and...

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U. Ulusoy, A review of particle shape effects on material properties for various engineering applications: from macro to nanoscale, Minerals 13, 91 (2023)

work page 2023