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REVIEW 6 major objections 5 minor 41 references

Using Machine Learning and Neural Networks to Analyze and Predict Chaos in Multi-Pendulum and Chaotic Systems

T0 review · 6 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that LSTM, GRU, and VRNN recurrent networks outperform classical machine-learning models at predicting multi-pendulum chaos, and that a time-step training approach can generalize to unseen initial angles in moderately…

desk verdict A student benchmark with a sensible time-step idea whose central generalization claim is contradicted by its own frictionless test and rescued only by switching to friction. read the letter →

arxiv 2504.13453 v1 pith:TTYBMCCO submitted 2025-04-18 cs.LG nlin.CDphysics.comp-ph

Vasista Ramachandruni , Sai Hruday Reddy Nara , Geo Lalu , Sabrina Yang , Mohit Ramesh Kumar , Aarjav Jain , Pratham Mehta , Hankyu Koo
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Jason Damonte Marx Akl
This is my paper · ORCID
classification cs.LGnlin.CDphysics.comp-ph
keywords chaoticsystemsmulti-pendulumLSTMtime-seriespredictionODE-RK4Lyapunovexponentsrecurrentneuralnetworkstime-stepapproach
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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 determine which of ten machine-learning models and neural networks can predict the chaotic motion of double- and triple-pendulum systems, using synthetic trajectories generated by fourth-order Runge-Kutta integration. Its central claim is that recurrent architectures, especially LSTM, with GRU and VRNN close behind, outperform classical regressors, and that a time-step based approach can generalize to unseen initial angles in moderately chaotic, friction-damped scenarios. The paper's motivation is that chaotic systems govern weather, disease, and financial markets, so a model that predicts them cheaply would be broadly useful. A sympathetic reading is that the ranking is reliable for short-horizon, low-dimensional, noise-free pendulum data; the same data also show that the sliding-window method is not truly predicting chaos.

What carries the argument

The central mechanism is the time-step based approach: train on 30 trajectories with initial angles $[120^\circ, 0^\circ]$ through $[120^\circ, 3.0^\circ]$ in $0.1^\circ$ increments, feed the time step as an input feature along with the initial angle, and then test on an untrained in-between angle such as $[120^\circ, 2.05^\circ]$. This turns prediction into an interpolation problem over initial-condition space rather than pure time-series extrapolation. It is carried by gated recurrent architectures, with LSTM's forget and input gates and GRU's update and reset gates credited for retaining the temporal information needed for chaotic dynamics while ignoring irrelevant fluctuations. The ODE-RK4 solver supplies the ground-truth trajectories, and Lyapunov-exponent and eigenvalue analyses are used to identify which initial-condition regions are truly chaotic and therefore harder to predict.

What would settle it

Run the paper's frictionless time-step protocol exactly: train ten models on double-pendulum trajectories for initial angles $[120^\circ, 0^\circ]$ through $[120^\circ, 3.0^\circ]$ in $0.1^\circ$ steps, with 2,000 steps per angle, then test on $[120^\circ, 2.05^\circ]$ and record LSTM RMSE. If the RMSE is near 0.26 with $R^2 \approx 0.23$, the generalization claim fails outside the damped regime; if it is near the friction-case value of $1.5\times10^{-2}$ with $R^2 \approx 0.996$, the claim stands. A complementary check is to compute the Lyapunov exponent for the unseen angle and see whether all failures occur where the exponent is positive.

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Extended reading notes

Core claim

On its own terms, the paper claims an empirical hierarchy: LSTM is the best model for the double pendulum in both the sliding-window and time-step approaches, with and without friction, reaching $R^2 = 0.998$ with RMSE $1.4\times10^{-2}$ in the sliding-window baseline and $R^2 = 0.991$ with RMSE $2.7\times10^{-2}$ on a trained angle in the time-step approach. For the triple pendulum, VRNN wins the sliding-window test, GRU wins the time-step test, and LSTM is best when friction is added. The paper's global conclusion is that LSTM can successfully predict chaotic systems with up to three features, including initial angles it has never seen, provided the chaos is moderate or damped. The frictionless test on the unseen angle is reported as a failure, with LSTM RMSE 0.26 and $R^2 = 0.23$, so the generalization claim is explicitly limited to reasonably chaotic scenarios.

Load-bearing premise

The paper's generalization claim rests on the assumption that a model trained on trajectories from discrete initial angles can interpolate to an untrained in-between angle even though chaotic systems amplify tiny differences in initial conditions; the paper's own frictionless test of this assumption fails, with LSTM RMSE 0.26 and $R^2 = 0.23$, so the assumption holds only for damped, moderately chaotic cases.

Editorial extensions

If this is right

  • For low-dimensional, friction-damped mechanical systems, LSTM is the first model to try, with GRU and VRNN as close alternatives.
  • The time-step approach allows full-interval trajectory visualization and testing on unseen initial angles, which the sliding-window approach cannot do.
  • The sliding-window approach should not be used as evidence about chaos prediction, since the paper finds it merely fits an erratic curve.
  • Models built on linear dependencies, such as autoregressive and feed-forward networks, are poor choices for chaotic dynamics.
  • Prediction quality degrades as the number of coupled variables grows: double-pendulum results are stronger than triple-pendulum results.

Reading between the lines

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

  • One consequence the authors leave implicit is that the in-between prediction task is only well-posed when the prediction horizon is short relative to the Lyapunov time; the frictionless failure with RMSE 0.26 is exactly what exponential divergence predicts, so the method's boundary could be stated in Lyapunov units.
  • A testable extension is to apply the same time-step protocol to other low-dimensional chaotic systems, such as the Lorenz or Rössler systems, and see whether LSTM retains the top ranking when the initial-condition grid is varied.
  • Because all models shared identical hyperparameters, the reported ranking is an upper bound on what tuned versions of each architecture might achieve; per-model tuning could plausibly change the ordering.
  • The paper's own Lyapunov heatmap could be turned into a quantitative predictor of model error by regressing RMSE on the local Lyapunov exponent to find the threshold separating predictable from unpredictable initial conditions.
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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

6 major / 5 minor

Summary. The paper evaluates ten machine learning and neural network models on synthetic double- and triple-pendulum trajectories generated with ODE-RK4. It first uses a single-step sliding-window approach and then proposes a 'time-step based approach' in which models are trained on trajectories at many initial angles and tested on an untrained 'in-between' angle (e.g., [120°, 2.05°]). The authors report RMSE and R² for frictionless and friction-damped systems, analyze chaoticity with Lyapunov exponents and eigenvalues, and conclude that LSTM best predicts chaotic systems with up to three features, with GRU and VRNN also performing well. The paper's central contribution is the claimed ability to generalize across initial angles in the time-step approach.

Significance. If the central claim were established, a reliable model ranking for predicting multi-pendulum trajectories at unseen initial angles would be a useful empirical contribution to machine learning for chaotic systems. The paper also offers a reproducible synthetic-data pipeline and openly states several limitations, including the noise-free simulation setting and uniform hyperparameters. However, the current evidence does not support the central generalization claim: the only frictionless in-between test fails, the friction-based success depends on damping equations that are never specified, results are single-run without uncertainty quantification, and the triple-pendulum equations contain unresolved symbolic placeholders. The significance of the work is therefore contingent on corrections that go beyond local presentation issues.

major comments (6)
  1. [§2, '10,000,000 synthetic data points'] The dataset size is internally inconsistent. A time interval from 0 to 1000 seconds at step size 0.001 yields 10^6 steps, not the stated 10^7 data points. Later in the same section, 10 seconds at step 0.001 yields 10,000 steps per trajectory, not the stated 2,000 data points per initial angle. Since the training/test split and the reported RMSE values depend on these counts, the experimental setup needs to be restated correctly and rerun if the reported counts are wrong.
  2. [§2.2 and §3.2, in-between angle test] The central generalization claim is contradicted by the paper's own frictionless result: the LSTM, the best model in the authors' ranking, achieves RMSE 0.26 and R² 0.23 on the unseen initial angle [120°, 2.05°] (Section 3.2, Figure 14B). This is essentially no predictive skill. The paper then switches to the friction-damped system and reports success there, but the frictionless failure is the direct test of the time-step approach as defined in Section 2.2. The Discussion's claim that LSTM predicts 'even ones it hasn't seen before' is supported only by a damped system whose equations are never given and by a single interpolation point without repeated trials.
  3. [§3.1, friction implementation] The friction model is not reproducible. The text says friction was introduced 'through the use of a damping constant, damping1 for the first pendulum, and damping2 for the second pendulum,' but the modified equations of motion and the numerical values of damping1 and damping2 are never provided. Without these, the friction-based results in Figures 12, 13, 15, and 16 cannot be verified or compared against the frictionless results.
  4. [§1, triple pendulum equations] The triple pendulum equations contain unresolved symbolic placeholders such as 'od1_1 ⋅ od1_2 ⋅ od1_3 ⋅ r2−4' and 'od3_3'. These are not mathematical equations. As a result, the triple-pendulum trajectory data cannot be reproduced or checked, and the triple-pendulum model rankings in Section 3.3 rest on an unverifiable simulation.
  5. [§3.3 and §4, numerical inconsistencies] The reported metrics for the triple pendulum are mutually inconsistent. In Section 3.3 the GRU on the time-step frictionless test has RMSE 1.688E-1, while the Conclusion reports RMSE 1.688E-2; for the friction in-between test Section 3.3 reports R² 0.98823 and RMSE 9.112E-3, while the Conclusion reports R² 0.99909 and RMSE 6.497E-3. These are not round-off differences. Because all metrics appear to be single-run values without seeds or error bars, the model ranking is unreliable even for the experiments that are described.
  6. [§3.5, Discussion claims] The Discussion states that 'LSTM was the best model to successfully predict chaotic systems with up to 3 features, even ones it hasn't seen before in moderately chaotic scenarios,' but the results in Section 3.3 name GRU as the best triple-pendulum model in the time-step approach, and the abstract says LSTM was best for triple pendulum with friction. The claim is also not qualified by the paper's own limitation statements about uniform hyperparameters and noise-free data. The conclusion overreaches the presented evidence.
minor comments (5)
  1. [§2, initial conditions list] The initial conditions list in Section 2 repeats 'm1 = 1 kg' twice and omits m2; this is likely a typographical error that should be corrected.
  2. [Figure 6 caption] The caption for Figure 6A says the time interval is (0, 100) while the body text and Figure 6B refer to (0, 1000); the interval should be stated consistently.
  3. [§3.5, GRU description] The Discussion describes GRU as a 'Convolutional Neural Network'; GRU is a gated recurrent architecture, not a convolutional network. This misclassification should be corrected.
  4. [References] Reference [17] appears twice with different titles, and several in-text citations (e.g., the 'score of' in the Introduction and the '(citation needed)' in Section 2) are incomplete placeholders.
  5. [Throughout] Several figure captions and section headings mix up 'trained angle' and 'untrained angle' (e.g., Figure 16A and 16B captions contradict the text). The captions should be checked against the reported experiments.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the time-step holdout test is a genuine out-of-sample interpolation, and the paper's own failed frictionless result confirms predictions are not forced by construction.

full rationale

The paper's central contribution is an empirical model comparison: train several ML models on simulated double- and triple-pendulum trajectories and evaluate on metrics such as RMSE and R^2. The time-step based approach trains on trajectories with initial angles [120°, 0°], [120°, 0.1°], ..., [120°, 3.0°] and then tests on an untrained 'in-between' angle [120°, 2.05°]. This is a genuine holdout along the initial-angle dimension: the test trajectory is not used in fitting, and no fitted parameter or model output is constructed to equal the test labels. The paper's own frictionless result for this holdout is poor (LSTM RMSE of 0.26 and R^2 of 0.23), which is strong evidence that the test is not circular; a setup where the prediction were equivalent to the input by construction would guarantee a near-perfect match. The later switch to friction-damped systems and the discussion claim that LSTM generalizes are empirical claims, not identities derived from the inputs. There are no load-bearing self-citations, no imported uniqueness theorems, and no ansatz smuggled in through prior work by the same authors. The equations of motion are standard ODE-RK4 solutions cited from external sources. The manuscript's internal inconsistencies, such as the triple-pendulum GRU RMSE differing between Section 3.3 (1.688E-1) and the Conclusion (1.688E-2), and the unsupported statement that LSTM was best for systems 'with up to 3 features' despite the triple-pendulum results favoring GRU, are correctness and reporting concerns, not circularity. Because no predicted quantity reduces by construction to a fitted parameter or to a self-citation chain, the circularity score is 0.

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

The core empirical claims rest on simulated data generated by RK4 integration, on the correctness of equations adopted from cited sources, on unspecified damping and hyperparameter choices, and on the assumption that training on a 0.1-degree-spaced grid of initial angles permits prediction at an intermediate angle in a chaotic system. None of these are derived in the paper; they are inputs. No new entities are postulated.

free parameters (4)
  • Damping constants for friction = Not reported in the paper.
    The friction-based results, which drive the main success claims, depend on unspecified damping coefficients introduced into the differential equations. No equation form or value is given.
  • Uniform model hyperparameters = Not reported in the paper.
    The paper states all models shared identical hyperparameters with Adam and minimum learning rate 1e-4, but layers, hidden sizes, sequence lengths, epochs, batch size, and regularization are not given. Rankings may change with tuning.
  • RK4 integration step size and interval = h = 0.001, intervals 1000 s and 10 s.
    Chosen by hand to balance computational cost and chaos development; these choices determine the data generating process and therefore all reported metrics.
  • Initial angle training grid for time-step approach = For double pendulum: theta2 from 0 to 3 degrees in 0.1 degree steps; for triple pendulum: theta3 from 0.1 to 3 degrees…
    The grid defines the 'in-between' interpolation problem. The choice of 0.1 degree spacing is asserted to give the most predictable deviation without evidence.
assumptions (4)
  • domain assumption The multi-pendulum equations of motion from cloud4science and Yesilyurt are correct and complete.
    The paper does not derive the equations; the double pendulum equations are displayed with formatting errors, and the triple pendulum equations contain unresolved variables. Referenced in Section 2 and Figure 2.
  • standard math RK4 numerical integration with step size 0.001 produces ground-truth trajectories suitable for training and evaluation.
    RK4 is a standard method, but no convergence check or validation against an independent solver is provided. This is the training target for all models.
  • ad hoc to paper A model trained on trajectories at 0.1 degree intervals can predict a trajectory at an unobserved intermediate initial angle within 10 seconds.
    This is the core assumption of the time-step approach in Section 2.2. The system is chaotic, so this interpolation is not guaranteed. The frictionless result for [120, 2.05] degrees violates the assumption, with LSTM RMSE 0.26 and R2 0.23.
  • standard math MinMax scaling and normalization preserve the temporal structure needed for prediction.
    Scaling is a monotone transform per feature and is standard, but the paper does not describe what was scaled or how inverse transforms were applied to RMSE values.

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Cite this review

Pith. "Pith review of Using Machine Learning and Neural Networks to Analyze and Predict Chaos in Multi-Pendulum and Chaotic Systems." pith.science (2026). https://pith.science/paper/TTYBMCCO

@misc{pith2026250413453,
  author       = {Pith},
  title        = {Pith review of: Using Machine Learning and Neural Networks to Analyze and Predict Chaos in Multi-Pendulum and Chaotic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTYBMCCO}},
  note         = {Machine review of arXiv:2504.13453}
}
read the original abstract

A chaotic system is a highly volatile system characterized by its sensitive dependence on initial conditions and outside factors. Chaotic systems are prevalent throughout the world today: in weather patterns, disease outbreaks, and even financial markets. Chaotic systems are seen in every field of science and humanities, so being able to predict these systems is greatly beneficial to society. In this study, we evaluate 10 different machine learning models and neural networks [1] based on Root Mean Squared Error (RMSE) and R^2 values for their ability to predict one of these systems, the multi-pendulum. We begin by generating synthetic data representing the angles of the pendulum over time using the Runge Kutta Method for solving 4th Order Differential Equations (ODE-RK4) [2]. At first, we used the single-step sliding window approach, predicting the 50st step after training for steps 0-49 and so forth. However, to more accurately cover chaotic motion and behavior in these systems, we transitioned to a time-step based approach. Here, we trained the model/network on many initial angles and tested it on a completely new set of initial angles, or 'in-between' to capture chaotic motion to its fullest extent. We also evaluated the stability of the system using Lyapunov exponents. We concluded that for a double pendulum, the best model was the Long Short Term Memory Network (LSTM)[3] for the sliding window and time step approaches in both friction and frictionless scenarios. For triple pendulum, the Vanilla Recurrent Neural Network (VRNN)[4] was the best for the sliding window and Gated Recurrent Network (GRU) [5] was the best for the time step approach, but for friction, LSTM was the best.

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Reviewed August 16, 2026 · model on record in the stance chip above.