Pith. sign in

REVIEW 4 major objections 7 minor 39 references

Learning Hamiltonian Dynamics with Bayesian Data Assimilation

T0 review · 4 major / 7 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Hamiltonian neural network trained on multi-step rollouts, coupled to an Unscented Kalman Filter, predicts long-horizon orbits while conserving energy.

desk verdict Useful combination of HNN+UKF for Molniya orbit prediction, but the evaluation needs error bars, a step-size check, and table cleanup before the claims are solid. read the letter →

arxiv 2501.18808 v1 pith:WABBFWML submitted 2025-01-31 cs.LG cs.ROcs.SYeess.SY

Taehyeun Kim , Tae-Geun Kim , Anouck Girard , Ilya Kolmanovsky This is my paper · ORCID
classification cs.LGcs.ROcs.SYeess.SY MSC 37M1570H0568T07
keywords HamiltonianneuralnetworksautoregressivetrainingunscentedKalmanfilterorbitdeterminationdataassimilationenergyconservationsymplecticintegratorlong-termprediction
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 claims that a Hamiltonian neural network trained without derivative labels can learn the dynamics of an unknown mechanical system and, when coupled with an Unscented Kalman Filter, deliver long-horizon predictions that conserve energy and stay close to the true trajectory. The core proposal is the Autoregressive Hamiltonian Neural Network (AHNN), whose loss unrolls the learned flow over a window of $W$ steps and penalizes the accumulated prediction error against the true sequence, rather than the one-step derivative mismatch used by standard HNNs. On a spring-mass system and on J2-perturbed Molniya orbits, the authors report that AHNN5 outperforms MLP, NODE, and standard HNN baselines in state RMSE, and that the AHNN-based UKF gives the lowest errors while producing uncertainty estimates. If correct, this matters because it shows that physically structured neural surrogates can serve as the process model inside a Bayesian filter for orbit determination and other Hamiltonian systems.

What carries the argument

The load-bearing object is the autoregressive Hamiltonian loss of Eq. (4), which treats the network as a map $F_\theta(x_k) = x_k + \int_{t_k}^{t_{k+1}} J\nabla H_\theta\,dt$ and trains it by unrolling $W$ composed copies of $F_\theta$ against the true trajectory segment. This makes the training objective match the closed-loop use of the model and penalizes error growth over exactly the horizon the model will be asked to predict. The second mechanism is the Unscented Kalman Filter, which propagates a set of $\sigma$ points through $F_\theta$, reconstructs mean and covariance, and corrects the state with measurements; this supplies both real-time refinement and uncertainty quantification. The paper also uses a Huber loss and a fourth-order Runge-Kutta integrator with a fixed 60-second step for the orbit experiments.

What would settle it

Re-run the orbit experiment on a test set where periapsis passages fall between the 60-second sample points, and compare AHNN5 predictions against a reference trajectory produced with an adaptive-step symplectic integrator; if the error spikes on those orbits or the energy drifts, the paper's claim that AHNN preserves invariants under its tested discretization fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Hamiltonian structure of an unknown system can be learned from state sequences alone and then used as a predictive model that both conserves energy and maintains accuracy over long rollouts. The authors replace the standard HNN loss on state derivatives with an autoregressive loss, Eq. (4), that applies the learned flow $F_\theta$ recursively for $W$ steps and averages the distance between the predicted and observed states. They then insert this flow into an Unscented Kalman Filter that propagates $\sigma$ points through $F_\theta$ and assimilates noisy position measurements. In their numerical experiments, AHNN5 achieves the best RMSE among MLP, NODE, HNN, and AHNN variants, with the UKF version AHNNKF5 best overall, and the reported energy RMSE for AHNN5 is around two orders of magnitude smaller than for MLP and NODE. The paper thereby claims a practical recipe for learning Hamiltonian dynamics where derivative measurements such as accelerations are unavailable.

Load-bearing premise

The whole method rests on the assumption that a fixed 60-second Runge-Kutta step resolves the sharp periapsis acceleration of the Molniya orbit, so the learned Hamiltonian is not biased by the discretization.

Editorial extensions

If this is right

  • If the central claim is correct, position-velocity telemetry alone is enough to train a prediction model that respects Hamiltonian structure, removing the need for noisy acceleration labels.
  • The best-performing model, AHNNKF5, provides both a low-error state estimate and a covariance, which is what real orbit determination and collision-assessment pipelines need.
  • Larger autoregressive windows improve prediction accuracy in the paper's ablations, at the cost of more expensive backpropagation through time; this gives practitioners a direct trade-off to tune.
  • The reported energy conservation means predicted trajectories will not exhibit the artificial spiral-in or spiral-out seen with MLP and NODE baselines over long horizons.

Reading between the lines

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

  • Editorial inference: the fixed 60-second Runge-Kutta step is part of what the network learns, so retraining with a symplectic or adaptive-step integrator is a direct test of whether the reported gains survive a different discretization.
  • Editorial inference: the same AHNN-UKF recipe should transfer to other systems with known symplectic structure but unknown potentials, such as rigid-body rotation or charged-particle motion in a magnetic field, though the paper does not demonstrate this.
  • Editorial inference: the paper's uncertainty bands come from the UKF's Gaussian approximation and may be optimistic under model mismatch; comparing the reported $3\sigma$ intervals against empirical error across many test orbits would quantify that.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The paper proposes a Hamiltonian Neural Network (HNN) training objective that does not require state-derivative labels, an autoregressive extension (AHNN) that trains over multi-step rollouts, and a UKF-based Bayesian data assimilation scheme (AHNNKF) for state estimation and uncertainty quantification. The authors validate the approach on a frictionless mass-spring system and on highly elliptic, J2-perturbed Molniya orbits, reporting RMSE and energy-conservation metrics. The central claim is that AHNN5 and AHNNKF5 achieve the best long-term prediction accuracy while preserving physical invariants such as energy, and that the UKF integration improves robustness and provides uncertainty estimates. The paper does not ship code, and the empirical validation has several gaps that currently prevent full confidence in the central claim.

Significance. If the claims are substantiated, the paper makes a useful algorithmic contribution: it replaces derivative labels with one-step transitions, adds an autoregressive training objective that is particularly relevant to long-horizon prediction, and embeds the learned surrogate in a Bayesian filter to obtain state estimates with uncertainty. The orbital benchmark on highly elliptic Molniya orbits is a meaningful and nontrivial test case, and the energy-RMSE comparisons against MLP and NODE baselines are informative. However, the load-bearing empirical claims are not yet fully supported: the fixed 60-second RK4 step may alias the fast periapsis dynamics, no error bars are reported around the headline RMSE values, and the gravitational potential used for data generation is underspecified. The paper is potentially valuable, but it needs substantial additional validation and clarification before the main claims can be accepted.

major comments (4)
  1. [§5.2 and Eq. (3)] The fixed 60 s RK4 step used to define and train Fθ may alias the periapsis segment of the Molniya orbits, so the claim that the learned Hθ preserves the physical Hamiltonian is not established. Section 5.1.2 reports periapsis altitudes of 540–560 km and eccentricities 0.7–0.8, and Figure 1 notes speed variations of an order of magnitude; with h=60 s the RK4 local error near periapsis is uncontrolled, and because the training samples are exactly 60 s apart, Hθ can absorb the discretization error into an effective step-dependent Hamiltonian. The reported energy RMSE (Figure 7) therefore does not distinguish a physical Hamiltonian from a step-tuned discrete surrogate. I request an explicit discretization-convergence study: retrain and evaluate with h=30 s and h=120 s, or with an adaptive integrator at a fixed tolerance, and show that the induced energy error and test RMSE are stable across these choices.
  2. [§5.3, Tables 1–2] No error bars or variance information are reported for the central comparisons. Section 5.2 states that models are evaluated with three random seeds, but Tables 1 and 2 report single RMSE values; it is not stated whether these are the best, median, or mean of the seeds, and no standard deviation or confidence interval is given. This matters because several advertised improvements are small: for example, in Table 1, AHNNKF5 position RMSE (0.6596) is only about 10% below HNNKF (0.7261) in the true-initial mass-spring case. Please report the mean and standard deviation over seeds and over test orbits for all entries in Tables 1 and 2, and similarly for the energy RMSE values reported in the text.
  3. [§4.2, Eq. (13)] The gravitational potential used to generate the orbital data is underspecified. Equation (13) presents a full spherical-harmonic expansion with unspecified degrees Nz and Nt, defines longitude via arccos, and does not specify the rotation between the Earth-centered inertial frame in which (q,p) are expressed and the body-fixed frame in which the potential is evaluated; the text then states that only J2 is used, but does not say whether the training and evaluation data were generated with only J2 or with additional GGM05C terms. Please state precisely which potential was used for data generation and training, give the J2 value and frame conventions, and replace the arccos longitude definition with atan2(y,x) if any longitude-dependent tesseral terms are retained.
  4. [Table 2, NODE row] In Table 2, the NODE row reports perturbed-initial RMSE values of 14374 and 25121, which are two orders of magnitude larger than the true-initial values (124.66 and 84.067) and appear inconsistent with the scaling column header; the text discusses this only qualitatively as a lack of robustness. Please clarify how these values were computed, whether any divergent filtered trajectories were excluded or clipped, and whether the scaling of the table entries is the same for all models.
minor comments (7)
  1. [Tables 1–2] Tables 1 and 2 contain repeated digit strings such as '1.55281.55281.5528' and '1.73811.73811.7381'; if these are not formatting artifacts in the manuscript, the table entries need to be corrected.
  2. [§5.2] The sentence 'We use a fourth-order Runge-Kutta integrator with a fixed time step of 60 s' appears to apply to the orbital experiments, but the mass-spring experiment uses a 0.01 s sampling interval; clarify that the 60 s step applies only to the orbital experiments and specify the integration step used for the mass-spring system.
  3. [References] The reference for Brown (2004) is incomplete because it lacks a title, and the AdamW citation lists only Loshchilov; please provide the full references.
  4. [§5.2] The 'ExpHyperbolicLR' scheduler is cited to a preprint by the authors but is not described in the paper; either define the scheduler or remove the dependency on an unpublished external method.
  5. [Eq. (7)] In Eq. (7), Yk is described as the set of measurements 'until the time instant tk+1'; it should be tk, and the quantity δxk+1 should be defined before it is used in the covariance expression.
  6. [Algorithm 1] Algorithm 1 has no return statement; specify the outputs (x̂k+1, Pk+1) to make the pseudocode self-contained.
  7. [§3.2.2] The sigma-point generation is described as taking 'the i-th row of the matrix square root'; specify whether a Cholesky factor or a symmetric square root is used, since this affects the spread of the sigma points.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the neural surrogates are trained on external trajectory data and benchmarked against independent mass-spring and J2-perturbed orbit tests; the only self-citation is a peripheral learning-rate scheduler.

full rationale

The central derivation chain is self-contained. The surrogate Fθ in Eq. (3) is trained by minimizing the metric d(Fθ(xk), xk+1) in Eq. (2) and its autoregressive extension in Eq. (4) against externally generated trajectory pairs; no target output is reused to define the loss. The benchmarks are independent: the true mass-spring Hamiltonian (Eq. 11) and the J2-perturbed two-body Hamiltonian (Eqs. 12-13) are fixed simulation models, with training data produced by GL4/KahanLi8 symplectic integrators and test RMSE computed against true states and energies. Energy conservation is not a fitted target; it is a structural property of the J∇Hθ form, and the reported energy RMSE is evaluated against the known physical energy, so the HNN/AHNN conservation results are external evidence rather than reverse-engineered outputs. The autoregressive recursion appears in both the loss (Eq. 4) and the evaluation, but that is the intended multi-step training objective, not a hidden reuse of test labels. UKF tuning of (α,β,κ) on validation RMSE is standard hyperparameter selection, and the final comparisons are on held-out test trajectories. The only self-citation is the ExpHyperbolicLR scheduler (Kim, 2024), which is peripheral to the dynamics-learning claim and not load-bearing. No load-bearing step reduces to its own inputs, so the paper does not exhibit circularity.

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

No new physical entities are introduced. The central claim rests on the Hamiltonian structural assumption, the discretization choices, and the known observation model; the free parameters are hyperparameters tuned on validation data.

free parameters (5)
  • Autoregressive window W = 5 (selected after ablation over {1,3,5})
    Controls the number of unrolled steps in LAHNN; the paper chooses W=5 as the best performer, a hand-picked hyperparameter that the final claims depend on.
  • UKF sigma-point parameters (alpha, beta, kappa) = not reported; tuned via TPE over 200 trials on validation RMSE
    Fitted to minimize validation RMSE; values not given, so the reported AHNNKF5 gains are conditioned on this tuning.
  • Neural network architecture and optimizer hyperparameters = not reported; chosen by search over layers, nodes, learning rates
    Final configuration not disclosed; results depend on the selected model size and training schedule.
  • Fixed time step h for RK4 in Ftheta = 60 s
    Chosen for all orbit experiments; the surrogate dynamics and the autoregressive unrolling both depend on this discretization, and periapsis accuracy is unverified.
  • Initial covariance P0 = diag(10,10,1e-3,1e-3) for orbit; 1e-7*I for mass-spring
    Chosen for the perturbed-initial-state experiments; affects the UKF transient and the width of the reported uncertainty bands.
assumptions (5)
  • domain assumption Hamilton's equations (1) with a time-independent Hamiltonian govern the unknown system; generalized coordinates q and conjugate momenta p are measured or inferred.
    The whole method and loss (2)-(4) is built on this canonical structure; used in Section 3.1.
  • ad hoc to paper The one-step map Ftheta computed by RK4 integration of J*grad(Htheta) over [t_k, t_{k+1}] is an accurate discrete-time surrogate of the true continuous flow.
    The training loss (2) and the autoregressive loss (4) both evaluate Ftheta; the 60s fixed step in Section 5.2 is assumed to resolve periapsis behavior.
  • domain assumption Observation model H(.) is known and noise omega, eta are zero-mean Gaussian; state prior is Gaussian.
    Assumed in Section 3.2.1 for the UKF; not validated against actual telemetry noise or measurement geometry.
  • domain assumption The J2-only gravitational potential adequately represents the training data and the test scenario.
    Section 4.2 sets J2=1.0826e-3 and declares other harmonics negligible; if the surrogate must reproduce higher-order GGM05C terms, the model is misspecified.
  • ad hoc to paper UKF covariance parameters (alpha, beta, kappa) selected by TPE yield a well-calibrated posterior.
    The uncertainty bands in Figures 6 and 8 are not evaluated for calibration; the 'uncertainty quantification' claim rests on this assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning Hamiltonian Dynamics with Bayesian Data Assimilation." pith.science (2026). https://pith.science/paper/WABBFWML

@misc{pith2026250118808,
  author       = {Pith},
  title        = {Pith review of: Learning Hamiltonian Dynamics with Bayesian Data Assimilation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WABBFWML}},
  note         = {Machine review of arXiv:2501.18808}
}
read the original abstract

In this paper, we develop a neural network-based approach for time-series prediction in unknown Hamiltonian dynamical systems. Our approach leverages a surrogate model and learns the system dynamics using generalized coordinates (positions) and their conjugate momenta while preserving a constant Hamiltonian. To further enhance long-term prediction accuracy, we introduce an Autoregressive Hamiltonian Neural Network, which incorporates autoregressive prediction errors into the training objective. Additionally, we employ Bayesian data assimilation to refine predictions in real-time using online measurement data. Numerical experiments on a spring-mass system and highly elliptic orbits under gravitational perturbations demonstrate the effectiveness of the proposed method, highlighting its potential for accurate and robust long-term predictions.

Figures

Figures reproduced from arXiv: 2501.18808 by the authors.

Figure 1
Figure 1. Learning unknown dynamics from spacecraft position and velocity data. The model performance is compared for a highly elliptical orbit, which poses significant challenges when the spacecraft is near the primary body due to rapid velocity changes, with speed variations up to an order of magnitude. problems (Pastor et al., 2021). Implementing missions for identifying parameters in such models requires considerable reso… view at source ↗
Figure 2
Figure 2. Illustration of the Hamiltonian Neural Network in (3). Our numerical experiments indicate that the HNN per￾forms well on low-dimensional problems, but not on high￾dimensional problems, e.g., n ≥ 3. For spacecraft orbit determination and prediction, in particular, we have to deal with six dimensional state space. Furthermore, existing HNN approaches may lack accuracy for highly elliptic or￾bits with large variations … view at source ↗
Figure 3
Figure 3. Illustration of the unscented transform for estimated mean and covariance propagation (blue) and true mean and covariance (black). training. In this paper, we use Huber loss (Huber, 1992) because of its robustness to extreme data in the training set and its everywhere differentiability properties. 3.2. State Estimator 3.2.1. CONVENTIONAL KALMAN FILTER The Kalman filter (KF) (Kalman, 1960) estimates the state of a di… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Accelerations from gravitational perturbations along the sample highly elliptic orbit [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: (a) The state prediction of the simple mass-spring system. (b) The energy evolution over time [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: The energy evolution of object along a highly elliptic orbit over time [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: demonstrates AHNNKF5 performance with a perturbed initial state and covariance estimate of P0 = diag(10, 10, 10−3 , 10−3 ). The UKF integration reduces state uncertainty (shown as red shading) and corrects the perturbed initial state prediction toward the true trajecto…
Figure 9
Figure 9. Figure 9: compares the SMA values for the frictionless mass-spring system using various NN-based methods without UKF. For all methods, the accumulated prediction errors lead to increases in SMA values over time. The SMA trajectory with NODE is ten times smaller than that with ML…
Figure 10
Figure 10. Figure 10: The SMAs of position and velocity RMSEs with a 240-step window using different NN models with UKF based on 125 test trajectories for the mass-spring system [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: SMA of position and velocity RMSE with a 240-step window for different neural network architectures in elliptic orbit. Position RMSE (km) and velocity RMSE (km/s) are shown for x, y, and z components, computed from 125 test orbits. 5.0 7.5 10.0 12.5 15.0 17.5 20.0 101…
Figure 12
Figure 12. Figure 12: SMA of position and velocity RMSE with a 240-step window for different neural network architectures with UKF in elliptic orbit. Position RMSE (km) and velocity RMSE (km/s) are shown for x, y, and z components, computed from 125 test orbits with 60-step observation fre…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 28 canonical work pages

  1. [1]

    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 format.date year duplicate empty "emp...

  2. [2]

    Iteration-free computation of gauss--legendre quadrature nodes and weights

    Bogaert, I. Iteration-free computation of gauss--legendre quadrature nodes and weights. SIAM Journal on Scientific Computing, 36 0 (3): 0 A1008--A1026, 2014

    work page 2014

  3. [3]

    Brown, R. G. Courier Corporation, 2004

    work page 2004

  4. [4]

    and Soares, C

    Caldas, F. and Soares, C. Machine learning in orbit estimation: A survey. Acta Astronautica, 2024

    work page 2024

  5. [5]

    T., Rubanova, Y., Bettencourt, J., and Duvenaud, D

    Chen, R. T., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018

    2018

  6. [6]

    C., and Tapley, B

    Cheng, M., Ries, J. C., and Tapley, B. D. Variations of the earth's figure axis from satellite laser ranging and grace. Journal of Geophysical Research: Solid Earth, 116 0 (B1), 2011. doi:10.1029/2010JB000850

    doi:10.1029/2010jb000850 2011

  7. [7]

    Structural schemes for hamiltonian systems

    Clain, S., Franck, E., and Michel-Dansac, V. Structural schemes for hamiltonian systems, 2025. URL https://arxiv.org/abs/2501.13515

    work page Pith review arXiv 2025

  8. [8]

    and Gorodetsky, A

    Galioto, N. and Gorodetsky, A. A. Bayesian identification of hamiltonian dynamics from symplectic data. In 2020 59th IEEE Conference on Decision and Control (CDC), pp.\ 1190--1195. IEEE, 2020

    work page 2020

Show all 39 references
  1. [9]

    Hamiltonian neural networks

    Greydanus, S., Dzamba, M., and Yosinski, J. Hamiltonian neural networks. Advances in neural information processing systems, 32, 2019

    2019

  2. [10]

    K., et al

    Gurfil, P., Seidelmann, P. K., et al. Celestial mechanics and astrodynamics: theory and practice, volume 436. Springer, 2016

    2016

  3. [11]

    Hamilton, W. R. On a General Method of Expressing the Paths of Light, & of the Planets, by the Coefficients of a Characteristic Function. PD Hardy, 1833

  4. [12]

    Huber, P. J. Robust estimation of a location parameter. In Breakthroughs in statistics: Methodology and distribution, pp.\ 492--518. Springer, 1992

    1992

  5. [13]

    Julier, S. J. and Uhlmann, J. K. New extension of the kalman filter to nonlinear systems. In Signal processing, sensor fusion, and target recognition VI, volume 3068, pp.\ 182--193. Spie, 1997

    1997

  6. [14]

    and Li, R.-C

    Kahan, W. and Li, R.-C. Composition constants for raising the orders of unconventional schemes for ordinary differential equations. Mathematics of computation, 66 0 (219): 0 1089--1099, 1997

    1997

  7. [15]

    Kalman, R. E. A new approach to linear filtering and prediction problems. Transactions of the ASME--Journal of Basic Engineering, 82 0 (Series D): 0 35--45, 1960

    1960

  8. [16]

    L., Scott, K

    Kazemi, S., Azad, N. L., Scott, K. A., Oqab, H. B., and Dietrich, G. B. Orbit determination for space situational awareness: A survey. Acta Astronautica, 2024

    2024

  9. [17]

    Hyperboliclr: Epoch insensitive learning rate scheduler

    Kim, T.-G. Hyperboliclr: Epoch insensitive learning rate scheduler. arXiv preprint arXiv:2407.15200, 2024

    2024 arXiv

  10. [18]

    and Waldherr, S

    K \"u per, A. and Waldherr, S. Numerical gaussian process kalman filtering for spatiotemporal systems. IEEE Transactions on Automatic Control, 68 0 (5): 0 3131--3138, 2022

    2022

  11. [19]

    Leonard, N. E. Mechanics and nonlinear control: Making underwater vehicles ride and glide. IFAC Proceedings Volumes, 31 0 (17): 0 1--6, 1998

    1998

  12. [20]

    Decoupled weight decay regularization

    Loshchilov, I. Decoupled weight decay regularization. arXiv preprint arXiv:1711.05101, 2017

    2017 arXiv

  13. [21]

    E., Sood, R., and Su, W

    Loshelder, G. E., Sood, R., and Su, W. Orbit determination using physics-informed neural networks. In AIAA SCITECH 2025 Forum, pp.\ 1931, 2025

    2025

  14. [22]

    Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations

    Lu, Y., Zhong, A., Li, Q., and Dong, B. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In International Conference on Machine Learning, pp.\ 3276--3285. PMLR, 2018

    2018

  15. [23]

    Hamiltonian mechanics and neural network technology for high stabilization of quadrotor uav on se (3)

    Mohamed, B., Abdelkader, M., Mohammed, B., Lamaari, H., et al. Hamiltonian mechanics and neural network technology for high stabilization of quadrotor uav on se (3). In 2022 IEEE 3rd International Conference on Electronics, Control, Optimization and Computer Science (ICECOCS),...

    2022

  16. [24]

    Initial orbit determination methods for track-to-track association

    Pastor, A., Sanjurjo-Rivo, M., and Escobar, D. Initial orbit determination methods for track-to-track association. Advances in Space Research, 68 0 (7): 0 2677--2694, 2021

    2021

  17. [25]

    Pytorch: An imperative style, high-performance deep learning library

    Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019

    2019

  18. [26]

    and Bai, X

    Peng, H. and Bai, X. Artificial neural network--based machine learning approach to improve orbit prediction accuracy. Journal of Spacecraft and Rockets, 55 0 (5): 0 1248--1260, 2018

    2018

  19. [27]

    and Bai, X

    Peng, H. and Bai, X. Comparative evaluation of three machine learning algorithms on improving orbit prediction accuracy. Astrodynamics, 3 0 (4): 0 325--343, 2019

    2019

  20. [28]

    and Karniadakis, G

    Raissi, M. and Karniadakis, G. E. Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics, 357: 0 125--141, 2018

    2018

  21. [29]

    Raissi, M., Perdikaris, P., and Karniadakis, G. E. Multistep neural networks for data-driven discovery of nonlinear dynamical systems. arXiv preprint arXiv:1801.01236, 2018

    2018 arXiv

  22. [30]

    Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378: 0 686--707, 2019

    2019

  23. [31]

    and Haber, E

    Ruthotto, L. and Haber, E. Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision, 62 0 (3): 0 352--364, 2020

    2020

  24. [32]

    Salleh, N., Yuhaniz, S., and Azmi, N. M. Modeling orbital propagation using regression technique and artificial neural network. Int. J. Adv. Sci. Eng. Inf. Technol, 12 0 (3): 0 1279, 2022

    2022

  25. [33]

    A physic-informed neural network approach to orbit determination

    Scorsoglio, A., Ghilardi, L., and Furfaro, R. A physic-informed neural network approach to orbit determination. The Journal of the Astronautical Sciences, 70 0 (4): 0 25, 2023

    2023

  26. [34]

    S., V \"a lisuo, P., Bhuiyan, M

    Selvan, K., Siemuri, A., Prol, F. S., V \"a lisuo, P., Bhuiyan, M. Z. H., and Kuusniemi, H. Precise orbit determination of leo satellites: a systematic review. GPS Solutions, 27 0 (4): 0 178, 2023

    2023

  27. [35]

    Orbit propagation from historical data using physics-informed neural odes

    Subramanian, S., Ramnani, R., Sengupta, S., and Yadav, S. Orbit propagation from historical data using physics-informed neural odes. In 2023 International Conference on Machine Learning and Applications (ICMLA), pp.\ 1643--1648. IEEE, 2023

    2023

  28. [36]

    Wan, E. A. and Van Der Merwe, R. The unscented kalman filter for nonlinear estimation. In Proceedings of the IEEE 2000 adaptive systems for signal processing, communications, and control symposium (Cat. No. 00EX373), pp.\ 153--158. Ieee, 2000

    2000

  29. [37]

    M., Wiese, D

    Watkins, M. M., Wiese, D. N., Yuan, D.-N., Boening, C., and Landerer, F. W. Improved methods for observing earth's time variable mass distribution with grace using spherical cap mascons. Journal of Geophysical Research: Solid Earth, 120 0 (4): 0 2648--2671, 2015

    2015

  30. [38]

    and Belda, K

    Z \'a da, V. and Belda, K. Mathematical modeling of industrial robots based on hamiltonian mechanics. In 2016 17th International Carpathian Control Conference (ICCC), pp.\ 813--818. IEEE, 2016

    2016

  31. [39]

    A lstm assisted orbit determination algorithm for spacecraft executing continuous maneuver

    Zhou, X., Qin, T., Ji, M., and Qiao, D. A lstm assisted orbit determination algorithm for spacecraft executing continuous maneuver. Acta Astronautica, 204: 0 568--582, 2023

    2023

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.