Pith. sign in

REVIEW 4 major objections 5 minor 32 references

Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read Univariate function sums let neural Hamiltonians hold energy longer

desk verdict KAN-HNN paper whose own tables contradict its energy-drift claim, and whose architecture comparison is confounded by unequal training setups. read the letter →

arxiv 2508.19410 v1 pith:WMYYPHYB submitted 2025-08-26 cs.LG physics.comp-ph

Zongyu Wu , Ruichen Xu , Luoyao Chen , Georgios Kementzidis , Siyao Wang , Yuefan Deng This is my paper · ORCID
classification cs.LGphysics.comp-ph MSC 37M1565P1068T07
keywords HamiltonianneuralnetworksKolmogorov-Arnoldrepresentationsymplecticstructureenergyconservationphysics-informedmachinelearningmulti-bodydynamicsunivariatefunctionapproximationlong-termstability
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 proposes a Hamiltonian Neural Network variant, KAR-HNN, that represents the learned energy function not with a multilayer perceptron but as sums and compositions of one-variable functions, following the Kolmogorov-Arnold representation theorem. The authors aim to show that this localized, univariate structure is better suited to the sharp gradients and multi-scale oscillations that appear in real Hamiltonian systems, and that it does so while preserving the symplectic geometry that keeps energy from drifting over long simulations. The paper argues that because the learned Hamiltonian remains a single scalar differentiable function, plugging it into Hamilton's equations yields a symplectic flow by construction. On four benchmark systems—spring-mass, pendulum, two-body, and three-body—they report lower test error in the learned derivatives and, in most cases, lower or competitive energy drift compared with MLP-based HNNs. If the claim holds, KAR-HNN would be a more parameter-efficient and stable starting point for learning unknown physics from trajectory data.

What carries the argument

The Kolmogorov-Arnold Representation: any continuous multivariate function can be written as a sum of 2n+1 outer univariate functions applied to sums of n inner univariate functions. KAR-HNN uses this decomposition as the Hamiltonian, with univariate blocks for the inner and outer functions. The mechanism carries the argument by concentrating each variable's contribution into separate one-dimensional channels, which the paper says yields localized approximation and smoother gradient fields than a fully connected MLP; because the Hamiltonian is still one scalar function, the induced vector field J∇Hθ is automatically Hamiltonian and hence symplectic.

What would settle it

Run the same four benchmarks with the architecture as the only variable—same optimizer, same number of training steps, same parameter budget, same data—comparing MLP-HNN against KAR-HNN. If KAR-HNN's test loss and energy drift advantages disappear or reverse under matched settings, the claim that univariate localized approximation drives the improvement is falsified.

Watch

Extended reading notes

Core claim

The central claim is that replacing the MLP inside an HNN with a Kolmogorov-Arnold Representation improves both accuracy and long-term physical consistency. Concretely, KAR-HNN writes the Hamiltonian as Hθ(q,p) = Σ_r Φ_r(Σ_s φ_{r,s}(z_s)) and uses automatic differentiation to form ∂Hθ/∂p and −∂Hθ/∂q. Because every continuously differentiable Hamiltonian generates a symplectic flow, the model inherits energy conservation and phase-space volume preservation without extra constraints. The paper's experiments show this architecture achieving the lowest test MSE on all four benchmarks—roughly 28.6 versus 35.9 for spring-mass, 34.3 versus 38.5 for the pendulum, 0.281 versus 1.92 for the two-body p

Load-bearing premise

The central claim assumes the observed accuracy and stability gains come from replacing the MLP with univariate function sums, but the experiments change optimizer, training length, and network size at the same time, so that specific cause is not separately tested.

Editorial extensions

If this is right

  • Learned dynamics are symplectic by construction: as long as the Hamiltonian is differentiable, trajectories preserve the phase-space 2-form, giving the same long-horizon energy behavior as classical Hamiltonian systems.
  • The architecture needs only small univariate layers (2–15 nodes) and hundreds rather than thousands of training steps on these benchmarks, suggesting a lower parameter cost than MLP-based HNNs.
  • Across the four benchmark systems, KAR-HNN reports the lowest test error in predicted derivatives, indicating better generalization to unseen trajectories.
  • The results expose a trade-off: MLP-HNN often enforces near-perfect energy at the cost of poorer derivatives, while KAR-HNN sacrifices a little energy exactness for much better local predictive fidelity.
  • Because the Kolmogorov-Arnold theorem covers continuous multivariate functions generally, the same construction extends in principle to higher-dimensional Hamiltonian systems than the four benchmarks shown.

Reading between the lines

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

  • The reported comparisons change optimizer, iteration count, and layer size at the same time as the architecture, so the experiments do not isolate univariate locality as the cause of the gains; a matched ablation that varies only the architecture would settle that.
  • If the architectural benefit holds under matched settings, KAR-HNN's small parameter count makes it a plausible backbone for Hamiltonian learning in high-dimensional or data-sparse settings, and its univariate channels could be inspected or symbolically regressed to recover interpretable energy terms.
  • A natural extension is to weight the loss to interpolate between exact energy conservation and derivative accuracy, formalizing the trade-off the paper observes rather than treating it as a side effect.
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 / 5 minor

Summary. The manuscript proposes KAR-HNN, a Hamiltonian Neural Network variant that replaces the usual MLP Hamiltonian with a KAN-style sum of univariate B-spline/function expansions. It argues that this localized, univariate representation better captures high-frequency and multi-scale dynamics, reduces energy drift, improves long-term stability, and preserves the symplectic form of Hamiltonian systems. The method is evaluated on four benchmark systems—spring–mass, simple pendulum, two-body, and three-body—against a baseline network and an MLP-based HNN, reporting train/test MSE and an energy-error metric.

Significance. If the empirical claims were established, the paper would provide a useful architectural improvement for Hamiltonian dynamics learning. The paper does present explicit Hamiltonian forms for the four benchmarks and reports mean-with-standard-deviation summaries, which is a helpful starting point for reproducibility. However, the central advertised advantage—reduced energy drift—is contradicted by the paper's own tables on two of the four benchmarks, and the comparisons are confounded by unequal training protocols. The theoretical section reproduces the standard fact that every differentiable scalar Hamiltonian generates a symplectic flow, so it does not isolate any KAR-HNN-specific benefit. As presented, the contribution is not supported.

major comments (4)
  1. [Abstract and §IV-C] The abstract and §IV-C claim that KAR-HNN reduces energy drift and improves long-term predictive stability relative to MLP-HNN. The paper's own Tables I and IV contradict this on two of the four benchmarks. In Table I (spring–mass), the MLP-HNN energy error is 0.376 ± 0.0798 while KAR-HNN is 1.63 ± 0.284, roughly 4× larger. In Table IV (three-body), HNN energy error is 2.97 ± 0.743 while KAR-HNN is 5.80 ± 1.54, roughly 2× larger. The claim as stated is therefore false on half of the reported benchmarks. The authors must either substantially weaken the claim or provide additional experiments that establish the advertised energy-drift reduction.
  2. [§V-A–D] The architecture comparison is confounded. KAR-HNN is trained with LBFGS for 200–4000 steps, uses 2–15 hidden nodes, and batch size 50, while the baseline and MLP-HNN use Adam for 2000–10000 steps, 200 hidden units, and different batch sizes. Any observed difference in test MSE or energy error could be due to optimizer, training budget, model capacity, or data handling rather than the univariate KAN-style representation. To support the central claim that KAR-HNN's architecture is responsible for superior accuracy, the authors need matched experiments or ablations in which only the architecture is varied and all training hyperparameters are held fixed.
  3. [§IV-C] The symplectic-preservation argument is tautological and does not distinguish KAR-HNN from MLP-HNN. The proof that X_H = J∇H generates a symplectic flow uses only the fact that H is a continuously differentiable scalar function; it holds identically for any MLP-based Hamiltonian as well. Thus the statement 'preserving the symplectic form' is not a KAR-HNN-specific property. The paper should either provide an architectural argument—for example, an approximation-theoretic or gradient-smoothness bound that depends on the univariate expansion—or explicitly acknowledge that both architectures share this property and focus the claimed advantage on approximation quality.
  4. [§IV-A] The use of the Kolmogorov–Arnold Representation is presented as a constructive advantage, but the paper provides no proof or analysis that the univariate decomposition yields better approximation of Hamiltonian dynamics, especially in high dimensions. The KAR theorem is existential, and the implemented B-spline KAN architecture is a heuristic approximation. All benchmarks are low-dimensional (2–6 dimensional phase spaces), so the claim of high-dimensional advantage is not tested. Without an approximation-error analysis or a high-dimensional experiment, the theoretical motivation remains unsupported.
minor comments (5)
  1. [Figure 1 caption, §III] The caption contains a self-referential artifact: 'I would like to check whether it is the good name of the caption of the graph.' This is clearly an editing leftover and should be removed before any formal submission.
  2. [§V-E] There is a typo at the end of the summary paragraph: 'priorities..' should be a single period.
  3. [§III] The text says 'Finally, Section II situates our approach...' but the preceding text is in Section III; this cross-reference appears incorrect.
  4. [References] Several references are incomplete or marked as forthcoming/in press without volume or page details (e.g., [12], [16], [17]). This makes verification difficult and should be corrected.
  5. [General] No code or data release is mentioned. Given the small scale of the experiments, releasing the data-generation and training code would greatly aid reproducibility.

Circularity Check

1 steps flagged · score 4.0 of 10

The empirical core is not circular, but the symplectic-preservation 'theoretical insight' is a generic Hamiltonian-flow property identical for MLP-HNN, so the KAR-HNN-specific claim reduces by construction.

  1. self definitional [Section IV-C, 'Symplectic Preservation']
    "Any continuously differentiable scalar function H(z) generates a symplectic flow because ιXH (ω0) = dH implies LXH ω0 = 0. In the Kolmogorov–Arnold representation-based Hamiltonian Neural Network (KAR-HNN), the Hamiltonian Hθ(z) is composed of univariate building blocks but remains a single, continuous scalar function: ... Substituting Hθ into the Hamiltonian equations ensures ˙z = J ∇z Hθ(z), thereby making the induced flow symplectic."

    The 'preservation of symplectic structure' is derived solely from Hθ being a continuously differentiable scalar function and the identity ˙z=J∇Hθ. It holds identically for the MLP-HNN baseline, whose Hθ is also a differentiable scalar. The Kolmogorov–Arnold univariate decomposition plays no role in the proof. Thus the claimed KAR-HNN-specific guarantee is equivalent, by construction, to the standard definition of a Hamiltonian flow; it cannot explain the paper's advertised reduction in energy drift or improved stability relative to MLP-HNN.

full rationale

The empirical part of the paper is self-contained: KAR-HNN and baselines are trained with a standard MSE loss on (q,p, qdot,pdot), and the reported train/test losses and energy errors are evaluated, not fitted parameters renamed as predictions. No load-bearing self-citation chain appears; the self-citations in the references are background and not used to justify the architecture. The central empirical comparison is, however, confounded by unequal training protocols (LBFGS vs Adam, 200–4000 vs 2000–10000 steps, 2–15 vs 200 hidden units), and the abstract's 'reducing energy drift' is contradicted by Tables I and IV where MLP-HNN has lower energy error than KAR-HNN. These are correctness risks, not circularity. The one genuine circularity-adjacent step is the theoretical 'symplectic preservation' contribution: it is a known property of every differentiable Hamiltonian flow, relabeled as a KAR-HNN advantage, so the architecture-specific claim reduces by construction. There is also an editing artifact in the Fig. 1 caption ('I would like to check whether it is the good name...'), which indicates lack of polish but has no scientific bearing. Overall score 4: one theoretical claim reduces by construction, while the empirical core retains independent content.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities, fields, or conserved quantities. Its free parameters are architecture and training choices made per benchmark. Its axioms are standard mathematical theorems and domain assumptions about the benchmark systems.

free parameters (6)
  • KAN univariate layer sizes (nodes per layer) = 2 (spring), 10/10 (two-body), 15/10 (three-body)
    Chosen per task; no sensitivity analysis.
  • B-spline order k = 5 (spring), 3 (pendulum, two-body, three-body)
    Hand-selected per task.
  • B-spline grid size = 2 (spring, three-body), 3 (two-body)
    Hand-selected.
  • Training steps and optimizer = LBFGS 200 steps (spring), 4000 (two-body), 200 (three-body) vs Adam 2000-10000 for baselines
    Optimizer and step count differ across methods, confounding comparison.
  • Noise variance sigma^2 = 0.1, 0.05, 0.2
    Chosen per dataset; ground truth derivative data presumably generated with these levels.
  • Dataset sizes = 25/25 trajectories (spring, pendulum), 1000 (two-body, 80/20 split), 5000 (three-body)
    Chosen per dataset; no error bars across seeds.
assumptions (4)
  • standard math Kolmogorov-Arnold Representation theorem (Kolmogorov 1957, Arnold 1957)
    Used in Section IV-A to justify representing H as sum of univariate functions.
  • standard math Any continuously differentiable scalar function H(z) on R^{2d} defines a Hamiltonian vector field J grad H whose flow preserves the standard symplectic form
    Invoked in Section IV-C to claim symplectic preservation for KAR-HNN; this property holds for any differentiable H, including MLP-HNN.
  • domain assumption B-spline basis functions can represent the univariate components
    The KAN implementation uses B-spline expansions; no proof that these capture the required univariate functions.
  • domain assumption The benchmark systems are exactly Hamiltonian and noise is zero-mean Gaussian
    Assumed in Section V; implies the training objective is well-posed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks." pith.science (2026). https://pith.science/paper/WMYYPHYB

@misc{pith2026250819410,
  author       = {Pith},
  title        = {Pith review of: Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMYYPHYB}},
  note         = {Machine review of arXiv:2508.19410}
}
read the original abstract

We propose a Kolmogorov-Arnold Representation-based Hamiltonian Neural Network (KAR-HNN) that replaces the Multilayer Perceptrons (MLPs) with univariate transformations. While Hamiltonian Neural Networks (HNNs) ensure energy conservation by learning Hamiltonian functions directly from data, existing implementations, often relying on MLPs, cause hypersensitivity to the hyperparameters while exploring complex energy landscapes. Our approach exploits the localized function approximations to better capture high-frequency and multi-scale dynamics, reducing energy drift and improving long-term predictive stability. The networks preserve the symplectic form of Hamiltonian systems, and thus maintain interpretability and physical consistency. After assessing KAR-HNN on four benchmark problems including spring-mass, simple pendulum, two- and three-body problem, we foresee its effectiveness for accurate and stable modeling of realistic physical processes often at high dimensions and with few known parameters.

Figures

Figures reproduced from arXiv: 2508.19410 by the authors.

Figure 1
Figure 1. The juxtaposition of Kolmogorov–Arnold Representation and MLP-Based Hamiltonian Neural Networks I would like to check whether it is the good name of the caption of the graph. The left panel (KAN-HNN) models the Hamiltonian Hθ as a superposition of univariate “kernel” or “basis” blocks (shown as small plots), following the Kolmogorov–Arnold Representation. Each block contributes to the final energy function in a loca… view at source ↗
Figure 2
Figure 2. Datasets for two classical systems. TABLE II SIMPLE PENDULUM RESULTS (SCALED BY 103 ). Model Train Loss Test Loss Energy Baseline 31.3 ± 1.76 38.0 ± 2.13 41.2 ± 7.33 HNN 31.7 ± 1.77 38.5 ± 2.10 28.8 ± 6.94 KAR-HNN 32.8 ± 1.76 34.3 ± 1.75 24.6 ± 6.94 The baseline model attains a training loss of about 31.3 but deteriorates on the test set to 38.0, accompanied by a sizeable energy drift of 41.2. By contrast, the HNN a… view at source ↗
Figure 3
Figure 3. Example trajectories for the two-body dataset (left) and the corre [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison of total energy over time in the two-body problem. KAR [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Sample trajectories in the three-body system. The baseline diverges early; HNN tracks partial motion but deviates significantly over time; KAR-HNN [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Hairer, C

    E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations , 2nd ed. Springer, 2006

    work page 2006

  2. [2]

    On the use of canonical transformations and symplectic integrators for hamiltonian systems,

    E. Celledoni, H. Z. Munthe-Kaas, B. Owren, and G. W. Wanner, “On the use of canonical transformations and symplectic integrators for hamiltonian systems,” Foundations of Computational Mathematics , vol. 21, no. 6, pp. 1605–1638, 2021

    work page 2021

  3. [3]

    Symplectic neural net- works in continuous and discrete time,

    O. Azencot, O. Vantzos, and M. Ovsjanikov, “Symplectic neural net- works in continuous and discrete time,” ACM Transactions on Graphics, vol. 41, no. 5, pp. 1–14, 2022

    work page 2022

  4. [4]

    Velocity-Inferred Hamiltonian Neural Networks: Learning Energy-Conserving Dynamics from Position-Only Data

    R. Xu, Z. Wu, L. Chen, G. Kementzidis, S. Wang, H. Wang, Y . Shi, and Y . Deng, “Velocity-inferred hamiltonian neural networks: Learning energy-conserving dynamics from position-only data,” arXiv preprint arXiv:2505.02321, 2025

    work page Pith review arXiv 2025

  5. [5]

    Neural ordinary differential equations,

    T. Chen, Y . Rubanova, J. Bettencourt, and D. K. Duvenaud, “Neural ordinary differential equations,” in Advances in Neural Information Processing Systems (NeurIPS) , 2018

    work page 2018

  6. [6]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear pdes,

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear pdes,” Journal of Computational Physics, vol. 378, pp. 686–707, 2019

    work page 2019

  7. [7]

    Hamiltonian neural networks,

    S. Greydanus, M. Dzamba, and J. Yosinski, “Hamiltonian neural networks,” in Advances in Neural Information Processing Systems (NeurIPS), 2019

    work page 2019

  8. [8]

    On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition,

    A. N. Kolmogorov, “On the representation of continuous functions of several variables by superposition of continuous functions of one variable and addition,” Doklady Akademii Nauk SSSR, vol. 114, pp. 953– 956, 1957

    work page 1957

Show all 32 references
  1. [9]

    On functions of three variables,

    V . I. Arnold, “On functions of three variables,” Doklady Akademii Nauk SSSR, vol. 114, pp. 679–681, 1957

    1957

  2. [10]

    Local kernel expansions for high-fidelity modeling of chaotic flows,

    G. Pan and D. Sattigeri, “Local kernel expansions for high-fidelity modeling of chaotic flows,” SIAM Journal on Scientific Computing , vol. 41, no. 6, pp. C633–C651, 2019

    2019

  3. [11]

    Radial basis function networks in physics-informed learn- ing,

    P. Mishra, “Radial basis function networks in physics-informed learn- ing,” Applied Mathematical Modelling , vol. 110, pp. 239–254, 2022

    2022

  4. [12]

    Kolmogorov–arnold networks for high-dimensional function ap- proximation,

    Z. Liu, “Kolmogorov–arnold networks for high-dimensional function ap- proximation,” Journal of Computational Methods , 2023, (forthcoming)

    2023

  5. [13]

    Bayesian kernelized learning for physics discovery,

    J. Tang, D. DeVito, and F. Ebert, “Bayesian kernelized learning for physics discovery,” in Conference on Uncertainty in Artificial Intelli- gence (UAI), 2020

    2020

  6. [14]

    J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. Springer, 1999

    1999

  7. [15]

    J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems . Chapman and Hall, 1994

    1994

  8. [16]

    Survey of physics-informed neural networks for scientific computing,

    G. Toth, “Survey of physics-informed neural networks for scientific computing,” Applied Computing and Informatics , 2023, (in press)

    2023

  9. [17]

    Active learning based sampling for high- dimensional nonlinear partial differential equations,

    W. Gao and C. Wang, “Active learning based sampling for high- dimensional nonlinear partial differential equations,” Journal of Com- putational Physics, vol. 475, p. 111848, 2023

    2023

  10. [18]

    Coordinate transform fourier neural operators for symmetries in physical modelings,

    W. Gao, R. Xu, H. Wang, and Y . Liu, “Coordinate transform fourier neural operators for symmetries in physical modelings,” Transactions on Machine Learning Research , 2024

    2024

  11. [19]

    Complex physics-informed neural network,

    C. Si, M. Yan, X. Li, and Z. Xia, “Complex physics-informed neural network,” arXiv preprint arXiv:2502.04917 , 2025

    2025 arXiv

  12. [20]

    Initialization-enhanced physics-informed neural net- work with domain decomposition (idpinn),

    C. Si and M. Yan, “Initialization-enhanced physics-informed neural net- work with domain decomposition (idpinn),” Journal of Computational Physics, vol. 530, p. 113914, 2025

    2025

  13. [21]

    Neural krylov iteration for accelerating linear system solving,

    J. Luo, J. Wang, H. Wang, Z. Geng, H. Chen, Y . Kuang et al., “Neural krylov iteration for accelerating linear system solving,” in The Thirty- eighth Annual Conference on Neural Information Processing Systems , 2025

    2025

  14. [22]

    Convolution-weighting method for the physics- informed neural network: A primal-dual optimization perspective,

    C. Si and M. Yan, “Convolution-weighting method for the physics- informed neural network: A primal-dual optimization perspective,”arXiv preprint arXiv:2506.19805, 2025

    2025 arXiv

  15. [23]

    Graph neural operators for multi-scale physical simulations,

    B. Freedman, K. Sutherland, and M. Mueller, “Graph neural operators for multi-scale physical simulations,” IEEE Transactions on Neural Networks and Learning Systems , 2022

    2022

  16. [24]

    Dynamic schwartz- fourier neural operator for enhanced expressive power,

    W. Gao, J. Luo, R. Xu, and Y . Liu, “Dynamic schwartz- fourier neural operator for enhanced expressive power,” Transactions on Machine Learning Research , 2025. [Online]. Available: https: //openreview.net/forum?id=B0E2yjrNb8

    2025

  17. [25]

    Holistic physics solver: Learning pdes in a unified spectral-physical space,

    X. Yue, Y . Yang, and L. Zhu, “Holistic physics solver: Learning pdes in a unified spectral-physical space,” 2025. [Online]. Available: https://arxiv.org/abs/2410.11382

    2025 arXiv

  18. [26]

    Deltaphi: Learning physical trajectory residual for pde solving,

    X. Yue, L. Zhu, and Y . Yang, “Deltaphi: Learning physical trajectory residual for pde solving,” 2024. [Online]. Available: https://arxiv.org/abs/2406.09795

    2024

  19. [27]

    Discretization-invariance? on the discretization mismatch errors in neural operators,

    W. Gao, R. Xu, Y . Deng, and Y . Liu, “Discretization-invariance? on the discretization mismatch errors in neural operators,” in The Thirteenth International Conference on Learning Representations , 2025

    2025

  20. [28]

    APOD: Adaptive PDE-observation diffusion for physics-constrained sampling,

    R. Xu, H. Wang, G. Kementzidis, C. Si, and Y . Deng, “APOD: Adaptive PDE-observation diffusion for physics-constrained sampling,” in ICML 2025 Workshop on Assessing World Models , 2025. [Online]. Available: https://openreview.net/forum?id=Z1J7LJGDxH

    2025

  21. [29]

    Symbolic learning of dynamical systems using mixed integer programming and neural approximations,

    Y . Xue, Q. Lu, and B. Gao, “Symbolic learning of dynamical systems using mixed integer programming and neural approximations,” Neural Computation, vol. 34, no. 11, pp. 2381–2405, 2022

    2022

  22. [30]

    Deep coordinate descent: Training neural networks with lagrangian mechanics,

    C. Jacobs and E. Bradley, “Deep coordinate descent: Training neural networks with lagrangian mechanics,” in International Conference on Learning Representations (ICLR) , 2021

    2021

  23. [31]

    Improved hamiltonian neural networks with extended energy conservation,

    X. Chen and S. Wang, “Improved hamiltonian neural networks with extended energy conservation,” in Proceedings of the 28th International Conference on Neural Information Processing , 2021

    2021

  24. [32]

    The impact of move schemes on simulated annealing performance,

    R. Xu, H. Wang, and Y . Deng, “The impact of move schemes on simulated annealing performance,” arXiv preprint arXiv:2504.17949 , 2025

    2025 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.