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Adding conservation and Lyapunov losses to a variational Gaussian-process model yields better predictions of conservative, dissipative, and port-Hamiltonian dynamics from noisy trajectory data than unregularized baselines.

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The paper extends symplectic spectral Gaussian processes with energy, volume, and Lyapunov regularizers to learn conservative, dissipative, and port-Hamiltonian dynamics from noisy data.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A useful, clearly written extension of Symplectic Spectrum GPs with physics regularizers, but the paper's own tables contradict the 'improved across systems' claim and the empirical story needs a major cleanup before the results can be taken at face value. the 4 major comments →

arxiv 2509.07280 v1 pith:5AK7CGAJ submitted 2025-09-08 cs.LG

Learning Generalized Hamiltonian Dynamics with Stability from Noisy Trajectory Data

Luke McLennan , Yi Wang , Ryan Farell , Minh Nguyen , Chandrajit Bajaj This is my paper
classification cs.LG
keywords Hamiltonian dynamicsGaussian processesrandom Fourier featuresvariational inferenceLyapunov stabilityenergy conservationdissipative systemsport-Hamiltonian systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 single probabilistic model can learn the dynamics of conservative, dissipative, and port-Hamiltonian systems from noisy position-momentum trajectories, without needing time-derivative labels. The model places a Gaussian-process prior on the Hamiltonian—the energy function that defines the motion—and approximates that prior with random Fourier features, so gradients of the energy are computed analytically. Training maximizes an evidence lower bound for data fit while adding soft loss terms for energy conservation, phase-space volume conservation, and Lyapunov stability. On nine benchmark systems, the paper reports lower test-trajectory error than Hamiltonian-neural-network baselines and than the unregularized symplectic Gaussian-process baseline in most settings, with the best results often coming when the noise level is known in advance. If the claim holds, the method offers a practical way to build uncertainty-aware, physically consistent models of damped and externally forced systems from noisy measurements.

Core claim

The paper's central claim is that the right inductive bias for generalized Hamiltonian learning is not a bigger network but a set of physics constraints layered onto a variational Gaussian process. It parameterizes the Hamiltonian as a random-Fourier-feature GP, separates the conservative term J∇H, the dissipation term D∇H, and the external force F(t) into distinct modules, and fits all of them by propagating sampled initial states through a differentiable ODE solver. The loss is the ELBO plus three regularizers—energy conservation, volume conservation (Liouville), and Lyapunov stability—with weights adapted by a gradient descent-ascent scheme. The paper finds that this regularized objective

What carries the argument

The load-bearing object is the random-Fourier-feature Gaussian process prior on H(q,p), whose cos/sin basis functions have learned frequencies and weights and give analytic gradients ∇H. Around this, the paper builds three modules: the symplectic matrix J for conservative flow, a diagonal non-positive dissipation matrix D=diag(0,...,0,−η₁²,...,−η_d²) for dissipative flow, and an MLP external forcing F(t) for port-Hamiltonian flow. These feed a multi-term loss: a variational ELBO (Monte-Carlo expected log-likelihood plus KL penalties on weights and initial states) and three physics regularizers—energy conservation along the flow, volume conservation via the characteristic function of a sample

Load-bearing premise

Everything about the dissipative and port-Hamiltonian versions rests on assuming damping acts only along each momentum coordinate and never couples coordinates: D=diag(0,...,0,−η₁²,...,−η_d²). If real dissipation mixes coordinates, the model cannot represent the physics.

What would settle it

Fit the model to data from a coupled damped oscillator whose dissipation matrix has nonzero off-diagonal entries, so damping in one coordinate drags the other. If the diagonal parameterization cannot represent the vector field, test-trajectory error will stay high and the measured energy dissipation will not match the true rate; that observation would pin the structural assumption as the bottleneck.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A single model, with one Hamiltonian module and one dissipation/forcing module, can cover all three classes of generalized Hamiltonian dynamics.
  • Users need only noisy (q,p) trajectories, not time-derivative labels, and the method returns predictive uncertainty.
  • When the noise level is known a priori, performance improves in most tested systems, because the model can separate observation noise from the true dynamics.
  • The volume regularizer suppresses spurious divergence in the learned vector field, which helps predictions stay faithful beyond the training horizon.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The diagonal dissipation assumption is the natural next test; a full dissipation matrix would extend the method to anisotropic or coupled damping, though it would break the clean energy and volume identities the regularizers currently rely on.
  • The volume-loss term is effectively the divergence term of the Onsager-Machlup functional, so the same machinery should extend to stochastic Hamiltonian systems with state-dependent noise, where the most probable path carries the physics.
  • Because the implemented Lyapunov loss keeps only the positivity penalty on H (the non-increase half is inherited from the energy regularizer on the conservative flow), the stability guarantee is empirical; a certified guarantee would need the Hamiltonian parameterized to be globally positive-definite by construction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a variational framework for learning generalized Hamiltonian dynamics (conservative, dissipative, and port-Hamiltonian) from noisy, sparse phase-space trajectories. The Hamiltonian is represented as a random Fourier feature approximation of a Gaussian process, and training combines an ELBO data-fit term with soft regularizers for energy conservation, volume conservation, and Lyapunov stability. Dissipation is parameterized as a diagonal matrix with a zero q-block and non-positive p-block, external forcing is restricted to the momentum coordinates and modeled by an MLP, and several multi-loss balancing schemes (equal weighting, GDA, MTAdam, Jacobian descent) are compared. Experiments are reported on pendulum, spring, Hénon–Heiles, and Duffing systems across the three classes, with baselines HNN/DHNN/PHNN and SSGP.

Significance. If the empirical claims held, the paper would provide a practical uncertainty-aware extension of symplectic spectral Gaussian processes to dissipative and port-Hamiltonian systems, with physics-informed regularizers and a public code release. The methodological ingredients are coherent and the experimental setup covers a useful range of systems and noise levels. However, the central claim of improved performance across systems is not supported by the paper's own tables: several reported numbers contradict the captions, and differences are generally within one standard deviation. The structural restriction to diagonal dissipation also limits the claimed generality. The contribution is incremental relative to SSGP, but the underlying idea is plausible and the failures are mostly in claim calibration and experimental reporting rather than in the derivation.

major comments (4)
  1. [§4.2, Tables 4 and 5] The headline claim in Section 4.2 — 'our method can achieve improved performance across systems in each of the various classes of generalized Hamiltonian dynamics' — is not supported by the reported numbers. In Table 4, the Forced Spring row shows SSGP at 0.0293±0.0102 versus Ours (Equal) 0.0311±0.0126 and Ours (GDA) 0.0319±0.0129, contradicting the caption's concession that the method loses on that system. In Table 5, for the damped pendulum at noise 0.1, SSGP is 0.0897±0.1240 while Ours (Equal) is 0.1097±0.1511 and Ours (GDA) is 0.1119±0.1540, directly contradicting the caption 'superior performance in each case of noise level.' Since these differences are within one standard deviation and no significance tests or repeated-seed statistics are reported, the evidence is at best mixed. The claims must be restricted to the systems and regimes where the method actually wins, or supported by
  2. [§3.2, Eq. (11)–(13); §3.4, Eq. (27)–(31)] The dissipative model is restricted to D = diag(0,...,0, -η_1^2, ..., -η_d^2). This is load-bearing: the separated conservative/dissipative parameterization, the energy-dissipation identity Eq. (27)–(29), the volume-dissipation identity Eq. (30)–(31), and the regularizers all depend on this diagonal form. For realistic dissipative or port-Hamiltonian systems with off-diagonal or state-dependent damping, the model cannot represent the true vector field, and the proposed regularizers would enforce the wrong dissipation laws. Since the title and abstract claim generalized Hamiltonian dynamics, the paper should either relax this assumption, or explicitly state and empirically test the limitation (e.g., with a coordinate-coupled damping example), rather than presenting the method as broadly applicable.
  3. [§3.6, Eq. (26)] Equation (26) defines the Lyapunov regularizer with two terms, but §3.6 states: 'in practice, since the first term is already satisfied by enforcing conservation of energy we omit it in the code implementation.' Thus the implemented Lyapunov term contains only ReLU(−H), i.e., a non-negativity penalty, not the stability condition dH/dt ≤ 0. The method description, Algorithm 1, and the stability claims in the abstract and introduction should be aligned with what is actually implemented. If the derivative part is omitted, the claim that the method enforces 'Lyapunov stability' is overstated.
  4. [§4.2, Table 8] The ablation study shows that adding the full set of regularizers can degrade test accuracy. For Windy Pendulum, the unregularized ELBO gives MSE 0.0493, while the full equal-weighted model gives 0.0670 and the full GDA-balanced model gives 0.2972; only the GDA energy-only variant (0.0303) improves over the unregularized model. This is an additional, concrete contradiction to the 'improved performance across systems' claim, and it highlights that the balancing strategy strongly determines whether regularizers help. The paper should report a principled selection rule for the balancing method and present this ablation as evidence of sensitivity rather than uniform benefit.
minor comments (5)
  1. [Figure 2, Table 1] The system name 'Hénon-Helies' should be 'Hénon-Heiles'.
  2. [§2.2] The dynamics equations labeled (dqdp-P), (dqdp-DP), (dqdp-S), etc. are not numbered in the main text. Numbering them would make cross-referencing easier.
  3. [§3.8, Tables 5–7] The treatment of the zero-noise case is only described in prose ('where σ is known to be zero, the negative log-likelihood is not well defined, so we use MSE'). Clarify how the 'noise 0' rows were computed for the SSGP baseline and for each 'Ours' variant.
  4. [Algorithm 1 and §3.3] The learning rate is denoted α in Algorithm 1, and α is also used for the exponential-stability exponent in §3.3. This notation collision should be fixed.
  5. [Tables 4–7] The captions use different levels of claims: Table 4 says 'in each system besides forced spring', Table 5 says 'superior performance in each case', Table 6 says 'majority of cases', and Table 7 says 'typically performs best'. Unify the claim language with what the numbers support.

Circularity Check

1 steps flagged

Physics regularizers are self-consistency terms that are identities of the model class; central trajectory-MSE evaluation is not manufactured.

specific steps
  1. self definitional [Section 3.3 (Conservation Law Regularizers), Eq. (23); Section 3.2, Eq. (1); Section 3.4, Eq. (27)]
    "which can be softly enforced with the following additional loss term: LEnergy = Eq,p,t [H(ρt(q,p))− H(q,p)]2. The Hamiltonian flow map ρt(·) = Φθ(t;·,w,ω) is the mapping from phase space at the initial time to a future time, defined by integrating along the Hamiltonian vector field."

    For conservative systems the dynamics are parameterized as ẋ=J∇H, so the flow ρt satisfies dH(ρt)/dt = ∇HᵀJ∇H = 0; hence H(ρt(q,p))−H(q,p) ≡ 0 and LEnergy is identically zero for any learned H. The volume regularizer is likewise zero because ∇·(J∇H)=0. For dissipative and port-Hamiltonian cases the paper states the conservation losses are evaluated on only the conservative part J∇H, so they remain identically zero. The Lyapunov term is also automatic by Eq. 27, dH/dt = −Σ(η_i ∂H/∂p_i)² ≤ 0, which the paper itself concedes by omitting that term in implementation. These 'physics constraints' are therefore restatements of the model's own parameterization rather than independent physical evidence. However, the central reported metric, held-out trajectory MSE, is an external benchmark and is no

full rationale

No load-bearing self-citations were found: the ELBO derivation follows the non-author SSGP paper [12], and the RFF basis is explicitly attributed to [12]; none of the cited references are by the present authors. The main empirical claim, improved test-trajectory MSE over HNN variants and SSGP, is evaluated on held-out noisy trajectories and is not a fitted quantity or a renamed training loss. The only notable circularity is in the physics-enforcement portion of the method: the conservation and Lyapunov regularizers are defined in terms of the model's own flow and parameterization, and for the model classes considered they reduce to algebraic identities (energy conservation along J∇H flow, zero divergence of J∇H, and dissipation law forced by the diagonal D parameterization). Thus the regularizers cannot provide independent evidence of physical correctness and are essentially vacuous as constraints; their inclusion in the reported improvements is therefore not substantiated. This is a partial self-consistency issue, not a circularity of the central predictive claim, so the score is 2 rather than higher.

Axiom & Free-Parameter Ledger

9 free parameters · 6 axioms · 0 invented entities

The framework rests on the standard Hamiltonian dynamics definitions (Eqs. 1-3), a zero-mean RFF GP prior over the Hamiltonian (Sec. 3.1), and the ELBO with mean-field variational family (Sec. 3.3). The concrete structural choices (diagonal dissipation, momentum-only forcing, Gaussian noise) are ad hoc to the paper and delimit the method's scope. There are no new physical entities introduced; the forcing MLP is a function approximator. The learned parameters (sigma_0, b, sqrt(C), Lambda, a, sigma, eta, vartheta) and the loss weights lambda are fitted to data.

free parameters (9)
  • sigma_0
    Prior standard deviation of RFF weights, in parameter vector theta (Eq. 8), fitted during training.
  • b
    Posterior mean of RFF weights, fitted (Eq. 6, 8).
  • sqrt(C)
    Posterior covariance factor, fitted via reparameterization to keep C PSD.
  • Lambda
    Diagonal length-scale matrix controlling RFF frequency distribution, fitted (Eq. 8).
  • a
    Standard deviation of initial condition prior, fitted unless noise level is known.
  • sigma
    Observation noise standard deviation, fitted unless known a priori.
  • eta
    Dissipation coefficients, parameterized to satisfy D_i = -eta_i^2, fitted for dissipative and port systems.
  • vartheta
    Weights of the MLP parameterizing external forcing F_vartheta(t), fitted for port systems.
  • lambda_1, lambda_2, lambda_3
    Loss weights for Lyapunov, energy, and volume regularizers, updated via gradient ascent in GDA (Eq. 42-43).
axioms (6)
  • domain assumption The dynamics belong to one of three known classes: conservative (1), dissipative (2), or port-Hamiltonian (3).
    Section 2.1 states the class is specified in advance; this is the standard mathematical definition of generalized Hamiltonian dynamics.
  • domain assumption The Hamiltonian H is a sample from a zero-mean Gaussian process with an RFF spectral kernel.
    Section 3.1 defines the surrogate; the representational power and smoothness of the learned H depend on this prior.
  • ad hoc to paper The dissipation matrix D is diagonal with zero q-block and nonpositive p-block.
    Section 3.2 assumes this structure; it limits the class of dissipative and port-Hamiltonian systems the model can represent.
  • ad hoc to paper The external forcing F(t) affects only the momentum coordinates.
    Section 3.2 sets F(t) = [0; F_vartheta(t)]; this excludes forcing directly on positions.
  • domain assumption Observations are noisy with i.i.d. Gaussian noise and known/variance-modeled sigma.
    Section 2.4 and the generative model in Section 3.3 assume y = x + N(0, sigma^2 I).
  • standard math Trajectories are computed by deterministic ODE integration from a latent noisy initial condition.
    Section 3.3 treats x_0 as latent and x_ij = Phi_theta(t_ij; x_0, W); this is a standard Neural ODE assumption.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of Learning Generalized Hamiltonian Dynamics with Stability from Noisy Trajectory Data." pith.science (2026). https://pith.science/paper/5AK7CGAJ

@misc{pith2026250907280,
  author       = {Pith},
  title        = {Pith review of: Learning Generalized Hamiltonian Dynamics with Stability from Noisy Trajectory Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5AK7CGAJ}},
  note         = {Machine review of arXiv:2509.07280}
}
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read the original abstract

We introduce a robust framework for learning various generalized Hamiltonian dynamics from noisy, sparse phase-space data and in an unsupervised manner based on variational Bayesian inference. Although conservative, dissipative, and port-Hamiltonian systems might share the same initial total energy of a closed system, it is challenging for a single Hamiltonian network model to capture the distinctive and varying motion dynamics and physics of a phase space, from sampled observational phase space trajectories. To address this complicated Hamiltonian manifold learning challenge, we extend sparse symplectic, random Fourier Gaussian processes learning with predictive successive numerical estimations of the Hamiltonian landscape, using a generalized form of state and conjugate momentum Hamiltonian dynamics, appropriate to different classes of conservative, dissipative and port-Hamiltonian physical systems. In addition to the kernelized evidence lower bound (ELBO) loss for data fidelity, we incorporate stability and conservation constraints as additional hyper-parameter balanced loss terms to regularize the model's multi-gradients, enforcing physics correctness for improved prediction accuracy with bounded uncertainty.

Figures

Figures reproduced from arXiv: 2509.07280 by Chandrajit Bajaj, Luke McLennan, Minh Nguyen, Ryan Farell, Yi Wang.

Figure 1
Figure 1. Figure 1: A visualization of our framework for generalized Hamiltonian learning. The Hamiltonian network queries the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Phase map diagrams with sampled trajectories for various conservative (top row), dissipative (middle row), [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The breakdown of our loss function into the ELBO, energy and volume conservation, and Lyapunov stability [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The 1st column is the true phase maps with Hamiltonian energy contours three different dissipative Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The 1st column is the true phase maps with Hamiltonian energy contours three different port-Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Volume preservation comparison between unregularized and volume-regularized models. Starting from initial [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Phase map comparison for the damped pendulum system with various noise levels. Subfigure (a) shows the [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

discussion (0)

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Forward citations

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.