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arxiv: 2605.23510 · v1 · pith:6WTF65A3new · submitted 2026-05-22 · 💻 cs.LG

Learning partially observed systems with neural Hamiltonian ordinary differential equations

Sunniva Meltzer , S{\o}lve Eidnes , Alexander Johannes Stasik This is my paper

Pith reviewed 2026-05-25 05:06 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural Hamiltonian ODEpartially observed dynamical systemsenergy conservationphysics-informed learninglatent dynamicsHamiltonian neural networksneural ODElong-horizon prediction
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The pith

Neural Hamiltonian ODEs learn partially observed dynamical systems by enforcing energy conservation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents NHODE as a way to learn dynamical systems when some state variables are never observed directly. It embeds Hamiltonian structure so that energy is conserved by construction, allowing the loss to be computed only on the visible variables while the model still infers the hidden ones. The authors test the approach on mass-spring systems and the chaotic three-body problem, showing that adding more physical constraints steadily raises accuracy and prevents instability over long time horizons. Purely data-driven baselines diverge in the same regimes. The central point is that the conserved-energy prior supplies enough inductive bias to recover latent dynamics from partial data alone.

Core claim

NHODE merges Hamiltonian neural networks with neural ODEs so that the learned vector field respects energy conservation even when only a subset of coordinates is observed. The training loss is defined solely on the visible variables; the Hamiltonian formulation plus optional symmetry transformations and separable energy terms then determine the evolution of the unobserved variables. On linear and nonlinear oscillators as well as the three-body problem, models with more embedded structure produce lower error and remain stable far beyond the training horizon, while unstructured neural ODEs become unstable.

What carries the argument

The NHODE model, which uses a neural network to parameterize a Hamiltonian whose gradients define the ODE right-hand side, combined with a flexible neural-ODE integrator that permits supervision only on observed coordinates.

If this is right

  • Accuracy and long-horizon stability increase as more physical structure (energy conservation, symmetries, separability) is embedded.
  • Latent variables can be reconstructed without any direct supervision on them.
  • The same framework remains stable on chaotic systems where purely data-driven models diverge.
  • Training remains possible when the loss is restricted to the observed subset of the state.

Where Pith is reading between the lines

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

  • The same inductive bias could be tested on systems that conserve quantities other than energy, such as angular momentum.
  • Real sensor data with measurement noise would provide a practical test of whether the learned Hamiltonian remains identifiable.
  • The approach may transfer to control tasks where only partial state feedback is available.
  • Adding dissipative terms or time-dependent Hamiltonians would be a direct next extension.

Load-bearing premise

The underlying system must obey Hamiltonian mechanics whose total energy can be recovered from the observed variables alone.

What would settle it

A direct comparison in which an unstructured neural ODE matches or exceeds NHODE long-horizon accuracy on the three-body problem after identical training would falsify the claimed benefit of the Hamiltonian constraint.

Figures

Figures reproduced from arXiv: 2605.23510 by Alexander Johannes Stasik, S{\o}lve Eidnes, Sunniva Meltzer.

Figure 1
Figure 1. Figure 1: Training pipeline for neural Hamiltonian ODEs. From an observed history, a neural [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dynamical systems investigated in this study. From left to right, we have a linear [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Predictions for the linear two-mass two-spring system. Panel (a) shows the predicted [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Model prediction for the triangular nonlinear mass-spring system. Panel (a) shows a grid [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Model prediction for the three body problem. Panel (a) shows a grid of snapshots, where [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Model predictions for the linear two–mass two–spring system when initial conditions of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Testing the optimal value of the Plummer softening parameter in the three body problem. [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Long-horizon rollouts for the three-body problem, showing the predicted position of the [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

When learning dynamical systems from data, embedding physical structure can constrain the solution space and improve generalization, but many physics-informed models assume access to the full system state. This limits their use in partially observed settings, where some state variables are completely unobserved and must be inferred without direct supervision. Here, we present neural Hamiltonian ordinary differential equations (NHODE), a framework that combines Hamiltonian neural networks (HNNs) with neural ordinary differential equations (neural ODEs) to learn partially observed dynamical systems from data. The Hamiltonian structure enforces energy conservation by construction, while the neural ODE framework enables a flexible training procedure that allows the loss to be defined only on observed variables. We also incorporate additional physical constraints through symmetry-aware coordinate transformations and separable energy formulations. The framework is evaluated on systems of increasing complexity, from linear and nonlinear mass-spring systems to the chaotic three-body problem. Across all examples, increasing the amount of embedded physical structure improves the accuracy and long-horizon stability of the predictions. Even in the most challenging regimes, the NHODE framework captures both observed and latent dynamics, whereas purely data-driven baselines become unstable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces neural Hamiltonian ordinary differential equations (NHODE), which integrate Hamiltonian neural networks with neural ODEs to learn partially observed dynamical systems. The Hamiltonian structure enforces energy conservation by construction, while the neural ODE formulation permits the training loss to be defined exclusively on observed variables. Additional inductive biases are incorporated via symmetry-aware coordinate transformations and separable energy formulations. The method is evaluated on systems of increasing complexity, including linear and nonlinear mass-spring oscillators and the chaotic three-body problem, with the central claim that greater embedding of physical structure yields improved accuracy and long-horizon stability, and that NHODE recovers both observed and latent dynamics where purely data-driven baselines become unstable.

Significance. If the empirical results are robust, the work provides a principled route to physics-informed learning under partial observability, a setting that arises frequently in applications such as robotics and experimental physics. The explicit combination of conservation-law constraints with flexible neural-ODE training on incomplete data is a clear technical contribution. The evaluation across a progression of Hamiltonian systems, including a chaotic example, supplies a useful test bed for assessing long-term stability.

minor comments (3)
  1. [Abstract] Abstract: the summary statement would be strengthened by the inclusion of at least one quantitative performance metric (e.g., long-horizon prediction error or stability horizon) together with a brief description of the baselines and training protocol.
  2. [§3] The manuscript should clarify in §3 or §4 whether the latent-state inference is performed by augmenting the observed state with additional neural-ODE variables or by an explicit encoder; the current description leaves this architectural choice ambiguous.
  3. [Figures 4–6] Figure captions for the three-body experiments should report the integration horizon and the precise definition of “stability” used to generate the reported trajectories.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its technical contribution, and the recommendation for minor revision. The report does not list any specific major comments.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The provided abstract and context describe NHODE as combining HNNs (energy conservation by construction) with neural ODEs for training on observed variables only, with added symmetry constraints. No equations, derivations, or self-citations are exhibited that reduce any prediction or result to fitted inputs or prior author work by construction. The framework uses standard optimization with physical inductive biases; the central claims remain independent of any internal reduction and are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The framework rests on the domain assumption that the target systems are Hamiltonian. Neural network parameters are the standard fitted objects. No new physical entities are postulated.

free parameters (1)
  • Neural network parameters
    Weights of the networks approximating the Hamiltonian and dynamics, optimized via gradient descent on observed data.
axioms (1)
  • domain assumption The system is Hamiltonian with conserved total energy
    Invoked to enforce conservation by construction and to enable inference of unobserved states.

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