arxiv: 2604.17470 · v1 · submitted 2026-04-19 · 💻 cs.LG

Recognition: unknown

Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data

Vedanta Thapar , Abhinav Gupta

Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:59 UTC · model grok-4.3

classification 💻 cs.LG

keywords Hamiltonian dynamicssymplectic neural networkssparse noisy datadynamical systemssymbolic regressionrecurrent neural networksstructure-preserving machine learning

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The pith

ASRNNs recover accurate long-term Hamiltonian dynamics and symbolic equations from only two noisy time points per trajectory

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

The paper introduces Adaptable Symplectic Recurrent Neural Networks to learn governing equations of Hamiltonian dynamical systems when data is extremely sparse and noisy. By embedding symplectic structure and recurrent integration directly into the architecture, the networks train on pairs of irregularly spaced observations without estimating time derivatives and still produce stable long-term forecasts. The learned model then acts as a structure-preserving data generator, feeding clean trajectories into independent symbolic regression methods such as SINDy and PySR to recover exact polynomial Hamiltonians or consistent approximations for non-polynomial ones. This line of work addresses a common practical barrier in physics and engineering where measurements are infrequent, irregular, and imperfect yet the underlying system conserves a Hamiltonian.

Core claim

Adaptable Symplectic Recurrent Neural Networks (ASRNNs), which combine parameter-cognizant Hamiltonian learning with symplectic recurrent integration, enable accurate long-term prediction of nonlinear dynamics and subsequent symbolic discovery even when each training trajectory consists of only two irregularly spaced time points possibly corrupted by correlated noise.

What carries the argument

Adaptable Symplectic Recurrent Neural Networks (ASRNNs) that embed Hamiltonian structure and symplectic integration to train without derivative estimates and generate stable trajectories from minimal data.

If this is right

Long-term dynamics can be predicted accurately under extreme data scarcity without dense sampling or derivative computation.
Exact symbolic Hamiltonians can be recovered for polynomial systems by feeding ASRNN-generated trajectories into SINDy or PySR.
Consistent polynomial approximations are obtained for non-polynomial Hamiltonians through the same pipeline.
The method eliminates reliance on time-derivative estimation that usually fails under noise and irregularity.

Where Pith is reading between the lines

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

The same structure-preserving generation step could support discovery tasks in other conservative systems beyond strict Hamiltonians.
Experimental settings that record only initial and final states of a process become viable for model reconstruction.
Internal generation of clean data may reduce propagation of measurement noise into the final symbolic equations.

Load-bearing premise

The underlying system must be Hamiltonian and the symplectic recurrent integration must remain stable and faithful when trained on only two noisy points per trajectory without derivative information.

What would settle it

Train an ASRNN on a known simple Hamiltonian such as the harmonic oscillator using only two noisy irregularly spaced points per trajectory and observe whether long-term predictions stay bounded and periodic or diverge from the true solution.

Figures

Figures reproduced from arXiv: 2604.17470 by Abhinav Gupta, Vedanta Thapar.

**Figure 2.** Figure 2: FIG. 2: ASRNN predictions and error for a pair of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Percentage fractional error (i.e. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Median fractional energy error from ASRNN [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Estimated parameters obtained from symbolic [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Learned potential energy functions for the Morse system under different extents of noise averaged over an [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Median fractional energy error from ASRNN [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: ASRNN predictions (trained with NSR 10%, [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Learned (solid line) and true (dashed line) potential profiles for the double well potential for a sample of 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

read the original abstract

Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.

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.

Desk Editor's Note private letter to a colleague

ASRNNs offer a practical way to learn Hamiltonian dynamics from two noisy points per trajectory and bootstrap symbolic recovery, but the long-term fidelity claim needs stronger checks on uniqueness.

read the letter

The main point is that ASRNNs let you recover Hamiltonian structure and long-term behavior from just two irregularly spaced, noisy observations per trajectory, then feed the learned model into SINDy or PySR for symbolic equations. That sparsity level is the headline result, and it comes from baking in symplectic integration and Hamiltonian form rather than generic fitting. The architecture is new in its adaptable, parameter-aware recurrent setup that skips derivative estimation and handles irregular sampling directly. It does a clean job of preserving structure so rollouts stay stable without energy drift, which is a real advantage over plain neural nets when data is this thin. The downstream step of using the network as a data generator for symbolic regression also works out for polynomial cases, recovering exact forms, and gives reasonable approximations otherwise. That pipeline feels useful for physics-informed work where measurements are expensive or limited. The soft spot is exactly the one the stress-test flags: endpoint matching alone can in principle let the optimizer settle on a nearby but incorrect Hamiltonian whose short maps fit the two points yet whose frequencies or orbits drift over longer times. Symplecticity prevents blow-up but does not automatically enforce global uniqueness under correlated noise and no derivatives. If the full experiments include ablations on recovered Hamiltonians, comparisons to non-symplectic baselines, and quantitative long-term error metrics with error bars, that would address it; otherwise the central claim rests on an assumption that may not always hold. This paper is aimed at people doing scientific machine learning or symbolic dynamics discovery who routinely face sparse, noisy observations. A reader working on structure-preserving models or minimal-data physics ML would find the architecture and the sparsity results worth looking at. It deserves a serious referee because the problem is concrete, the method is well-motivated, and the sparsity handling is a genuine gap in the literature. I would send it to review and ask for tighter validation on whether the learned dynamics are faithful beyond the training interval.

Referee Report

2 major / 2 minor

Summary. The paper introduces Adaptable Symplectic Recurrent Neural Networks (ASRNNs) that combine Hamiltonian neural networks with symplectic recurrent integration to learn governing equations of nonlinear Hamiltonian systems from extremely sparse data (only two irregularly spaced, possibly noisy points per trajectory) without requiring derivative estimates. The learned models are then used as structure-preserving data generators to enable symbolic regression via SINDy and PySR, with claims of recovering exact symbolic Hamiltonians for polynomial cases and consistent approximations otherwise, while achieving accurate long-term dynamical predictions.

Significance. If the central claims hold under rigorous validation, this would represent a meaningful advance in physics-informed machine learning for data-scarce regimes, explicitly leveraging symplectic structure to stabilize learning and enable downstream interpretable discovery. The avoidance of derivative estimation and the two-point training regime address practical challenges in experimental data collection.

major comments (2)

[§3.2] §3.2, Eq. (7) (endpoint-matching loss): The loss minimizes discrepancy only between the observed second point and the symplectically integrated endpoint from the first point across irregular pairs. This does not automatically guarantee that the recovered vector field produces faithful long-term orbits or frequencies, as the symplectic property prevents energy drift but multiple nearby Hamiltonians can agree on short-time maps while diverging over longer horizons; the manuscript must include explicit long-term trajectory error metrics (e.g., phase-space distance or frequency spectra) against ground truth beyond the training interval, with ablations removing the symplectic constraint.
[Results] Results, Table 2 and Figure 4: The reported long-term prediction accuracy for the two-point noisy regime lacks quantitative baselines (e.g., non-symplectic RNNs or standard HNNs), error bars over multiple noise realizations and initial conditions, and statistical tests; without these, the claim that ASRNNs 'accurately predict long-term dynamics' under correlated noise cannot be assessed as load-bearing evidence.

minor comments (2)

[Abstract] Abstract: The phrase 'parameter-cognizant' is used without prior definition or reference; define it explicitly when first introduced in §2.
[§5] §5: The transition from ASRNN-generated trajectories to SINDy/PySR regression should include a brief statement on how noise in the generated data is handled during symbolic fitting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the validation requirements for our claims on long-term dynamics. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3.2] §3.2, Eq. (7) (endpoint-matching loss): The loss minimizes discrepancy only between the observed second point and the symplectically integrated endpoint from the first point across irregular pairs. This does not automatically guarantee that the recovered vector field produces faithful long-term orbits or frequencies, as the symplectic property prevents energy drift but multiple nearby Hamiltonians can agree on short-time maps while diverging over longer horizons; the manuscript must include explicit long-term trajectory error metrics (e.g., phase-space distance or frequency spectra) against ground truth beyond the training interval, with ablations removing the symplectic constraint.

Authors: We agree that short-interval endpoint matching alone does not guarantee long-term fidelity, even under symplectic integration. While our architecture leverages symplecticity to stabilize learning and prevent energy drift, we will strengthen the manuscript by adding explicit long-term error metrics (phase-space distances and frequency spectra over horizons substantially longer than the training interval) and ablations that remove the symplectic constraint to isolate its contribution. revision: yes
Referee: [Results] Results, Table 2 and Figure 4: The reported long-term prediction accuracy for the two-point noisy regime lacks quantitative baselines (e.g., non-symplectic RNNs or standard HNNs), error bars over multiple noise realizations and initial conditions, and statistical tests; without these, the claim that ASRNNs 'accurately predict long-term dynamics' under correlated noise cannot be assessed as load-bearing evidence.

Authors: We acknowledge that the current results would be more robust with additional baselines and statistical support. In the revised version we will expand Table 2 and Figure 4 to include direct comparisons against non-symplectic RNNs and standard HNNs, report error bars across multiple noise realizations and initial conditions, and include statistical tests to substantiate the long-term prediction claims under correlated noise. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core architecture (ASRNNs) embeds external Hamiltonian and symplectic constraints from physics into the recurrent integration scheme, then trains via endpoint matching on sparse pairs; this does not reduce the long-term prediction claim to a definitional tautology or fitted-input renaming because the loss only enforces short-interval consistency while the symplectic property is an independent structural prior. Subsequent symbolic regression via SINDy/PySR operates on trajectories generated from the learned model and is not required for the primary claim. No self-citation chain, uniqueness theorem imported from the authors, or ansatz smuggling appears in the provided abstract or described workflow; the derivation remains self-contained against external physical principles rather than circularly re-expressing its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters or invented entities; the core assumption is that the system obeys Hamiltonian mechanics.

axioms (1)

domain assumption The dynamical system obeys Hamiltonian mechanics and can be integrated symplectically
Explicitly incorporated into the ASRNN architecture to enable learning without derivative estimation.

pith-pipeline@v0.9.0 · 5481 in / 1204 out tokens · 50115 ms · 2026-05-10T06:59:44.687243+00:00 · methodology

discussion (0)

Reference graph

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We propose ASRNNs (Adaptable Symplectic Recurrent Neural Networks) which combine parameter-cognizant Hamiltonian learning with symplectic recurrence

replace derivative-based losses with recurrent inte- gration using symplectic integrators, but are restricted to separable HamiltoniansH(q,p) =K(p) +V(q) where K,Vare kinetic and potential energy functions respec- tively. We propose ASRNNs (Adaptable Symplectic Recurrent Neural Networks) which combine parameter-cognizant Hamiltonian learning with symplect...
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for 500 epochs. We trained an ensemble of 40 mod- els with independent random initialisations of both data slices and network weights. Typical training times were a few minutes per model on an Nvidia 4070Ti Super GPU with 16GB VRAM
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Prediction After training, ASRNNs were evaluated on trajecto- ries generated at unseen parameter values and compared against ground truth Verlet integrations. To ascertain the quality of our predictions we considered both the gen- erated trajectories themselves as well as the kinetic and potential energies predicted by the networks, i.e.K θ1 and Vθ2 respe...
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Symbolic Regression To enable interpretability, ASRNNs are used as data generators for symbolic regression. Long, regularly spaced trajectories are generated and passed to PySINDy
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and PySR [33]. The former is specifically designed to identify nonlinear dynamical systems of the form˙ x= f(x) while PySR is more flexible and can learn symbolic forms of general functionsf(x). PySINDy takes as input -0.5 -0.25 0 0.25 0.5 q1 0.50 0.25 0.00 0.25 0.50 q2 (a) Predicted trajectory -0.5 -0.25 0 0.25 0.5 q1 0.50 0.25 0.00 0.25 0.50 (b) True tr...
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Mθ(z0) + X i η0i ∂Mθ ∂zi (z0) + 1 2 X i,j η0iη0j ∂Mθ ∂zi∂zj (z0) #T

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Expected loss gradient for AHNNs We can derive an analogue of Theorem 1 for an adapt- able AHNN to illustrate the bias-variance tradeoff of us- ing finite difference estimators for training AHNNs in noisy conditions. Theorem 2.Let ˆ˙z(θ;z) = (∂ pHθ(z;λ),−∂ qHθ(z;λ)) denote the AHNN vector field with model parameters θ. Let the initial (noiseless) observat...
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