Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes

Jan-Hendrik Ewering; Niklas Wahlstr\"om; Robin E. Herrmann; Thomas B. Sch\"on; Thomas Seel

arxiv: 2511.05330 · v2 · submitted 2025-11-07 · 💻 cs.LG · cs.SY· eess.SY

Learning Dynamics from Input-Output Data with Hamiltonian Gaussian Processes

Jan-Hendrik Ewering , Robin E. Herrmann , Niklas Wahlstr\"om , Thomas B. Sch\"on , Thomas Seel This is my paper

Pith reviewed 2026-05-17 23:58 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SY

keywords Hamiltonian Gaussian processesdynamics learninginput-output dataBayesian inferencenon-conservative Hamiltonian systemshidden state estimationphysically consistent modeling

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The pith

Hamiltonian Gaussian processes learn system dynamics from input-output data alone by adding non-conservative terms and Bayesian inference over hidden states.

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

The paper develops a method to build energy-consistent dynamics models from only input and output measurements, without needing velocity or momentum data. It extends Gaussian processes with a Hamiltonian structure that includes non-conservative effects such as damping or external forces. A fully Bayesian procedure then recovers probability distributions over the unknown states, GP hyperparameters, and structural parameters like damping coefficients. This approach is demonstrated on a nonlinear simulation example and compared against methods that require momentum measurements. The result supports physically consistent modeling for control tasks where full state information is unavailable.

Core claim

A non-conservative Hamiltonian Gaussian process formulation combined with fully Bayesian inference allows recovery of hidden states, hyperparameters, and structural parameters such as damping coefficients directly from input-output data, producing uncertainty-aware models that respect energy exchange with the environment.

What carries the argument

Non-conservative Hamiltonian Gaussian process with Bayesian joint inference over hidden states and structural hyperparameters such as damping coefficients.

If this is right

Models can be learned for control applications where only position and force measurements are available.
Uncertainty estimates remain available while enforcing energy consistency even with dissipation present.
The same inference pipeline can jointly estimate unknown physical parameters such as damping without separate experiments.

Where Pith is reading between the lines

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

The approach could extend to systems with partial observability in robotics or mechanics by treating the non-conservative terms as learnable corrections.
If the method generalizes to noisy real-world data, it would reduce the sensor requirements for identifying energy-based models.

Load-bearing premise

The true dynamics can be expressed as a Hamiltonian system plus non-conservative terms, and the Bayesian procedure can accurately recover the hidden states and parameters from input-output sequences alone.

What would settle it

A controlled experiment on a system whose trajectories violate Hamiltonian structure with non-conservative terms, or a direct comparison showing that the input-output method fails to match the accuracy of momentum-based baselines on the same data.

Figures

Figures reproduced from arXiv: 2511.05330 by Jan-Hendrik Ewering, Niklas Wahlstr\"om, Robin E. Herrmann, Thomas B. Sch\"on, Thomas Seel.

**Figure 1.** Figure 1: Test system & Hamiltonian. To evaluate the proposed method, we conduct a simulation case study with a non-harmonic oscillator, governed by the Hamiltonian H(q, p) = q 2 2 + p 2 2 + 2 cos q, with the position q and the momentum p (see [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

**Figure 2.** Figure 2: Flow maps following from the true and learned Hamiltonians. The reduced-rank Hamilto [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: True and estimated system behavior in the training data set (left), as well as density [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: True system behavior and forward predictions of the learned Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Flow maps following from the true and learned Hamiltonians of the non-harmonic oscil [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

read the original abstract

Embedding non-restrictive prior knowledge, such as energy conservation laws, into learning methods is a key motive to construct physically consistent dynamics models from limited data, relevant for, e.g., model-based control. Recent work incorporates Hamiltonian dynamics into Gaussian Processes (GPs) to obtain uncertainty-quantifying, energy-consistent models, but these methods rely on -- rarely available -- velocity or momentum data. In this paper, we study dynamics learning using Hamiltonian GPs and focus on learning solely from input-output data, without relying on velocity or momentum measurements. Adopting a non-conservative formulation, energy exchange with the environment, e.g., through external forces or dissipation, can be captured. We provide a fully Bayesian scheme for estimating probability densities of unknown hidden states, GP hyperparameters, as well as structural hyperparameters, such as damping coefficients. The proposed method is evaluated in a nonlinear simulation case study and compared to a state-of-the-art approach that relies on momentum measurements.

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

This extends Hamiltonian GPs to input-output data only by adding non-conservative terms and inferring hidden states plus damping in a fully Bayesian way, but the simulation results leave open whether the latent velocities are reliably recovered.

read the letter

The main point is that the authors have a workable scheme for learning Hamiltonian-style dynamics from positions and inputs alone. They drop the usual requirement for velocity or momentum measurements by splitting the dynamics into a conservative GP-modeled part and explicit non-conservative terms (damping and external forces), then run joint Bayesian inference over the hidden velocity trajectories, GP hyperparameters, and structural parameters like damping coefficients. That is the concrete advance over earlier Hamiltonian GP papers that needed momentum data. The simulation case study on a nonlinear system and the comparison to a momentum-based baseline are reasonable first checks, and the fully Bayesian treatment is a clean way to quantify uncertainty in the recovered quantities. Credit to them for targeting a setting that actually occurs in robotics and system identification, where you often only get position sensors and known inputs. The math appears to build on established Hamiltonian GP foundations without obvious circularity. That said, the identifiability concern is real and not fully dispelled by the current evidence. Without direct velocity observations the mapping from observed positions to latent velocities is under-constrained, and the extra flexibility from the GP plus the damping parameters can trade off against each other. A single nonlinear simulation does not stress-test this enough, especially under realistic noise levels or model mismatch. It would be useful to see either a theoretical identifiability argument or experiments that deliberately vary the amount of data and check posterior consistency on the hidden states. Overall this is aimed at people working on physically consistent learning for control who already know the Hamiltonian GP literature. A reader looking for a practical Bayesian method that avoids velocity sensors would find the formulation useful. I would send it to peer review; the core idea fills a documented gap and the technical steps are clear enough that referees can evaluate the derivations and ask for stronger validation.

Referee Report

2 major / 2 minor

Summary. The paper introduces a non-conservative Hamiltonian Gaussian Process formulation for learning dynamical systems from input-output trajectories alone, without velocity or momentum observations. It develops a fully Bayesian inference procedure to jointly estimate hidden states, GP hyperparameters, and structural parameters such as damping coefficients, and evaluates the approach on a nonlinear simulation case study against a momentum-measurement baseline.

Significance. If the Bayesian recovery of hidden velocities and damping parameters proves reliable from position-input data, the method would meaningfully extend Hamiltonian GP models to settings where full-state measurements are unavailable, supporting uncertainty-aware, energy-consistent dynamics learning for control. The simulation evaluation provides initial evidence but does not yet establish robustness or identifiability guarantees.

major comments (2)

[§3.3] §3.3 (Non-conservative Hamiltonian GP): The joint posterior over latent velocities, GP hyperparameters, and damping coefficients is claimed to be recoverable from position-input data, yet the formulation allows trade-offs between the conservative GP vector field and the dissipative terms; no identifiability analysis or sensitivity study is provided to show that the posterior concentrates on the true parameters rather than equivalent explanations of the same trajectories.
[§4.2] §4.2 (Simulation case study): The reported state and parameter recovery metrics are obtained under a specific nonlinear system with known structure; the paper does not test scenarios with weaker excitation, higher noise, or model mismatch that would stress the weak-identifiability concern raised by the non-conservative split.

minor comments (2)

[§3.1] Notation for the non-conservative force term is introduced without an explicit equation reference in the main text; adding a numbered equation would improve readability.
[§4.1] The comparison baseline is described only as 'state-of-the-art'; citing the specific prior work and its exact assumptions would clarify the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment below, clarifying our approach and outlining revisions to strengthen the manuscript's discussion of identifiability and robustness.

read point-by-point responses

Referee: [§3.3] §3.3 (Non-conservative Hamiltonian GP): The joint posterior over latent velocities, GP hyperparameters, and damping coefficients is claimed to be recoverable from position-input data, yet the formulation allows trade-offs between the conservative GP vector field and the dissipative terms; no identifiability analysis or sensitivity study is provided to show that the posterior concentrates on the true parameters rather than equivalent explanations of the same trajectories.

Authors: We acknowledge that the non-conservative formulation permits potential trade-offs between the GP-modeled conservative vector field and the explicit dissipative terms. Our fully Bayesian procedure jointly infers latent velocities, GP hyperparameters, and damping coefficients under priors that encode expected physical behavior. The reported simulation demonstrates accurate recovery of ground-truth values, suggesting the posterior is informative in the tested regime. We agree, however, that a dedicated identifiability analysis is absent. In the revised manuscript we will add a subsection discussing possible degeneracies and include a sensitivity study that perturbs the damping coefficients and examines posterior concentration. revision: yes
Referee: [§4.2] §4.2 (Simulation case study): The reported state and parameter recovery metrics are obtained under a specific nonlinear system with known structure; the paper does not test scenarios with weaker excitation, higher noise, or model mismatch that would stress the weak-identifiability concern raised by the non-conservative split.

Authors: The presented case study uses a nonlinear system with known structure to enable direct quantitative comparison against the momentum-measurement baseline. We recognize that this single setting does not fully probe robustness under reduced excitation, elevated noise, or structural mismatch. To address the identifiability concern, the revised evaluation section will incorporate additional experiments that systematically vary input richness, noise intensity, and model mismatch while reporting posterior diagnostics. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on external Hamiltonian and GP foundations

full rationale

The paper's central contribution is a Bayesian inference procedure for recovering hidden states, GP hyperparameters, and structural parameters (e.g., damping) from input-output trajectories under a non-conservative Hamiltonian formulation. This scheme is constructed from standard external priors on Hamiltonian dynamics and Gaussian processes rather than by redefining or fitting quantities that are then relabeled as predictions. No load-bearing step reduces by construction to the inputs via self-definition, fitted-input renaming, or a self-citation chain that itself lacks independent verification. The evaluation on nonlinear simulation data further separates the method from tautological reproduction of its modeling assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The method rests on standard Gaussian Process assumptions and the domain assumption that dynamics admit a Hamiltonian structure with non-conservative extensions; several hyperparameters are estimated from data.

free parameters (2)

damping coefficients
Structural hyperparameters estimated via the Bayesian scheme as examples of parameters capturing energy exchange.
GP hyperparameters
Estimated jointly with hidden states in the fully Bayesian procedure.

axioms (1)

domain assumption System dynamics can be represented using a Hamiltonian structure that permits non-conservative energy exchange with the environment.
Explicitly adopted non-conservative formulation to capture external forces or dissipation.

pith-pipeline@v0.9.0 · 5483 in / 1240 out tokens · 33287 ms · 2026-05-17T23:58:29.635913+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We consider learning of non-conservative dynamics with Hamiltonian GPs... from input-output data only... damping coefficient d=:ϑ_S
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

˙x = (J(x)−R(x))∇xH(x) + G(x)u + w

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

doi: 10.1146/annurev-control-042920-020211. Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud. Neural ordinary differential equations. InAdvances in Neural Information Processing Systems, volume 31. Curran Associates, Inc, 2018. Miles Cranmer, Sam Greydanus, Stephan Hoyer, Peter Battaglia, David Spergel, and Shirley Ho. Lagrangian...

work page doi:10.1146/annurev-control-042920-020211 2018
[2]

Gabriel Riutort-Mayol, Paul-Christian B¨urkner, Michael R

doi: 10.1063/5.0048129. Gabriel Riutort-Mayol, Paul-Christian B¨urkner, Michael R. Andersen, Arno Solin, and Aki Vehtari. Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming. Statistics and Computing, 33(1), 2023. doi: 10.1007/s11222-022-10167-2. 12 LEARNINGDYNAMICS FROMINPUT-OUTPUTDATA WITHHAMILTONIANGPS Gareth O...

work page doi:10.1063/5.0048129 2023

[1] [1]

doi: 10.1146/annurev-control-042920-020211. Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud. Neural ordinary differential equations. InAdvances in Neural Information Processing Systems, volume 31. Curran Associates, Inc, 2018. Miles Cranmer, Sam Greydanus, Stephan Hoyer, Peter Battaglia, David Spergel, and Shirley Ho. Lagrangian...

work page doi:10.1146/annurev-control-042920-020211 2018

[2] [2]

Gabriel Riutort-Mayol, Paul-Christian B¨urkner, Michael R

doi: 10.1063/5.0048129. Gabriel Riutort-Mayol, Paul-Christian B¨urkner, Michael R. Andersen, Arno Solin, and Aki Vehtari. Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming. Statistics and Computing, 33(1), 2023. doi: 10.1007/s11222-022-10167-2. 12 LEARNINGDYNAMICS FROMINPUT-OUTPUTDATA WITHHAMILTONIANGPS Gareth O...

work page doi:10.1063/5.0048129 2023