Hamiltonian Dynamics

Follow this topic Ask about this topic

Momentum

1 paper in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Jul 13Week of Sep 28

Latest papers 32

Oct 5, 2026quant-ph

Out-of-control Hamiltonian Learning

Learning the Hamiltonian of a many-body system from its dynamics is a central task in quantum science, yet the algorithms with the strongest provable guarantees assume some level of quantum control--fast, arbitrary single-qubit gates interleaved with time evolution, and measurements in arbitrary bases--that is beyond the capabilities of near-term analog quantum simulators. Motivated by analog atom- and ion-based platforms, we study Hamiltonian learning under minimal access models. Uniform state preparation and measurements: We first consider the setting where in every experiment, one can rotate each qubit to the same state, perform short-time evolution, and measure every qubit in the same basis. Surprisingly, we show that for generic 2-local Hamiltonians on any interaction graph, all of the parameters can be reconstructed from such experiments. Computational basis state preparation and measurements: We then consider a similarly constrained setting, but where state preparation and measurement are restricted to the computational basis. For nearest-neighbor Hamiltonians with only Pauli

X/Z

interactions, a class which captures contemporary Rydberg atom platforms, we show that over 1D and 2D rectangular lattices, all of the parameters can be reconstructed from such experiments up to unavoidable gauges. Our protocols introduce new techniques for solving structured polynomial systems over an extensive number of parameters. Taken together, our results suggest that one can learn a great deal from the dynamics of quantum many-body systems even under the most stringent experimental constraints.

Weiyuan Gong, Muzhou Ma, Sitan Chen +2

Harvard University. · Caltech. · Oratomic.

Quantum Many-Body Physics Hamiltonian Dynamics

Sep 24, 2026math.NA

Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs

The Brinkman penalisation method embeds boundary-value problems on complex domains into a simple computational box by modeling the solid region as a strongly dissipative medium, avoiding body-fitted mesh generation. We show that multi-symplectic Hamiltonian PDEs regularised by Brinkman-type penalisation retain a multi-conformal symplectic structure under a compatibility condition linking the symplectic matrix and the penalisation projection. This yields an exact local conservation law, under which the multi-symplectic two-form is conserved in the fluid region and decays exponentially inside the solid. The linear wave equation with Brinkman friction and Maxwell's equations with artificial Ohmic conductivity satisfy this condition, with explicit modified Hamiltonian densities. Building on this, we propose (i) structure-preserving numerical integrators via Strang splitting that satisfy a discrete conformal conservation law, and (ii) conformal symplectic neural operators that interleave exact dissipative flows with learnable multi-symplectic evolution operators, allowing geometry-dependent operator learning. Numerical experiments on wave and electromagnetic scattering demonstrate that our methods reproduce correct local energy budgets and avoid unphysical energy drift, providing a principled framework for physics-consistent scientific machine learning on complex domains.

Teo Deveney, Baige Xu, Takaharu Yaguchi

Department of Computer Science, University of Bath, United Kingdom · Center for Advanced Intelligence Project, RIKEN, Japan · Kyushu University / Center for Advanced Intelligence Project, RIKEN, Japan

PDE Operator Learning Neural Operators Hamiltonian Dynamics

Aug 10, 2026cs.LG

A matched-integrator evaluation of Hamiltonian neural networks on pendulum and Kepler dynamics

Hamiltonian Neural Networks (HNNs) parameterize conservative dynamics through a learned scalar Hamiltonian, providing an architectural prior that is absent from generic vector-field neural networks. We evaluate this prior under a controlled protocol in which an HNN and a parameter-matched feedforward baseline are trained on the same RK4-generated trajectories, use the same central-difference derivative targets and optimization settings, and are integrated at inference with the same RK4 scheme. Results are reported over five independent training seeds. On the nonlinear pendulum, the HNN reduces mean energy drift by 42-fold and mean trajectory MSE by 15.8-fold at T = 100, approximately 16 pendulum periods. Its energy drift also remains bounded and exhibits substantially lower seed-to-seed variability than the standard-network baseline. An energy-stratified analysis shows that the difference becomes more pronounced as trajectories explore more nonlinear regions of phase space. As an additional diagnostic, we examine an explicit Störmer--Verlet-style rollout of the learned HNN. Because the learned Hamiltonian is not constrained to the separable form H(q,p) = T(p) + V(q), the standard symplecticity guarantee of velocity Verlet does not directly apply. We further apply the same matched-integrator protocol to the three-dimensional Kepler two-body problem. The HNN again exhibits lower trajectory, energy, and angular-momentum drift than the parameter-matched baseline. These experiments provide a controlled study of how Hamiltonian parameterization affects long-horizon prediction and physical consistency across two conservative dynamical systems.

Lenick Kemunto Nyabuto, Yae Ulrich Gaba, Birahim Tewe

African Institute for Mathematical Sciences (AIMS) Senegal

Dynamical Systems Hamiltonian Neural Networks Hamiltonian Dynamics

Jul 31, 2026nlin.CD

Extrapolating the emergence of Hamiltonian chaos with random-feature Hamiltonian neural networks

Machine learning of Hamiltonian dynamics has driven growing interest in Hamiltonian neural networks (HNNs), which encode Hamilton's equations of motion into the learning architecture. Despite this progress, it remains unknown whether such networks can predict dynamical regimes absent from their training data, in particular the broad chaotic sea that emerges beyond the observed parameter interval. We address this question using a parameter-aware random-feature Hamiltonian neural network (RF-HNN). Trained using data from only a small number of control-parameter values at which invariant tori dominate, the RF-HNN predicts autonomous long-time dynamics at unseen parameter values where mixed phase space develops and chaotic regions expand, with no data from that regime used in training or model selection. The method is demonstrated across four two-degree-of-freedom Hamiltonian families, including the Hénon-Heiles system. Using Poincaré-section geometry and finite-time Lyapunov exponents, we show that the RF-HNN reproduces the breakup of regular structures and the emergence and growth of chaotic regions, whereas conventionally trained HNNs with the same Hamiltonian structure remain too regular. These results show that what decides parameter extrapolation is not Hamiltonian structure alone but how the fitted Hamiltonian continues in the control parameter. To our knowledge, this is the first demonstration that a learned Hamiltonian can qualitatively extrapolate from predominantly regular dynamics into a broad chaotic sea absent from training.

Jaesung Choi

Center for Artificial Intelligence and Natural Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

Neural Surrogate Modeling Dynamical Systems Hamiltonian Neural Networks Chaotic Dynamical Systems Hamiltonian Dynamics

Jul 31, 2026cs.LG

Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics

Structure-preserving neural networks are essential for the long-term prediction of Hamiltonian systems from data. Many important Hamiltonian systems in mechanics and control admit symmetry reduction to Lie--Poisson systems, including rigid bodies, underwater vehicles, fluids, plasmas, and optimal control problems. A fundamental challenge in learning such systems is that their dynamics evolve in momentum variables that are typically unobservable, while available data consist only of observable quantities such as configurations and velocities. In optimal control applications, the situation is further complicated because the latent variables contain unobservable co-states and the Hamiltonian may be degenerate, preventing the existence of a corresponding Lagrangian and rendering the encoder-decoder approaches inapplicable. We introduce Latent Lie--Poisson Neural Networks (LLPNNs), a structure-preserving framework for learning Lie--Poisson dynamics directly from observable data. The proposed approach exploits three geometric ingredients: (i) learning either a Hamiltonian decoder or a pseudo-Lagrangian encoder on the active variables, (ii) constructing latent trajectories through a universal Noether invariant arising from Lie--Poisson symmetry reduction, and (iii) reconstructing observable and latent dynamics through Lie--Poisson flows combined with Magnus-based Lie-group updates. The resulting method preserves the geometric structure and is applicable to both regular and degenerate Hamiltonian systems. We demonstrate the method on three examples: a generalized rigid body on SO(3), Kirchhoff's underwater vehicle on SE(3), and an optimal-control problem for interacting vehicles on

SE(2)^N

. Numerical experiments show excellent long-term predictive accuracy, strong robustness to noise, and competitive performance using only modest datasets and lightweight neural-network architectures.

Vakhtang Putkaradze

Department of Mathematics, The University of Alabama, Gordon Palmer Hall, Tuscaloosa, AL, 35401, USA

Latent Dynamics Modeling Hamiltonian Neural Networks Hamiltonian Dynamics

Jul 26, 2026math.OC

When Rates Are Geometric: Rate-Certificate Transfer for Contact Splittings in Optimization

Discrete optimization algorithms are often analyzed through continuous-time limiting ODEs, but a convergence certificate for the ODE is not automatically one for the discrete algorithm. We develop contact Hamiltonian systems as a setting where the transfer can be made precise. A contact Hamiltonian

H

on

J^1(\mathbb{R}^n)

obeys the intrinsic decay identity

\dot H = -H\,\partial_s H

, so an augmented energy

\mathcal{E}

built from

H

, together with the conformal rate

\partial_s H

, is a continuous-time rate certificate whenever

\mathcal{E}

controls the objective gap. Our main theorem states, under three named and independently checkable hypotheses, that an order-

r

contact splitting with step

h

transfers this certificate over the finite horizon set by backward error analysis. The discrete decay envelope is governed by the modified conformal factor up to

O(h^r)

perturbations plus a backward-error shadowing defect, and the mechanism is inherited exactly because the modified Hamiltonian is itself a contact Hamiltonian. Quadratic heavy ball is a fully solvable example: its projected dissipative-leapfrog spectrum agrees with established conformal-symplectic optimization theory, while the augmented contact Hamiltonian yields a sharp objective-to-certificate comparison that verifies the transfer hypotheses. For strongly convex objectives with state-dependent damping, an explicit Bregman-type Lyapunov certificate instead transfers by an auxiliary-shadowing corollary. The decomposition

H=K+V+D

into kinetic, objective-encoding potential, and dissipation terms serves as a design template, with a catalogue of closed-form sub-flows including contact-specific damping families. Numerical experiments confirm the predicted conformal-factor tracking orders and show competitive performance on ill-conditioned benchmarks and deep-learning tasks.

George A Kevrekidis

Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218, USA

Optimization Convergence Analysis Hamiltonian Dynamics

Jul 22, 2026physics.comp-ph

Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics

Classical regularization removes the binary-collision singularity from the Kepler problem, but its value as a representation for learned Hamiltonian dynamics has not been systematically isolated. We compare Cartesian and planar Levi--Civita formulations of a perturbed Kepler system with a smooth quadrupole potential. With the perturbation supplied analytically, a Levi--Civita Hamiltonian splitting holds the maximum relative energy error near

2.1\times10^{-5}

through eccentricity

e=0.99

, while the Cartesian splitting becomes unstable. This advantage persists at matched physical horizon and force-evaluation budget, where the regularized baseline is

3\times10^{-5}

, about

4.7

--

8.3

orders of magnitude below the Cartesian arm depending on eccentricity. In held-out high-eccentricity tests with matched sampling, regularized models produce finite rollouts in

40/40

runs versus

0/40

for Cartesian. However, the fixed-shell construction supplies the regularized model with the exact initial orbit energy, and survival still carries

\mathcal{O}(1)

energy error. Four neural residual objectives fail to approach the analytic result. Exact-feature controls show that the regularized residual is a four-monomial degree-6 polynomial that a direct least-squares solve fits to the baseline. The remaining exact-feature gap is due to severe raw-basis ill-conditioning: orthogonalization restores baseline fitting for L-BFGS in two iterations. Small MLPs remain at

\mathcal{O}(1)

rollout error even after gauge symmetrization. Levi--Civita coordinates therefore improve dynamical conditioning while worsening raw-basis optimization conditioning; accurate neural residual learning remains unresolved. This is a controlled falsification-plus-trade-off study, not a solution to learned close-encounter dynamics.

Abhishek Shankar

Residual Learning Hamiltonian Dynamics

Jul 14, 2026cs.LG

Quantum Port-Hamiltonian Neural Networks: Learning Conservative and Dissipative Dynamics via Measurement-Induced Nonlinearity

We introduce Quantum Port-Hamiltonian Neural Networks (Q-pHNNs), a family of parameterised quantum circuits that learn classical dynamics in a structure-preserving manner. The framework relies on the Isomorphic Hamiltonian Mapping (IHM): the skew-symmetric interconnection matrix

\mathbf{J}

corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix

\mathbf{R}

corresponds to Measurement-Induced NonLinearity (MINL) realised via mid-circuit measurement and classical feedforward. This ensures conservation and passivity are enforced by construction rather than penalty terms. We instantiate the IHM in four architectures: (1) a Quantum HNN that learns conservative energy manifolds and extracts Hamilton's equations exactly via the Parameter-Shift Rule; (2) a Q-pHNN using Born-rule measurement for dissipation; (3) a Q-pHNN jointly learning the energy ansatz and damping coefficient; and (4) a topology-entangled Quantum Graph Neural Network for

N

-node coupled-phasor networks. Experiments on the nonlinear pendulum and damped harmonic oscillator demonstrate: (i) ~~$1.35\%$ relative energy drift with a symplectic integrator and scale correction; (ii)~~

100\%

energy monotonicity for the MINL circuit; and (iii)~

12.1\%

error in damping-coefficient identification from vector-field snapshots with no direct supervision on the damping coefficient.

Dibakar Sigdel

1Mindverse Computing LLC, Lynnwood, WA 98087

Hamiltonian Neural Networks Port-Hamiltonian Systems Hamiltonian Dynamics Quantum Neural Networks

Jul 8, 2026cs.LG

Unlocking Temporal Generalization in Hamiltonian Video Dynamics Models

World models are typically trained to predict discrete-time physical dynamics with a fixed step size baked into the model weights, preventing prediction at variable temporal resolutions. This matters for hierarchical planning, sim-to-real transfer, and scientific or game-engine applications that must query the same dynamics at multiple timescales. Hamiltonian Generative Networks (HGN) offer a principled path forward, grounding predictions in a continuous-time energy function that is, in principle, independent of the observation frame rate. In practice, however, their temporal generalization breaks down in non-conservative settings. We show that in externally forced, dissipative environments, HGN rollouts at step sizes beyond the training regime fail due to distinct failure modes, including latent magnitude growth driven by an unconstrained action-force map, and global truncation error accumulation from an under-resolved integrator. We identify a targeted fix for each mechanism and demonstrate stable dynamics prediction at temporal resolutions well outside the training distribution. In a detailed analysis, we recommend several strategies for enabling temporal generalization in continuous-time video generation.

Eli Laird, Corey Clark

Department of Computer Science, Southern Methodist University, Dallas, TX, USA

Physical Consistency in Video Generation Latent Dynamics Modeling Hamiltonian Neural Networks Generative Modeling Temporal Consistency in Video Generation Hamiltonian Dynamics

Jul 3, 2026cs.LG

CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems

Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical symplectic form by a scalar factor that reflects the net dissipation. We propose Conformal Symplectic Networks with damping identification (CSympNet-ID), a discrete-time map-learning framework that learns the one-step flow map directly from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, without penalty terms or projection. The architecture composes an exact symplectic neural core with explicit diagonal scaling layers whose factors are parameterized exponentially by a scalar damping-rate parameter, thereby guaranteeing positivity and interpretability of the learned dissipation factor. We establish a scaling-conjugacy factorization for conformal symplectic maps and derive a pointwise-in-step density result for CSympNet-ID. We evaluate an irregular-step damped oscillator, a damped spring-mass chain, a damped nonlinear cubic oscillator, and additional high-dimensional extensions. CSympNet-ID gives the most favorable overall results among the compared models in the reported experiments, particularly in data-scarce regimes, target contraction-law recovery, and high-dimensional tests where unstructured baselines degrade rapidly.

Jiale Gong, Pengzhan Jin, Dongyang Kuang +2

School of Mathematics · School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China · National Engineering Laboratory for Big Data Analysis and Applications, Peking University, Beijing, China +3

Dynamical Systems Nonlinear System Identification System Identification Hamiltonian Dynamics

Jun 27, 2026math.CT

Compositional Dynamics in Learning and Mechanics

We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are smooth adaptive arrangements, which consist of a reactive parameter space, a lens given by smooth output and input maps, and a real-valued potential. The main technical result of the paper is what we call lens internalization, a lax symmetric monoidal functor Lens(C)

\to

C associated to any symmetric monoidal closed category C. Using it, we provide two functors

Φ_\text{phase}

,

Φ_\text{conf}

: Arr

\to

PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category.

Φ_\text{phase}

stores both position and momentum, whereas

Φ_\text{conf}

stores only position. When applied to a parameterized function,

Φ_\text{conf}

recovers the gradient descent training algorithm, with backpropagation as the lens' backward pass. When applied to harmonic particles wired together -- in series, or according to any finite directed graph -- one diagram yields two different regimes, both of which are governed by the graph Laplacian:

Φ_\text{phase}

gives the discrete wave equation, which is conservative and second-order, and

Φ_\text{conf}

gives the discrete heat equation, which is dissipative and first-order. They are two semantics of one adaptive arrangement, e.g. with the same potential in each case. And because Arr is an operad, such diagrams nest -- larger systems wired from smaller ones -- and each semantics assembles a system's dynamics functorially from its parts. These dynamics are moreover executable: a parameterized neural network and a graph of particles both compile, by the same construction, to explicit state machines one can run.

David I. Spivak

Dynamical Systems Hamiltonian Dynamics

Jun 25, 2026cs.LG

Symplectic Neural Networks for learning Generalized Hamiltonians

Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta +2

Computer Science Department, Indraprasth Institute of Information Technology, New Delhi, India 110020 · Athena Research and Innovation Center, Marousi, Greece 15125 · HSBC Holdings, Singapore

Dynamical Systems Hamiltonian Neural Networks System Identification Neural ODEs Hamiltonian Dynamics

Jun 23, 2026cs.LG

When Do Conservation Laws Survive Learned Representations? Certified Horizons for Latent World Models

We ask a representation-learning question about physical world models: when does a conservation law remain certifiable after a model learns a latent representation? A certified horizon bounds -- in advance, from measurable model defects -- how many steps a rollout provably stays on a physical invariant's level set. The key design choice is what is certified: not a learned latent Hamiltonian or a learned scalar witness (a model can conserve either while drifting in true energy), but the decoded physical invariant obtained by decoding the latent state and evaluating the known invariant. Around this object we derive shell-horizon certificates whose budget decomposes into representation, readout, and latent-dynamics defects, with a monotone alignment bridge through which a soft learned witness yields a certified horizon for the decoded invariant, and test them across state, learned-lift, and pixel observations on conservative systems. Conservation certificates can survive learned representation, but not all geometric priors survive equally. Hard canonical symplectic structure yields the longest horizons in known phase coordinates yet does not cross a learned chart, whereas a controlled-Lipschitz-aligned soft invariant survives in the nonlinear learned-representation settings we test -- two lift systems, with the gain growing with nonlinearity, and pixels. Pixel certification is recovered on a readout-stable sub-tube, and the Kepler problem exposes a geometric boundary. The central object is therefore not a latent Hamiltonian, but a decoded physical invariant whose robustness to representation learning can be measured, certified, and falsified.

Hongbo Wang

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA

Dynamical Systems Latent World Models Certified Robustness Hamiltonian Dynamics

Jun 15, 2026math.OC

Accelerated Convex Optimization via Hamiltonian Dynamics with Deterministic Integration Time

We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence. By exploiting contraction of averaged Hamiltonian flow trajectories rather than requiring contraction at trajectory endpoints, we show that Hamiltonian dynamics-based optimization methods admit deterministic and accelerated convergence guarantees, extending prior work that is limited to quadratic objectives or holds only in expectation. We analyze an idealized continuous-time algorithm and derive practical discrete-time implementations with optimal first-order complexity, thereby establishing Hamiltonian dynamics as a useful algorithmic primitive for deterministic accelerated convex optimization.

Xiuyuan Wang, Vishwak Srinivasan, Qiang Fu +3

Department of Computer Science, Yale University · Department of EECS, Massachusetts Institute of Technology

Hamiltonian Dynamics

Jun 3, 2026cs.LG

Learning symplectic model reduction based on a approximation theorem of symplectic embeddings

High-dimensional Hamiltonian systems play a central role in many scientific and engineering disciplines, with dynamics evolving on symplectic manifolds. Although deep learning provides powerful tools for constructing low-dimensional surrogates from data, the intrinsic symplectic structure is easily destroyed during model reduction. As a result, a standard autoencoder may produce latent coordinates that do not support a Hamiltonian flow, leading to unstable long-time prediction. In this paper, we first establish a universal approximation theorem for symplectic embeddings. Based on this theory, we propose symplecticity-preserving autoencoders (SpAE), in which the decoder is parameterized as a symplectic embedding and the encoder is constructed as the corresponding symplectic projection. This architecture is expressive enough to approximate nonlinear symplectic embeddings and the associated symplectic projections, preserves the symplectic structure exactly by construction, and can be trained by standard unconstrained optimization, thereby improving both reconstruction and prediction accuracy. Extensive experiments on high-dimensional lattice and particle systems demonstrate the effectiveness of the proposed method.

Liyi Feng, Yifa Tang, Yulin Xie +2

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China · State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China · Department of Mathematics, National University of Singapore, 119076, Singapore

Dynamical Systems Autoencoders Reduced-Order Modeling Hamiltonian Dynamics

May 22, 2026cs.LG

Learning partially observed systems with neural Hamiltonian ordinary differential equations

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.

Sunniva Meltzer, Sølve Eidnes, Alexander Johannes Stasik

Department of Mathematics and Cybernetics, SINTEF Digital, Norway · Department of Physics, University of Oslo, Norway · Department of Data Science, Norwegian University of Life Science, Norway

Dynamical Systems Latent Dynamics Modeling Nonlinear System Identification Hamiltonian Neural Networks Physics-Informed ML Neural ODEs+1

May 19, 2026cs.LG

Beyond Isotropy in JEPAs: Hamiltonian Geometry and Symplectic Prediction

JEPAs often regularize one-view embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation. We show that this is not merely a benign default. For a known structured downstream geometry

H\succ0

, the minimax and maximum-entropy covariance under a Hamiltonian energy budget is

(c/d)H^{-1}

, and Euclidean isotropy incurs a closed-form price of isotropy. More importantly, when the downstream geometry is unknown, no geometry-independent fixed marginal target is canonical: every fixed covariance shape can be maximally misaligned for some structured geometry. We further show that even oracle one-view marginals do not identify the JEPA view-to-view predictive coupling. These results suggest that the structural bias in JEPAs should enter the cross-view coupling rather than a fixed encoder marginal. We instantiate this principle with \textbf{HamJEPA}, which encodes each view as a phase-space state

(q,p)

and predicts view-to-view transitions with a learned Hamiltonian leapfrog map, while non-isotropic scale and spectral floors prevent collapse. In a deliberately headless token protocol, HamJEPA improves over SIGReg on CIFAR-100 by

+4.89

kNN@20 and

+3.52

linear-probe points at 30 epochs, and by

+6.45

kNN@20 and

+10.64

linear-probe points at 80 epochs, while a matched MLP predictor ablation shows that the symplectic coupling is the ingredient driving the neighborhood-geometry gain. On ImageNet-100, HamJEPA-

q

improves by

+4.82

kNN@20 and

+7.52

linear-probe points at 45 epochs.

Robert Jenkinson Alvarez

Joint-Embedding Predictive Architecture Geometric Representation Learning Self-Supervised Visual Representation Learning Hamiltonian Dynamics

May 18, 2026cs.LG

PH-Dreamer: A Physics-Driven World Model via Port-Hamiltonian Generative Dynamics

World models built on recurrent state space architectures enable efficient latent imagination, yet remain physically unstructured, producing dynamics that violate conservation and dissipative principles. We introduce a unified Port-Hamiltonian framework that remedies this through three synergistic mechanisms. First, we embed implicit physical priors into recurrent transitions by modeling projected latent evolution as action controlled energy routing governed by flow and dissipation, biasing the projected PH phase space toward a more compact and physically structured representation. Second, we develop a kinematics aware energy world model that estimates the Hamiltonian and power balance from proprioceptive observations, providing an explicit physical signal for thermodynamic reasoning. Third, leveraging these energy gradients, we establish an energy guided Actor-Critic that uses Lagrangian multipliers to regularize policy optimization toward lower energy and smoother control. Across visual control benchmarks, this paradigm not only attains superior asymptotic returns but also elevates internal simulator fidelity by establishing a tighter, lower variance alignment between imagined and real rewards, all while reducing latent phase space volume by 4.18-8.41%, energy consumption by up to 7.80%, and mean squared jerk by up to 9.38%.

Xueyu Luan, Chenwei Shi

School of Electronic and Information Engineering Tongji University Shanghai 200092, China · Shanghai Research Institute for Intelligent Autonomous Systems Tongji University Shanghai 200092, China

Dynamical Systems Reinforcement Learning Physics-Informed ML Latent World Models Port-Hamiltonian Systems Hamiltonian Dynamics+1

May 16, 2026cs.LG

Prediction Is Not Physics: Learning and Evaluating Conserved Quantities in Neural Simulators

A diffusion model trained on Hamiltonian trajectories can achieve rollout MSE near

10^{-3}

, but the standard deviation of its energy over time is between 7500 and 36000 times larger than the ground-truth energy standard deviation, indicating a failure to preserve conservation laws. This gap motivates our central question of whether neural networks can learn or select globally conserved quantities from physical trajectories. We investigate this across three Hamiltonian systems: projectile motion, pendulum, and spring-mass. We use a structured

T(v)+V(q)

energy model, a black-box Conservation Discovery Network (CDN), a polynomial CDN, and a conditional diffusion baseline. The structured network reaches

R^2 \geq 0.9999

against analytical energy on clean data, while the black-box CDN reaches

R^2 \geq 0.996

when trained with temporal consistency plus a small alignment loss to analytical energy at

t=0

(

λ_{\mathrm{align}}=0.2

). With

λ_{\mathrm{align}}=0

, CDN Pearson

R^2

collapses on pendulum and spring-mass (

< 10^{-3}

), showing that temporal consistency alone is not enough to reliably identify the true energy. Under

1\%

additive Gaussian noise, the CDN outperforms the structured model on the projectile and spring-mass systems, suggesting that the CDN may be more robust to noisy inputs in this setting. However, the polynomial CDN is sensitive to training configuration: it achieves

R^2=0.78

under a short training schedule on the pendulum system, but reaches

R^2=0.9998

with more training time and data, regardless of whether noise is added.

Andrew Bukowski, Aditya Kothari, Simba Shi +1

Yale University

Neural Surrogate Modeling Dynamical Systems Scientific Discovery Hamiltonian Dynamics Diffusion Models

May 15, 2026cs.LG

Identify Then Project: Contrastive Learning of Latent Dynamics from Partial Observations with Port-Hamiltonian Structure

Identifying latent state representations and dynamics is essential when direct modeling in observation space is infeasible, particularly under partial and high-dimensional observations. In such settings, representation learning and physics-aware modeling are inherently coupled. We study this problem for latent port-Hamiltonian systems, a structured class encompassing both conservative and dissipative dynamics. We propose a two-stage identify-then-project framework. First, a contrastive teacher learns continuous-time latent dynamics from partial observations. Then, a student projects the identified teacher representation and dynamics onto a port-Hamiltonian submanifold via a learned affine chart, yielding a physically consistent realization. As a conceptual counterfactual, we also consider a single-stage variant that jointly learns latent identification and port-Hamiltonian structure, but find it to be less reliable, motivating the proposed two-stage teacher-student framework. We show theoretically that affine projection is the natural bridge between the affine gauge of contrastive latent identification and the port-Hamiltonian systems. Empirically, we demonstrate that the proposed two-stage approach preserves the teacher's dynamics while enforcing physical structure, and performs more reliably than the single-stage alternative, particularly in dissipative regimes and high-dimensional visual settings.

Peilun Li, Kaiyuan Tan, Daniel Moyer +1

Department of Computer Science Vanderbilt University Nashville, TN 37235

Contrastive Learning Latent Dynamics Modeling System Identification Physics-Informed ML State Representation Learning Port-Hamiltonian Systems+1

May 15, 2026math.DS

Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems

The modeling and simulation of infinite-dimensional Hamiltonian systems are central problems in mathematical physics and engineering, however they pose significant computational and structural challenges for standard data-driven architectures. In this work, we introduce the Symplectic Neural Operator, a neural operator architecture designed to preserve the symplectic structure intrinsic to Hamiltonian PDEs. We provide a theoretical characterization of their symplecticity and establish a rigorous long-term stability result based on the combination of symplectic structure preservation and learning accuracy. Numerical experiments on canonical Hamiltonian PDEs corroborate this theoretical result and show that SNOs exhibit improved energy behavior compared with non-structure-preserving neural operators.

Yeang Makara, Yusuke Tanaka, Takashi Matsubara +1

Graduate School of Science Kobe University 1-1 Rokkodai-cho, Nada-ku, Kobe, Japan · NTT Communication Science Laboratories 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto, Japan · Faculty of Information Science and Technology Hokkaido University Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, Japan +1

PDE Operator Learning Neural Operators Hamiltonian Dynamics

May 13, 2026quant-ph

Physics Guided Generative Optimization for Trotter Suzuki Decomposition

Trotter Suzuki product formulas are the standard route to Hamiltonian evolution on noisy intermediate-scale quantum (\NISQ{}) hardware, but their accuracy depends on three coupled choices: term grouping, product-formula order, and time-step allocation. Grouping and order are discrete, which makes direct gradient optimization infeasible and forces existing compilers to rely on static heuristics. We describe P-GONE, a method that combines a conditional diffusion model (D3PM + DDPM), a graph neural network (\GNN{}) encoder, and closed-loop REINFORCE fine-tuning to jointly learn grouping, order, and time-step optimization over a mixed discrete-continuous space. Under fidelity-matched conditions (

F \geq 0.95

), the method achieves circuit depth 86 versus 1673 for Qiskit fourth-order (ungrouped, Suzuki-4), about

19.4\times

compression, and 141 for Paulihedral (first-order Trotter), about

1.6\times

compression. At

T=0.90

the method also beats the Qiskit group-commuting teacher (65 vs 103,

1.6\times

compression), though at

T=0.95

the teacher still leads -- a stratified pattern that points toward fidelity-aware fine-tuning. Under a standard depolarizing noise model, the method achieves noisy fidelity roughly

2\times

the Qiskit fourth-order baseline (0.743 vs 0.380). Ablation shows a clear hierarchy: order learning

>

time allocation

>

grouping. Best-of-N sampling (

N=32

is a practical sweet spot) and CFG guidance give flexible fidelity-depth trade-offs at inference. The method works well on structured Hamiltonians (TFIM, Heisenberg), but random Pauli Hamiltonians fail entirely at

T \geq 0.95

-- a boundary that defines where the method applies.

WenBin Yan

University of Colorado Boulder[quant-ph]

Quantum Circuit Compilation Hamiltonian Dynamics

May 7, 2026cs.LG

When Descent Is Too Stable: Event-Triggered Hamiltonian Learning to Optimize

Fixed-budget nonconvex optimization can fail not because local descent is unstable, but because it is too stable: after reaching a nearby stationary point, an optimizer may spend the remaining evaluations refining an uninformative local minimum. We formulate this failure mode as a control problem over optimizer dynamics, where the learner must decide when to descend, when to exploit a promising basin, and when stagnation should trigger movement elsewhere. We introduce SHAPE, a structured adaptive port-Hamiltonian task-family optimizer for event-triggered minima hunting under local information. Starting from gradient-descent dynamics, SHAPE lifts optimization to an augmented phase space

(q, p)

, where the primal state

q

represents the candidate solution, the cotangent variable

p

carries directional sensitivity, and a controller

u

provides processed information from current gradient oracle. Within each stage, a learned Hamiltonian vector field induces structured local descent; across stages, a fixed event clock in the implementation updates ports and memory when local equilibria are detected, with stage-dependent horizons treated in the analysis as a direct generalization. This design preserves a passivity-compatible structure while allowing the same trained policy to use clean, stochastic, or estimated gradient inputs. Experiments on fixed-budget nonconvex optimization tasks show that SHAPE improves best-so-far performance compared with fixed-policy optimizers. These results suggest that adaptive Hamiltonian energy shaping provides a principled mechanism for balancing descent, exploration, and budget allocation in difficult optimization landscapes.

Yi Wang, Chandrajit Bajaj

1Oden Institute · Department of Computer Science

Dynamical Systems Learning to Optimize Nonconvex Optimization Event-Triggered Control Port-Hamiltonian Systems Hamiltonian Dynamics

May 7, 2026cs.AI

HaM-World: Soft-Hamiltonian World Models with Selective Memory for Planning

World models enable model-based planning through learned latent dynamics, but imagined rollouts become unstable as the planning horizon grows or the dynamics distribution shifts. We argue that this instability reflects two missing structures in planner-facing latents: history-conditioned memory for approximate Markov completeness, and geometric organization that separates configuration, momentum, and task semantics. We propose HaM-World (HMW), a structured world model that decomposes the latent state into a canonical (q, p) subspace and a context subspace c, while using Mamba selective state-space memory as the history-conditioned input to the same latent dynamics. Within this interface, (q, p) evolves through an energy-derived Hamiltonian vector field plus learnable residual/control dynamics, while c captures semantic, dissipative, and non-conservative factors. This gives the planner a single latent state shared by dynamics prediction, reward/value estimation, imagined rollouts, and CEM action search. On four DeepMind Control Suite tasks, HaM-World reaches the highest Avg. AUC (117.9, +9.5%), reduces long-horizon rollout error to 45% of a strong baseline model, and wins 11/12 k in {3,5,7} MSE cells. Under 12 OOD perturbations spanning dynamics shifts, action delay, and observation masking, HaM-World achieves the highest return in every condition, with average OOD-return gains of 10.2% on Finger Spin and 13.6% on Reacher Easy. Mechanism diagnostics further show bounded action-free Hamiltonian-energy drift, structured energy variation under policy rollouts, and coherent control-induced energy transfer, supporting the intended Soft-Hamiltonian dynamics design.

Haoyun Tang, Haodong Cui, Keyao Xu +2

1Xi’an Jiaotong University · 2Huazhong University of Science and Technology · 3Nankai University +1

World Model Learning Latent Dynamics Modeling World Model-Based Planning Hamiltonian Neural Networks Model-Based Planning Hamiltonian Dynamics

May 6, 2026cs.CV

Detecting Deepfakes via Hamiltonian Dynamics

Driven by the rapid development of generative AI models, deepfake detectors are compelled to undergo periodic recalibration to capture newly developed synthetic artifacts. To break this cycle, we propose a new perspective on deepfake detection: moving from static pattern recognition to dynamical stability analysis. Specifically, our approach is motivated by physics-inspired priors: we hypothesize that natural images, as products of dissipative physical processes, tend to settle near stable, low-energy equilibria. In contrast, generative models optimize for statistical similarity to real images but do not explicitly enforce structural constraints such as geometric smoothness, leaving deepfakes more likely to occupy unstable, high-energy states. To operationalize this, we introduce Hamiltonian Action Anomaly Detection (HAAD), comprising three contributions: \textbf{i)} We model the image latent manifold as a potential energy surface. Under this hypothesis, real images are expected to produce basin-like low-energy responses, whereas fake images are more likely to induce high-potential, high-gradient responses. \textbf{ii)} We employ Hamiltonian-inspired dynamics as a stability probe. By releasing latent states from rest, samples near stable regions remain bounded, while high-gradient samples produce larger trajectory responses. \textbf{iii)} We quantify these dynamic behaviors through two trajectory statistics, \ie, Hamiltonian action and energy dissipation. Extensive experiments show that HAAD outperforms evaluated state-of-the-art baselines on challenging cross-dataset transfer benchmarks, supporting a physics-inspired stability prior for digital forensics.

Harry Cheng, Ming-Hui Liu, Tianyi Wang +3

School of Computing, National University of Singapore, Singapore · School of Software, Shandong University, China · School of Information Science and Technology, Harbin Institute of Technology (Shenzhen), China +1

Digital Image Forensics Deepfake Detection Hamiltonian Dynamics

May 1, 2026cs.AI

Physically Native World Models: A Hamiltonian Perspective on Generative World Modeling

World models have recently re-emerged as a central paradigm for embodied intelligence, robotics, autonomous driving, and model-based reinforcement learning. However, current world model research is often dominated by three partially separated routes: 2D video-generative models that emphasize visual future synthesis, 3D scene-centric models that emphasize spatial reconstruction, and JEPA-like latent models that emphasize abstract predictive representations. While each route has made important progress, they still struggle to provide physically reliable, action-controllable, and long-horizon stable predictions for embodied decision making. In this paper, we argue that the bottleneck of world models is no longer only whether they can generate realistic futures, but whether those futures are physically meaningful and useful for action. We propose \emph{Hamiltonian World Models} as a physically grounded perspective on world modeling. The key idea is to encode observations into a structured latent phase space, evolve the latent state through Hamiltonian-inspired dynamics with control, dissipation, and residual terms, decode the predicted trajectory into future observations, and use the resulting rollouts for planning. We discuss how Hamiltonian structure may improve interpretability, data efficiency, and long-horizon stability, while also noting practical challenges in real-world robotic scenes involving friction, contact, non-conservative forces, and deformable objects.

Sen Cui, Jingheng Ma

Tsinghua University

Latent Dynamics Modeling World Models World Model-Based Planning Model-Based Planning Hamiltonian Dynamics Embodied World Models

Apr 26, 2026cs.LG

Hamiltonian Graph Inference Networks: Joint structure discovery and dynamics prediction for lattice Hamiltonian systems from trajectory data

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in

α

-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via

k

-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

Ru Geng, Panayotis Kevrekidis, Yixian Gao +2

School of Mathematics and Statistics, Changchun University, Changchun, 130022, China · Department of Mathematics and Statistics, University of Massachusetts, Amherst, 01003, MA, USA · Department of Physics, University of Massachusetts, Amherst, 01003, MA, USA +2

Dynamical Systems Graph Structure Learning Graph Neural Networks Graph Representation Learning System Identification Physics-Informed ML+2

Apr 19, 2026cs.LG

Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data

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.

Vedanta Thapar, Abhinav Gupta

The Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom · Center for Theoretical Physics, St. Stephen’s College, University of Delhi, Delhi 110007, India

Nonlinear System Identification Recurrent Neural Networks Hamiltonian Neural Networks System Identification Physics-Informed ML Hamiltonian Dynamics+1

Mar 17, 2026cs.RO

Early-Terminable Energy-Safe Iterative Coupling for Parallel Simulation of Partitioned Port-Hamiltonian Systems

Parallel simulation of robotic systems requires partitioning the dynamics into coupled subsystems. Finite-iteration coupling across the partition boundary can inject spurious energy, even when each subsystem is passive. We propose an early-terminable, energy-safe coupling interface for port-Hamiltonian subsystems based on Douglas--Rachford splitting in wave (scattering) coordinates. The wave-domain formulation reduces passivity to norm inequalities and coupling to orthogonality. Within this setting, the deep correspondence between monotone operator theory and discrete passivity can be exploited to construct a Douglas--Rachford inner iteration whose Fejér monotonicity provides algorithmic dissipation. Under passivity of the subsystem integrators and an impedance-tuning condition, the proposed method guarantees discrete passivity of the augmented storage for any finite inner-iteration budget and converges to the monolithic discretization as the budget increases. Experiments on a linear--Duffing coupled-oscillator benchmark support the finite-iteration energy inequality at numerical roundoff (1e-14 in double precision), with state-error metrics decreasing over the tested inner-iteration budgets.

Qi Wei, Jianfeng Tao, Hongyu Nie +1

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Dynamical Systems Port-Hamiltonian Systems Physics-Based Simulation Hamiltonian Dynamics

Jan 29, 2026cs.LG

Learning Hamiltonian Flow Maps: Mean Flow Consistency for Large-Timestep Molecular Dynamics

Simulating the long-time evolution of Hamiltonian systems is limited by the small timesteps required for stable numerical integration. To overcome this constraint, we introduce a framework to learn Hamiltonian Flow Maps by predicting the mean phase-space evolution over a chosen time span, enabling stable large-timestep updates far beyond the stability limits of classical integrators. To this end, we impose a Mean Flow consistency condition for time-averaged Hamiltonian dynamics. Unlike prior approaches, this allows training on independent phase-space samples without access to future states, avoiding expensive trajectory generation. Validated across diverse Hamiltonian systems, our method in particular improves upon molecular dynamics simulations using machine-learned force fields (MLFF). Our models maintain comparable training and inference cost, but support significantly larger integration timesteps while trained directly on widely-available trajectory-free MLFF datasets.

Winfried Ripken, Michael Plainer, Gregor Lied +5

Technical University Berlin · BIFOLD · Free University of Berlin +6

Dynamical Systems Flow Map Learning Mean Flow Molecular Dynamics Simulation Physics-Based Simulation Hamiltonian Dynamics