Hamiltonian Physics

Finite-dimensional Koopman models enable efficient linear prediction and control of nonlinear robotic systems. However, models learned purely from trajectory data may violate the energetic structure of the underlying mechanics, producing predictions that exhibit artificial energy growth and diverge under recursive propagation. This work presents a structure-preserving Koopman framework for Euler-Lagrange systems built on generalized-momentum coordinates. The momentum transformation exposes the mechanical actuation as a known, state-independent port, which is preserved explicitly in the lifted dynamics. A structure-constrained neural architecture is developed to jointly learn the lifting functions and a port-Hamiltonian Koopman generator, rendering the learned dynamics passive by construction rather than through penalty terms or post-hoc projection. A Cayley-midpoint discretization further preserves the corresponding storage-dissipation balance exactly in discrete time. These properties are established analytically by deriving the discrete storage balance and associated stability guarantees of the learned predictor. Simulation and experimental studies demonstrate improved prediction accuracy, data efficiency, and closed-loop tracking over Koopman baselines, with increasing gains for higher-dimensional systems.

This paper presents an energy-based controller for a multiagent robotic system designed to achieve and maintain a specific formation while moving on a plane and avoiding collisions between agents. The controller emulates a network of elementary spring-damper modules connecting pairs of agents. This network, with its de-energized states representing the desired formation, determines the system's dynamics, which is fully encapsulated by a bond graph model. The modeling is further enhanced through the introduction of a formation matrix, using a graph-theoretic approach, that describes both the distances and relative velocities among the agents of the arrangement. This matrix mathematically represents the interconnection and energy-exchange structure of the bond graph, allowing us to put it in correspondence with the control-by-interconnection CbI-scheme of the IDA-PBC theory, facilitating the solution of the formation control problem within the port-Hamiltonian system framework. Furthermore, the paper presents leader-following and position-based formation control systems based on the CbI scheme, including a stability analysis of the corresponding closed-loop systems. The theoretical findings are validated through numerical simulations across various scenarios.

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.

Learning solution operators for differential equations is a central problem in scientific machine learning. However, many neural operator methods optimize prediction accuracy without explicitly enforcing the geometric structure of the dynamics. Structure-preserving models such as SympNets and Symplectic Neural Flows address this issue for conservative Hamiltonian systems by preserving the symplectic form. In dissipative Hamiltonian systems with conformal symplectic structure, however, the symplectic form evolves according to a conformal factor determined by the dissipation. We propose CoSynFlow, a conformal symplectic neural flow for learning continuous-time solution maps of dissipative Hamiltonian dynamics. CoSynFlow composes symplectic shear maps with explicit conformal scaling, preserving the conformal symplectic structure by construction. By conditioning it on a finite-dimensional Hamiltonian descriptor and the dissipation parameter, a single trained model predicts solution maps for unseen systems without retraining. CoSynFlow keeps the structure error at machine precision, attains the lowest long-horizon error, and admits physics-informed training.

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.

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.

We model human motor cortex during a wrist-extension BCI task as a port-Hamiltonian system (pHS): a conservative interconnection (gyroscopic coupling between neural phasors) plus a dissipative port (power-law energy decay driven by a GNN surrogate). A metriplectic integrator evolves the phasor state; a Fluctuation--Dissipation-consistent noise channel produces stochastic trajectories at body temperature. Training on \FitTrainN\ real EEG cycles (PhysioNet EEGMMIDB, 3 held-out subjects) reaches a test MSE of \FitTestMSE\ and passes three scale-free criticality rungs: near-critical branching ratio ($σ\approx1$), $1/f$ power-law spectrum, and long-range DFA correlations. The model generates closed-loop neuromodulation signals that restore phase-locking in silico when applied to de-synchronised inputs, suggesting a path toward structure-preserving BCI decoders.

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.

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.

Modeling soft robot dynamics is challenging due to their continuum structure and typically nonlinear dynamics. Creating models based on first-order principles is typically time-demanding, and their expressiveness is limited, whereas data-driven models lack interpretability and physical consistency. This work aims to overcome these challenges by introducing a port-Hamiltonian Gaussian Process Regression framework for learning and simulating the dynamics of planar, rod-like soft robots. In detail, the proposed model integrates Cosserat rod theory and Hamiltonian physics with data-driven inference to preserve the system's energy structure while accurately learning the rod dynamics. Numerical simulations show that we can achieve accurate and energy-consistent representations of a rod-like soft robot, showing the potential for a robust and interpretable pathway for modeling complex continuum mechanics.

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.

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.

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%.

We present a novel parameterized 4-qubit Eisert-Wilkens-Lewenstein (EWL) quantum game circuit for recommender systems in quadruple helix innovation ecosystems (academia, industry, government, and civil society). The local strategy operators $U_{i} = R_y(θ_{i})$ for each helix actor are directly tuned by normalized dominance weights extracted from real participant funding data (\texit{ecContribution}) in the European Commission CORDIS Horizon Europe database (project COVend, ID 101045956). The circuit employs a multi-qubit EWL entangler followed by parameterized local rotations, inverse entangler, and full measurement, achieving only 22 gates and circuit depth 11 while scaling as $O(n)$ for $n$-round helix communications. Measurement probabilities after the quantum game serve as recommender scores for disruptive versus sustaining innovation trends. These scores are subsequently mapped into the diagonal Dirac potential of a Dirac-Solow-Swan Hamiltonian, enabling time-evolution simulation of capital accumulation and bifurcation dynamics under disruptive innovation. Numerical experiments on real CORDIS quadruple-helix collaboration networks demonstrate the circuit's NISQ compatibility and its ability to forecast disruptive capital trajectories with high fidelity. The proposed framework bridges quantum game theory, parameterized quantum circuits, and relativistic economic growth models, offering a computationally efficient tool for innovation policy and strategic decision-making in complex socio-economic ecosystems. Complexity analysis and reproducibility are provided through open Qiskit implementations.

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.

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.

Hybrid machine learning combines physical knowledge with data-driven models to enhance interpretability and performance. In this context, Port-Hamiltonian Systems (PHS), which generalize Hamiltonian mechanics to describe open, non-autonomous dynamical systems, have been successfully integrated with neural networks under the name Port-Hamiltonian Neural Networks (PHNNs). While the ability of PHNNs to identify Hamiltonian ordinary differential equation (ODE) systems has already been demonstrated, their application to learning Hamiltonian partial differential equation (PDE) systems remains largely unexplored. This limitation restricts their use in musical acoustics, where instruments are typically modeled as distributed parameter systems governed by PDEs. In this work, we demonstrate how to learn the nonlinear string dynamics from data in a physically-consistent framework through a PHNN extension to PDEs. By constructing structured neural network architectures based on PHS, we can recover both the Hamiltonian governing the string and the dissipation affecting it. This approach outperforms baseline, non-physics-informed methods in terms of both accuracy and interpretability. Numerical experiments using synthetic data demonstrate the ability of the proposed PHNN model to identify and emulate the nonlinear dynamics of the system.

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.

We present Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net: a structure-preserving framework for mesh-based continuum physics that cleanly separates the physics' topological structure from its metric structure. Imposing minimal physical principles (locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequality), we prove a reduction theorem for mesh-based physics. Under these conditions, the physical dynamics admit a local factorization into a port-Hamiltonian form: the conservative interconnection is fixed uniquely by mesh topology, whereas metric effects enter only through constitutive relations and dissipation. This reduction clarifies what must be fixed and what should be learned, directly informing MeshFT-Net's design. Across evaluations on analytic and realistic datasets, physics-consistency tests, and out-of-distribution validation, MeshFT-Net achieves near-zero energy drift and strong physical fidelity (correct dispersion and momentum conservation) along with robust extrapolation and high data efficiency. By eliminating non-physical degrees of freedom and learning only metric-dependent structure, MeshFT provides a principled inductive bias for stable, faithful, and data-efficient learning-based physical simulation.

We develop a physics-informed learning framework for energy-shaping control of port-Hamiltonian (pH) systems from trajectory data. The proposed approach co-learns a pH system model and an optimal energy-balancing passivity-based controller (EB-PBC) through alternating optimization with policy-aware data collection. At each iteration, the system model is refined using trajectory data collected under the current control policy, and the controller is re-optimized on the updated model. Both components are parameterized by neural networks that embed the pH dynamics and EB-PBC structure, ensuring interpretability in terms of energy interactions. The learned controller renders the closed-loop system inherently passive and provably stable, and exploits passive plant dynamics without canceling the natural potential. A dissipation regularization enforces strict energy decay during training, thereby enhancing robustness to sim-to-real gaps. The proposed framework is validated on state-regulation and swing-up tasks for planar and torsional pendulum systems.

We introduce HAML (Hamiltonian Adaptation via Meta-Learning), a framework for fast online adaptation of effective Hamiltonian models of superconducting quantum processors. HAML proceeds in two phases. A supervised training phase uses an ensemble of simulated devices to learn an offline map from control inputs and device parameters to effective Hamiltonian coefficients. An online adaptation phase then uses a small number of hardware-accessible measurements to identify the unknown parameters of a new device. By training directly against effective two-qubit coefficients extracted from full multi-mode simulations, HAML implicitly learns the reduction from full multi-mode Hamiltonians to effective qubit descriptions without invoking perturbation theory. We further show that a variance-maximizing greedy selection of measurement configurations boosts online adaptation efficiency. We demonstrate HAML on a transmon-coupler-transmon system, recovering effective two-qubit coefficients across a wide range of operating regimes, including parameter regions where Schrieffer-Wolff perturbation theory (SWPT) breaks down. This establishes a scalable, sample-efficient approach to Hamiltonian reduction and characterization for near-term quantum processors, with direct implications for calibration, control, and error mitigation.

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.

This work focuses on learning non-canonical Hamiltonian dynamics from data, where long-term predictions require the preservation of structure both in the learned model and in numerical schemes. Previous research focused on either facet, respectively with a potential-based architecture and with degenerate variational integrators, but new issues arise when combining both. In experiments, the learnt model is sometimes numerically unstable due to the gauge dependency of the scheme, rendering long-time simulations impossible. In this paper, we identify this problem and propose two different training strategies to address it, either by directly learning the vector field or by learning a time-discrete dynamics through the scheme. Several numerical test cases assess the ability of the methods to learn complex physical dynamics, like the guiding center from gyrokinetic plasma physics.

Hamiltonian dynamics describe a wide range of physical systems. As such, data-driven simulations of Hamiltonian systems are important for many scientific and engineering problems. In this work, we propose kernel-based methods for identifying and forecasting Hamiltonian systems directly from trajectory data. We present two approaches: a 2-step method that reconstructs trajectories before learning the Hamiltonian, and a 1-step method that jointly infers both. Across several benchmark systems, including mass-spring dynamics, a nonlinear pendulum, and the Henon-Heiles system, we demonstrate that our framework achieves accurate, data-efficient predictions and outperforms 2-step kernel-based baselines, particularly in scarce-data regimes, while preserving the Hamiltonian structure. Moreover, we prove a priori error estimates, ensuring reliability of the learned models. We also provide a more general, problem-agnostic numerical framework that goes beyond Hamiltonian systems and can be used for data-driven learning of arbitrary dynamical systems.