Geometric Contact Flows: Contactomorphisms for Dynamics and Control
Authors: Andrea Testa
Søren Hauberg
Tamim Asfour
Leonel Rozo
Venue: ICML 2025 poster (2025)
Keywords: Contact geometry
Riemannian geometry
contact Hamiltonian dynamics
Dynamical systems
Diffeomorphisms
Primary area: general_machine_learning->sequential_network_and_time_series_modeling
TL;DR: Leveraging Riemannian and Contact geometry as inductive biases to learn and control complex dynamical systems.

Abstract: Accurately modeling and predicting complex dynamical systems, particularly those involving force exchange and dissipation, is crucial for applications ranging from fluid dynamics to robotics, but presents significant challenges due to the intricate interplay of geometric constraints and energy transfer. This paper introduces Geometric Contact Flows (GFC), a novel framework leveraging Riemannian and Contact geometry as inductive biases to learn such systems. GCF constructs a latent contact Hamiltonian model encoding desirable properties like stability or energy conservation. An ensemble of contactomorphisms then adapts this model to the target dynamics while preserving these properties. This ensemble allows for uncertainty-aware geodesics that attract the system’s behavior toward the data support, enabling robust generalization and adaptation to unseen scenarios. Experiments on learning dynamics for physical systems and for controlling robots on interaction tasks demonstrate the effectiveness of our approach.

Geometric Contact Flows: Contactomorphismss for Dynamics and Control
Andrea Testa Soren Hauberg
Tamim Asfour
Leonel Rozo
Abstract
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Coefficient
Accurately modeling and predicting complex dynamical systems, particularly those involving foree exchange and dissipation, is crucial for applications ranging from fluid dynamics to robotics, but presents significant challenges due to the intricate interplay of geometric constraints and energy transfer. This paper introduces Ge- ometric Contact Flows (GFC), a novel framework leveraging Riemannian and Contact geometry as inductive biases to learn such systems. GCF constructs latent contact Hamiltonian model encoding desirable properties like stability energy conservation. An ensemble of contacto- morphisms then adapts this model to the target dynamics while preserving these properties. This ensemble allows for uncertainty- geodesics that attract the system's behavior toward the data support, enabling robust generalization and adaptation to unseen scenarios. Experiments learning dynamics for physical systems and for controlling robots on interaction tasks demonstrate the effectiveness of our approach.
Momentum
Momentum
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Damppnn
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Ground Truth
Recanstruction
Reconstruction
MLP Reconstruction 2
GCF Reconstruction
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Figure I. A dynamical system modeled via an MLP (left) and our GCF (right). The first prediction (orange) starts within the data support and the second (red) outside. The MLP fails to reconstruct the dynamics does not converge to attractor, while GCF provides reliable and its contact Hamiltonian structure.
predictions due
support (orange trajectory). This stems from the model's inability encode the physical relationships between variables, which are governed by underlying energy function. The model also extrapolates unreliably in data- sparse regions (red trajectory), showing the lack of physical interpretability and generalization capacity in black-bon approaches (Wang et al., 2020). To overcome these limitations, physics-inforee informed models incorporate physical principles into the model architecture, to ensure consistency with the underlying dynamics. These physical inductive biases appear as geometric structures, such as symplectic and contact manifolds (Blair & Blair, 2010), providing a powerful differential geometric perspective that we exploit.
1. Introduction
Modeling dynamical systems is fundamental to many scientific and engineering disciplines, enabling the analysis and prediction of system behaviors across diverse applica- tions (Yu Wang, 2024). This involves generating state- space trajectories forecast the system evolution. While the increasing availability of data has accelerated the adop- tion of ddiven methods, purely black-box models face substantial challenges (Chen et al., 2018). As exemplified in Fig. 1, a naive MLP network fails to capture the true dynam- ics, leading to inaccurate predictions even within the data
This paper introduces GEOMETRIC CONTACT FLOWS (GFC), a novel framework leveraging Riemannian and con- tact geometry inductive biases to learn complex dynami- cal systems. By integrating these geometries (Sec. 3) with learning model, GCF provides robust, interpretable, and dynamics modeling, overcoming the limitations black-box methods. First, baseline latent dy- namics encodes desired properties about the target dynamics (e.g., stability, energy conservation) via contact structure and an associated Riemannian metric. We then adapt this baseline model to the target dynamics through a contact dif- feomorphism (Sec. 4). We improve the GCF generalization robustness via an ensemble of contact diffcomorphisms to identify uncertain regions, allowing the model to adapt its behavior accordingly. This is achieved by modifying the Riemannian metric to incorporate uncertainty, resulting in
Bosch Center for Artificial Intelligence, Renningen, Germany Institute for Anthropomatics and Robotics. Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany "Section of Cognitive Sys- tems, Technical University of Denmark (DTU), Lyngby, Denmark. Correspondence to: Andrea Testa <andrea.testa de. bosch.com>.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
(M. m)
(N.1)
dynamical systems is define a simple (stable) dynamical system in latent space and use diffeomorphism to map it to the target dynamics, while preserving key latent properties like periodicity or convergence to an attractor (Perrin & Schlehuber-Caissier, 2016; Rana et al., 2020; Zhang et al., 2022). Extensions of this approach incorporate contraction properties to ensure stronger stability guarantees (Mohammadi et al., 2024; Jaffe et al., 2024). However, these methods are limited to first-order dynamical systems and cannot model complex behaviors such as self-crossing trajectories, physical interactions, and asymmetric obstacle avoidance. While Fichera & Billard (2024) extend diffeomorphisms to purely dissipative second-order dynamical systems, their approach remains limited, as it cannot model dynamics with limit cycles or non-zero curl components. We overcome these limitations by introducing contact Hamiltonian structure in the diffeomorphisms to preserve complex latent system's 's physical properties, even for general second-order systems.
Ensemble
=(ta)
a(f2)
=((ess)
Pr
((tap))
Contactomorphism (T)
(r) o (r)
Contact dynamics integration
Contact Hamiltonian
Hr, 2000 (D (7)(a) + (a)s
Figure 2. Geometric Contact Flows: A state x(txx in the ambient space (M, is mapped to a corresponding state a(le) in the latent space (W,n') via the ensemble of contactomophhssmssq.... (T)) The uncertainty as esti- mated by the ensemble is depicted as colored patch (dark and light colors denote low and high uncertainty). The latent dynamics integrated by following a contact Hamiltonian model resulting into the latent state (, which is then projected back into the ambient space =I(++1) via (T)) (T)))n())))) Each contacto- morphism in the ensemble is a composition of transformations (Eq. (10a)), where each transformation results from integrating the flow of a contact Hamiltonian vector field (Eq. (11)).
Riemannian Metric Learning. Inspired by the manifold hypothesis (Bengio et al., 2013), latent models can access and represent data manifolds in high-dimensional spaces. This can be exploited to learn Riemannian metric from dynamic trajectories, subsequently represented as geodesics (Amold, 1966). This metric also distinguishes the data support from the surrounding ambient space (Hauberg, 2018). Building on this, Beik-Mohammadi et al. (2023) learned a Riemannian metric that takes large values at the boundaries of the data manifold, confining the learned dynamics to a safe, data-informed region. By adjusting the metric, the system's ss behavior can directly be adapted to specific environments, such as navigating around obstacles (Zhi et al., 2022). Building on this idea, we estimate an uncertainty- based metric to guide the system's 's geodesics towards safe regions.
geodesics that avoid uncertain regions. Furthermore, this Riemannian metric reshaping principle is leveraged to avoid unsafe regions or unexpected interactions (Sec. 5). Figure 2 illustrates the overall GCF architecture. We demonstrate the effectiveness of GCF on two physical dynamical sys- tem reconstruction experiments (spring mesh and quantum system), on handwriting dynamics reconstruction using two datasets (Lemme et al., 2015; Fabi et al., 2022), and on two real-world robotic tasks (Sec. 6).
In summary, our contributions are: (1) Design of latent contact Hamiltonian dynamics with desirable properties that are transferred to the target dynamics; (2) An expressive class of neural networks to represent contactomorphisms, preserving the dynamic and geometric properties of the latent dynamies while adapting them to the observed data; (3) An ensemble of contactomorphisms to quantify the uncertainty in dynamics prediction; (4) A generalization mechanism based on Riemannian geodesics and ensemble uncer- tainty to guide the dynamical system's behavior and (5) Experiments including comparisons to state-of-thee art approaches and a methodology to transform dynamic data to canonical contact Hamiltonian coordinates.
Hamiltonian Learning. The Hamiltonian framework models second-order dynamics by describing how conjugate variables (e.g., position and momentum) evolve through a symplectic structure, which encodes energy conservation geometrically. While many learning approaches parametrize the energy function to model dynamical systems (Grey- danus et al., 2019; Zhong et al., 2020a), or incorporate a symplectic structure into neural networks (Jin et al., 2020; Li et al., 2020; Tong et al., 2021), modeling systems with friction and energy exchange requires extending the Hamiltonian formulation. Several studies (Sosanya & Greydanus, 2022; Zhong et al., 2020b; Chen et al., 2021) incorporate forces and damping as external disturbances on the symplectic manifold. However, these modifications disrupt the symplectic structure and prevent us from conserving the physical properties of the latent system solely through the preservation of the geometric structure via diffeomorphisms. Canizares et al. (2024) maintain the
2. Related Work
Diffeomorphisms Learning, Differential geometry gives a formal framework to connect the target dynamics to a simpler latent space representation through a diffeomor- phism (Franks, 1971). A prevalent approach for learning
2

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
symplectic structure by doubling the manifold's dimension represent non-conservative dynamics, the cost of introducing syysicalllymmaningkkss variables. We approach this problem by leveraging contact Hamiltonian dynamics.
Contact Hamiltonian Learning. Contact Hamiltonian theory (Kholodenko, 2013) extends the symplectic manifold by introducing a single, physicalll meaningful variable, to account for energy dissipation and generation. This makes contact geometry particularly well-suited for modeling non- conservative systems (Bravetti et al., 2017). The term con- tact refers to the geometric relationship between the dynam- ical solutions and the energy constraints, where solutions "touch" the constraints but do not violate them (Geiges, 2001). Zadra (2023) proposes to parametrize the contact Hamiltonian with a neural network to model a broad class of dynamical systems, but this is unstable beyond the training data, hindering its generalization capabilities. Our approach overcomes these limitations using the contact Hamiltonian framework as an inductive bias to design con- tact diffcoomorphimss These are later employed to design uncertainty-awwt aware geodesies for robust generalization.
91
91
Figure 3. distinct vector fields (shown as streamlines) under different geo- metric structures. In Riemannian geometry (left), the streamlines correspond to the gradient flow, following the direction of steepest ascent or descent f. acting as a potential. In contrast, in contact geometry (right), the function f describes an energy, and the angle between the streamlines and the level curves of f. depends on the energy dissipation in the dynamics, governed by Eq. (8).
The same scalar function f. via its 1-form df. yields
such cases, it defines a Finslerian structure, generalizing the Riemannian structure (Chern, 1996). The Riemannian metric g establishes correspondence between differential I-form df and vector field X, through the relation,
df(X) 9(X,, X); AX TM.
(1)
3. Preliminaries
The vector field X, is orthogonal to the level sets of f. which can be interpreted potential function (Fig. 3).
Vector Fields and 1-Forms. Let M be smooth compact manifold. The tangent space at x E _M, denoted T.M, consists of all tangent vectors v to M at x. The tangent bundle TM is the union all tangent spaces. An element is TM pair (z, v). The assignment of a specific tangent vector to every point in the manifold is described by vector field X .M- T.M. Given a function f . - R, the 1-form df: TM- + & generalizes the gradient from Euclidean spaces by measuring the variation of f in x along a direction identified by v. The set of all the I-forms df V (x, v) is the cotangent bundle T*M, the union of all the cotangent spaces T M. The contact and Riemannian geometries provide two distinct mechanisms to associate 1-form to vector field, thereby establishing different connections between the tangent and cotangent bundles. In simpler terms, these geometries link the same function f to different vector fields X, as illustrated in Fig. 3.
Contact Geometry. A I-form 1) on a (2d + --dimennsional manifold M is called a contact form if it satisfies the maxi- mal non-integrability condition (Geiges, 2008). This means that the wedge product of i with the d-times repeated wedge product of its exterior derivative dn, is non-zero everywhere on M: 7 A (dn) d + O. A contact manifold is then the pair (M, Given the I-form df, the contact form 1j uniquely determines vector field Xy through the relation,
= X) ((x, A V in TM, (2)
where Lx, denotes the Lie derivative along X2 (Geiges, 2001). The non-integrability condition ensures that 1) imposes nonholonomic constraints, restricting the allowable directions of motion without confining the dynamics to submanifold. Consequently, unlike the symplectic case, motion is not constrained to the level sets of f (Cruz, 2018). In both symplectic and contact geometry, f is typically referred to as the Hamiltonian and interpreted as energy, with the associated vector field written as X h- Consequently, while symplectic geometry is limited to describing conservative dynamical systems, contact geometry offers a more general framework for modeling broader range of dynamical behaviors (Bravetti et al., 2017).
Riemannian Geometry. Riemannian metric g: TMX TM- + R is a smooth, symmetric, and dositive-definte bilinear map that defines an inner product on the tangent spaces. The length of smooth curve ztt) [to, t1] + M w.r.t. the metric g is i(t) E M M is the vector tangent to the curve at xtt). The curve minimizing this length between two points xfo) and ztt) on M is called geodesic. Geodesics generalize straight lines in Euclidean space to curved spaces, repre- senting the shortest paths in the geometry induced by g. The geometric structure gi)(t), symmetric with respect to è to measure curve lengths. In
Sth (g(#(t),*(0) (t))dt, where
The integral flow x<t), generated by the vector field X from an initial point zo. defines a one-parameter family of transformations sp(t). mapping zo - x(t) within the contact manifold (M,
(t)) does not need to be
xt1) = (z) [0, C CR- + (M,1).
(3)
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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
Each transformation p(l): (M,n) - (M,n') ) is smooth, invertible, and preserves the contact structure up to conformal factor, i.e., '01) anj, E R (Zadra, 2023). So. (t(t constitutes a one-parameter family of contact- preserving diffeomorphisms, a.k.a. contactomorphisms. More generally, a contactomorphism maps between distinct contact manifolds xp: (M, + (f, In GCFs, contactomorphisms are employed both to represent integral dynamic contact flows xxf) within a single manifold and as mapping between ambient and latent spaces.
fixing this value (Udrişte, 2000). In symplectic geometry for conservative systems, 8 remains constant and is not dynamic variable. In contrast, contact geometry captures energy dissipation or external input through the evolution of 8. Our proposed Geometric Contact Flows build on this dual perspective modeling the dynamics by harnessing the contact interpretation while generalizing it through the Riemannian perspective. Further details on these concepts, along with broader comparison between contact and symplectic structures, are provided in App. A.
Contact Flows as Riemannian Geodesics. Notably, the
dynamic contact flow xxt) on the contact manifold (M, n) can be interpreted as a reparametrization of a geodesic on
4. Contactomorphisms Learning
Latent Contact Hamiltonian Dynamics. Let (N, be a (2d+ 1)-dimensional contact manifold, locally de- fined by a set of canonical coordinates q (g).....):: p {pI....pul pu)- In classical mechan- ics, the conjugate variables q: and pt respectively represent position and momentum, while s is the Lagrangian action. As detailed in Sec. 3, the contact manifold can be inter- preted as the augmented cotangent bundle of Riemannian manifold R.v, i.e., A = T'Ry x R. with q E Rw, {q. P) E T and s E RR. A contact Hamiltonian dy- namics the latent space (N,')') is defined via contact Hamiltonian function Hg (z), generating a flow V3 such that, ztt) (t)
an augmented space-time Riemannian manifold (R x IR. g). where M = T R (Abraham & Marsden, 2008). In this setting, we denote the state variables as q E R, q RyR, P E R, and E IR, with x {g. P, 8). This means that the augmented Riemannian manifold is equipped with a cotangent bundle TRR endowed with a contact struc- ture. This connection between the contact and the Rieman- nian geometries arises from Maupertuis' principle (L.ppez & Martínez, 2000), which first adopts a variational perspec- tive by expressing the contact flow se(() as the solution of dynamic optimization problem,
{q. s.. with
[" p(t) ((t) - H(((t),((t),(()) dt. (4)
min g.P.8
t E [0,
(6)
The choice of (z) induces a metric 9 on Ry, making Pg follow its geodesics.
This formulation, defined T x RR can be trans- ferred to TR I by mapping the 1-form p to vector i via the Riemannian connection in Eq. (1), resulting in ming 1h L(g(t), q(t)) dt, where L is the transformed cost functional, i.e., the Lagrangian. This dynamic optimization can be reparametrized using as the independent variable, with ds- - /11(u.p, p, s) Hv (q) dt, where Hv denotes the potential energy component of the Hamiltonian (Abraham & Marsden, 2008; Udrişte, 2000). In this setting, 8 plays the role of arc-length to be minimized on the Riemannian manifold and corresponds to the action associated with the Lagrangian L. The problem then reduces to geodesic computation on R x IR
We propose to preserve the properties of the dynamical system H, in the ambient space through a suitable coordi- nates change, i.e., via contactomorphism. The choice of the latent dynamics does not affect the reconstruction of the target dynamics on the data support, but instead serves as an inductive bias to guide the generalization. Possible choices for Hg are,
g = p p 1 2
q q.
(7a)
2
1
1
H 2D p+ 29 i94
(7b)
min/ 1') V(() q
Hye = 2D
p + 2 g
s².
(7c)
(5)
For instance, to ensure periodic orbits in the target dynamics, we can use the simple harmonic oscillator (7a), as our latent dynamies. Other dynamical systems, such as the example shown in Fig. 1, naturally converge to attractor. This behavior can be induced in the latent dynamics by incorporating damping term. We use the dependence of the contact Hamiltonian on the Lagrangian action 8, to govern the variation of the system's 's total energy. This relationship is expressed as follows (Bravetti et al., 2017),
The modified Riemannian metric 9 is induced by the Hamil- tonian H and depends on the variable 8. This dependence reflects how the shape of trajectories is affected by their own length, capturing the influence of non-conservative forces the system's 's evolution. By modifying the metric g. one can alter the system's dynamics, providing a geometric approach for controlling or shaping its trajectories (Udrişte, 2003; López& & Martínez, 2000). The resulting geodesics lie on R x R. as they depend on the reference level of s. They can be projected onto the configuration space R by
H(t) (()) e 08 dr.
(8)
4

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
where /0 represents the damping coefficient of the dy- namics. Consequently, the + term in Eq. 7b ensures the de- pletion of the system energy and the convergence toward the attractor {q. p) } = O. Instead, the dynamics (7c) is suit- able for safety-critical scenarios requiring stopping the sys- tem near unsafe regions of the state space (N, r/)). By scal- ing all terms of the contact Hamiltonian with 8, the energy H can reach zero while q and p remain non-zero, stopping the system before it reaches the attractor. Unsafe regions are here characterized by low values of the Lagrangian action
(0)))) we integrate the latent dynamics to obtain the predicted latent trajectories (z)(t), t E [0, fol (e(1.) (3) We then map the predicted latent trajectories to the am- bient space using the inverse contactomorphism sp, (T), yielding (xx(t), t E [0, (s) bE[L.B]; (4) Finally, we compute the estimation error in both spaces using the average loss,
B
L
W (x)()) (t(tt
B b=1
tb
(12)
(s(t) - Zu (t)1l2)
W
Contactomorphismms Design. While diffeomorphisms are commonly used to align latent and target dynamics in first- order systems, they fail to preserve the intrinsic structure of second-order systems, in particular the interplay between conjugate variables encoded in the contact form. To address this, we leverage dontactomorphismss, which preserve ní and thus maintain the latent space structure. The practical impli- cations of this inductive bias are examined in the ablation study presented in App. C.I. A contactomorphism trans- forms coordinates z from the latent space (N, v/') into a new set of coordinates x that describe the ambient space (M, 17)- As described in Sec. 3, we model this transformation as the flow (1) of contact Hamiltonian system evaluated at time = T.
where wa and w2 are weights balancing the ambient and latent space components. Typically, wy > Wz. as we focus on capturing the ambient dynamics. However, the latent error term acts as a useful regularizer, as shown in App. C.2. A Algorithm in App. B.1 outlines the training phase.
After training, predicting the ambient dynamics from an initial point Xy involves mapping it to the latent space via the learned contactomorphism yr. integrating the latent dy- namics 9g. and then mapping the result back to the ambient space via the inverse contactomorphism yp, Importantly, integrating the latent dynamics does not require evaluating the learned contactomorphism at every step. This enables ef- ficient long-horizon predictions using only single forward and inverse mapping. Formally, this process is expressed as composition of contact Hamiltonian flows,
(x)) x x= x-);((T)(2)
(9)
For expressivity, we implement this contact transformation as a composition of K sequential parametrized networks, 4p, (T) = Prew (T) o...... o Pro (T) (r).....00 o Pro, (T), (10a)
x(1) = -. (T) o (p)(() o ypr (T)))), t E [0,4]- (13)
Details on the physical interpretation of this composition are provided in App. B.4.
(T) = Pro, 1 (7) o...... Pro (7) o... o Prox (T).
(10b)
Each individual contactomorphism Pra, (r) updates the ini- tial point x by integrating, for duration r, the vector field associated to the contact Hamiltonian Hrm defined as,
5. Contactomorphisms Generalization
Ensemble of contactomorphisms. Relying on a single contactomorphism to map the latent dynamics to the ambi- ent space neglects predictive uncertainty. While predictions may closely align with the training trajectories on the data manifold, extrapolation in data-sparse regions tends to be unreliable. To quantify uncertainty and identify the bound- aries of the data manifold, we propose to use an ensemble of N contactomorphisms { (T)) pEELL.N] each randomly initialized and trained on the same dataset (Syrota et al., 2024). In data-rich regions, the predictions from the ensem- ble closely match, while in regions with limited data, the predictions diverge, resulting in high model uncertainty.
1 Hrup -22 Mo, (p)p + Vo, (q) Fo, (q)s,
(11)
where Ma, (p), Vo. (q), Fe, (q) are scalar functions of the conjugate variables, parametrized by random Fourier fea- tures networks (Rahimi & Recht, 2007). The structure of Hro, and the choice of network architecture are elaborated in the ablation studies of App. C.3. The integration of the associated vector field is performed using contact split- ting integrator (Zadra, 2023), as detailed in App. B.2. Our architecture is analytically invertible, minimizing the com- putational cost of computing Proy 1 (7).
The ensemble-based uncertainty quantification, whether in ambient or latent space, is computed by transforming the starting point of the trajectories using the ensemble of con- tactomorphisms to obtain N points:
Training contactomorphism (p, (T), given a dataset of B trajectories from the target dynamics (x), I E [0, to] involves the following steps: (1) We map the training trajectories to their corresponding latent tra- jectories (t)( t E [0, talleee via the contactomor- phism (T); (2) Starting from the initial latent states
[z]Ne|I,N (T))
(14a)
[x]nE[LNN] = (T)) (z).
(14b)
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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
1.00
I' which is generally Finsterian (see Sec. 3), due its poten- tial asymmetry in q. To solve this, bring back the opti- mization to the contact manifold M = RN by adopting the geodesic interpretation of dynamical systems (Udriste & Udriste, 2000). This involves the reformulation of the prob- lem as a dynamic optimization under a time reparametriza- tion (López & Martínez, 2000). Specifically, we define a perturbation u to the velocity field of the learned geodesic 4 under metric 9. resulting in a perturbed direction 4g' under a modified metric g'.
0.8
0.75
Uncertainty
uncertalngs
0.6
0.50
0.4
0.25
0.2
0.00
Figure 4. Optimal control problem (18) in both latent (left) and ambient spaces (right). The learned dynamics 2 = Zu, is ex- tended by the control input u (represented by the golden arrow) to minimize uncertainty and quickly converge the data support.
min J" (ox(gg Lg 8) + [|(())]13) di,
(17)
The average prediction is computed and mapped back to the original space to estimate the predictive variance of the resulting N points,
s.t. 9,' 8 م + u(t)
The control term u locally deforms the trajectory to avoid uncertain regions. Since the learned geodesic flow is en- coded by the contact Hamiltonian vector field Zı,, we lift the optimization to the latent state space via the Riemannian connection in Eq. (1),
xx = = (T) (u(z)ne(L.NI),
(15a)
Ox = oppm (T) IT) (m(x)ne()NN))
(15b)
where ls and a denote the mean and variance of the pre- dictions. Note that this computation requires all contacto- morphisms to share the same latent space. We ensure this by jointly training the ensemble as follows: (I) Transform all B training trajectories into N x B latent trajectories {zz (t), t [0, bel using all N contactomorphisms ppy: (2) Integrate the latent dynamics starting from the N B initial latent states {zī (0) [E] to compute the [n
min I/ (d)(z(t)) ||u(t)lP) dt to
(18)
s.t. 2(t) = ZH, (z(t)) + u(t.
In regions where the uncertainty is low (i.e., O2 0), the control action vanishes (i.e., latent dynamics z(t) = ZH, (z) are unchanged, correspond- ing to the original latent metric, i.e., D' x 9. Conversely, in high-uncertainty regions, the control action u modifies the dynamics by bending the latent trajectories to guide them away from data-poor regions. Figure illustrates the optimal control problem (18) in the reconstruction of the damped harmonic oscillator dynamics (Figure 1).
0), and the resulting
predicted latent trajectories {z, (t), t E [0, tul Melneel be (3) Randomly select N inverse maps with replacement from the ensemble to compute the ambient predictions (xī( (t), E
[E, (4) Compute the average loss over the N [0, 4b) [E[L.
contactomorphisms as le 1/2 EN=1 l, with l defined in Eq. 12. Algorithm 2 in App. B.1 summarizes this process.
Designing safety-aware geodesics. The optimization problem (18) can be extended with other types of energy functions aimed at penalizing the crossing of unsafe re- gions. To illustrate this, consider learning a contact Hamil- tonian dynamics for a controlled physical system. Here, an unsafe region can be described by obstacles in the system workspace. Formally, an obstacle in the position space R5 is represented by set T. consisting of posi- tions qa E Y uniformly sampled from the spatial distri- bution of points occupied by the obstacle. This set ex- tends to the contact ambient space as Tx, where each state x,, = (go, s) E Tx C (M, is obtained by augmenting q. with the momentum p and action variable s that the dynamical system can acquire. This formula- tion implies that an obstacle at qo cannot be traversed for any combination of p and 8. The sampled points x,, are mapped to the latent space (N,,)') via the ensemble, as Y, = (x.)) (xee))) In this latent space, the en- ergy assigned to each bin is determined by the density of mapped points it contains. This energy distribution is then
Designing uncertainty-awast -ware geodesics. We leverage the ensemble uncertainty to drive the system dynamics to remain within or converge toward well-informed, data-rich regions. As discussed in Sec. 3, the contact Hamiltonian dynamics ztt) on the latent space r/'), governed by the Hamiltonian function Hg. can be interpreted as geodesies q(s) on the augmented Riemannian manifold (Rx x IR, g) with respect to the metric 9 in Eq. (5). To guide the dynamics away from uncertain regions, we propose to reshape using the ensemble uncertainty az- This increases the traversal cost outside the data manifold, thereby steering the latent dynamics away from data-poor regions. The geodesic qss) obtained by solving the optimization problem,
911 min I'h 40
(Vola.q.s) +2. s)) ds, (16)
where canonical coordinates z = s E RR x IR are converted to {q. q. $} E TRx x R. This corresponds to computing a geodesic with respect to an augmented metric
6

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
incorporated into the optimization (18) as a penalty term to enforce obstacle avoidance,
Quantum Dynamics Reconstruction. A single-mode bosonic system is quantum mechanical oscillator whose deterministic dynamics are governed by contact Hamil- tonian operator. The evolution of its wave function is de- scribed by the quantum operators q and ý. representing the system's 's stochastic position and momentum, along with the quantum phase s, which governs interference between wave components. We conducted experiments on 20 synthetic sys- tems with varying parameters, reconstructing the dynamics of the state variables' expected values using the GCF model and baseline methods, over an 8 seconds horizon. HNN and DHNN are inapplicable here due to their limitation to even- wven-dimensional phase spaces, which prevents modeling the additional odd variable 8. Results are summarized in Table 2, showing that GCF achieves a 60% reduction in re- construction error. Further details are provided in App. E.2,
min /." (Fr(z(()) (2(2(()) + [|(())])) di u(.)
(19)
s.t. 2(t) ZH, (z() + (t),
where (r(z(t) denotes the energy term associated with the unsafe region, i.e., the latent space obstacle distribution. This ensures that unsafe latent space regions are associated with high energy values, which penalize trajectories that cross these regions, resulting in e.g., obstacle avoidance.
6. Results
We test our approach on established baselines that incor- porate inductive biases in learning dynamical systems. Eu- clideanizing Flows (EF) (Rana al., 2020) use diffeomor- phisms to encode desirable properties like periodicity or target convergence. Neural Contractive Dynamical Systems (NCDS) (Mohammadi et al., 2024) improve upon EF by enforcing global contraction. Hamiltonian Neural Networks (HNN) (Greydanus et al., 2019) embed physical structure by modeling systems with Hamiltonian dynamics. Dissipative Hamiltonian Neural Networks (DHNN) (Sosanya & Grey- danus, 2022) extend HNNs non-conservative systems by introducing dissipation. Details on the experimental setup are provided in App. D. Scalability across target dimen- sions and model sizes is evaluated in App. E.5. In classical mechanics, the state of physical systems is typically de- scribed by positions and velocities (4. which we convert to canonical coordinates {q. p. s). required in the contact Hamiltonian formulation (App. D.3). The Lagrangian action is not directly observable but it is estimated by comparing the system's 's behavior as described by Maupertuis' prin- ciple with that derived from Noether's theorem, ensuring consistency between the two representations.
Table 2. Reconstruction error via DTWD the quantum system
GCF DHNN HNN EF NCDS MLP 0. 0.29 N/A N/A 0. 0.72 0.70+0.11 0.41+0.06
Handwriting Datasets. The LASA (Lemme et al., 2015) and DigiLeT's (Fabi et al., 2022) datasets are widely used benchmarks for motion generation, featuring 2D handwrit- ten trajectories. LASA involves simpler, first-order dynam- ics, while DigiLeTTs includes more complex, second-order motions with self-crossing paths. The learned dynamics are treated as a skill that an actuated system must not only imitate but also generalize and adapt to new scenarios. In these tasks, the dynamics converge to a target, so we exclude HNN, which models only periodic behaviors. Figure 5 com- pares the remanent approaches two characters. EF and NCDS model position-dependen vector field, enabling flow lines to be plotted in the background. In contrast, DHNN includes position and momentum, and GCF incor- porates the Lagrangian action, making global flow lines infeasible. Instead, circular patches depict local dynamics around trajectory points with fixed extra-variable values. GCF's patches represent the model uncertainty.
Spring Mesh Dynamics Reconstruction. We consider a 60-dimensional dataset (Otness et al., 2021) deseribing the dynamics of 2D square grid of nodes connected by springs. The interaction among multiple springs results in complex, large-scale deformations and oscillations. Pre- dicting the motion of mesh nodes closely resembles finite element modeling of material deformation. Table 1 summa- rizes the dynamics reconstruction results, reporting the Dy- namic Time Warping Distance (DTWD) (Berndt& Clifford, 1994) across 20 trials on systems simulated for 8 seconds from varying initial conditions. Our GCF model achieves a 57% reduction in reconstruction error. Experimental set- tings and predicted dynamics plots are in given in App. 1.
Table 3. Reconstruction error via DTWD on handwriting datasets Character
EF
NCDS
DHNN
GCF
00.1
0.65 0.44
0.44
l
2.22 20.03
0.82 2.79
0.70
EF and NCDS are limited to first-order dynamics and cannot capture self-crossing trajectories. DHNN lacks convergence guarantees, often failing to reach the target and to generalize beyond the training data. In contrast, GCF accurately recon- structs the dynamics, converges to the data manifold, and reduces uncertainty, as shown by streamlines avoiding high- uncertainty areas. Table 3 reports the reconstruction error of the position dynamics. GCF achieves performance com- parable to EF on LASA, while significantly outperforming
Table Reconstruction error via DTWD on the spring mesh
GCF DHNN HNN EF
NCDS
MLP
0.50 19 1.24 02 1.49+.7 74 30.9 24.9+2.
1.
1.71
7

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
1.0
0.8
Potendal Puncoon
0.8
Potentill Function
0.8
Function
0.8
Potenbal Function
0.6
0,6
0.4
0.4
0.4
0.2
0.2
0.2
x10-
x10
x10-1
x10
x10
x
x10
2.0
Uncertainty
Lagrangian Action
Uncertainty
1.5
2.0
1.0
1.5
Uncertainty
Lagrangian Action
0.0
Lagran Egrrangian
Lagrangian
-0.5
1.0
1.0
0.5
-1.5
Figure 5. Results on the handwriting datasets: Reconstruction and generalization of the LEAF _2 character from the LASA dataset and the ELLE character from the DigiLeT's dataset. From top bottom rows, EF, NCDS, DHNN and GCF methods, Demonstrations model predictions are shown as dashed and solid green lines.
other methods on DigiLeTss. Comprehensive results with additional characters are provided in App. E.3. Figure 6 shows generalization tests two characters, with predic- tions initialized from states outside the training distribution. A grid of points in the position space is used, with other state variables set to zero. The violin plots provide a statistical analysis of these predictions. Each point, representing a pre- dicted trajectory, computed as the ratio between time steps spent within the data manifold (in the position space) and the total time steps. Due the lack of constraints outside the data manifold, only few DHNN trajectories converge,
shown by the high proportion of points with a ratio EF ensures asymptotic stability, guiding trajectories to the target but not toward the data manifold, resulting uniform ratio distributions. NCDS performs better with contractive be- havior, but neither method supports recond-orde dynamics. In contrast, GCF shows greater reliability, ensuring all the dynamics predictions for the two characters converge to the data manifold and spend most of their time within it. Table reports quantitative metrics for the two characters. Figure 7 illustrates how GCF adapts to unseen scenarios. Obsta- cles, absent during training and representing unsafe are depicted as red points in the position space. Obstacle- induced energy steers the system dynamics away from col- lisions, as shown by higher cost and deflected streamlines near the obstacle. The right plot compares distances to the data support and the obstacle. The former distance peaks the obstacle, then drops, marking exit and reentry into
regions,
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
EF
NCDS DHNN
GCF
EF
NCDS DHNN GCF
Figure 6. Generalization results for the LEAF 2 (left) and ELLE (right) characters. The violin plots depict the distribution of conver- gence rates for predictions initialized from grid of points the state space. While NCDS performs well on LEAP 2, it struggles with ELLE. In contrast, GCF shows greater reliability, achieving the highest average convergence ratio and lowest variance.
Table 4. Generalization measured average ratio of convergence
Character
EF
NCDS
DHNN
GCF
0.49
0.94+0.19
0. 0.18
0.69
l
0.61
0.54+
0.17+0.15
0.6640.12
8

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
Table S. Reproduction error (DTWD) in WRAP-AND-PUL task.
the data region. Appendix E.3 provides extended results on additional characters and ablation study on replacing the contactomorphism ensemble with single map.
EF
NCDS
DHNN GCF Safe
GCF Stable
1. 1..00
3.37 4.25 1..12
10.12
0.62+t0.2
0.61 +0.25
Table 6. Energy Consumption (J) in the WRAP-AND- -UNL Task: Comparison of baselines with stable and safe GCF EF, NCDS and DHNN values are included despite task failure.
x10-1 X10-1
-1.5
Obstacle Distance
1.0
Uncertainyy
0.05
0.8
Scenario
EF
NCDS
DHNN GCF Safe
GCF St.
Lagrang anggian
2
0..
Unloaded
0.54+.04 04
1.52+.18
1.01
0.60 07
0.59+.07
0.00
Data Distance
Loaded
0.55+.u6
2.73+.44
43
0.63 2.83+.58 022
0.81+.09
100
200 300
400
0.01
1.21
1.82+.72
0.03+.07
0.22
Time steps
Figure 7. Obstacle avoidance by LEAF_2 (red dot as obstacle).
handle deviations from training states. We test GCF's gener- alization to unseen energy exchanges by adding a load to the TRAP-AND-PULL. task. Table 6 shows energy expenditure for both loaded and unloaded cases. The GCF implementa- tion with stable latent dynamics reproduces the trajectory learned in the unloaded scenario, even in the loaded case, by consuming more energy. In contrast, the safe GCF vari- ant halts motion when the Lagrangian action approaches zero. As shown in Fig.9, the added load increases energy demands, causing to vanish before task completion. This triggers an early stop, preventing energy use beyond what was observed during training. This mechanism enables energy-basee based task completion and acts as a safety stop in the presence of unexpected interactions, as further demonstrated in the DISHWASHER LOADING task. For details that task and additional analysis of the robot's robustness to physical disturbances in this proof-of-concept, see App.E.4. A full video of the robot experiments, including adaptability to unseen obstacles, and the repository implementing the GCF framework are available at com/vies http:///gemerrccccoccee conntacttfloss......
Figure 8. Snapshots of the unloaded (top) and loaded (bottom) WRAP-AND-PUL robotic task, showing rope pickup, wrapping around a drum, and pulling as the robot moves forward.
WRAP-AND-PUS PULL Robotic Task. In robotics, learning interaction tasks from demonstrations is more challenging than learning free-space motions (Scherzinger et al., 2019; Le et al., 2021). GCF's ability to model energy behav- iors and flexibly generalize unforeseen states, makes it well-suited for online robot control in interaction tasks. We apply GCF to two robotics tasks: A WRAP- APP-AND-PULL task (detailed below) and a realistic DISHWASHER LOADING task (Simmoteit et al., 2025) (See App. F.4). The first task, a proof of concept, showcases all the robot capabilities en- abled by the framework. Three kinesthetic demonstrations were recorded for training, in which the robot grasps rope, wraps it around drum, and pulls it along a specific tra- jectory (see Fig. 8). The ensemble generates a reference dynamic trajectory using the measured robot state, which is tracked by a low-level Cartesian impedance controller. Setup details are in App. D.I.
http://ssssss..gge...... google.
0.5
Unloaded Pull
Lagrangian Action
Loaded Pull
Loaded Pull
Unloaded Pull
Demonstrations
0.0
0.5
Ambient Space
1.0
Latent Space
200
400
600
800
Time Steps
Figure 9. Position and Lagrangian action trajectories (left and right plots) of the WRAP-AND-PUL task, for both the unloaded and loaded scenarios, using safe GCF.
7. Conclusion
Geometric Contact Flows (GCF) model dynamical systems leveraging geometric inductive biases. It starts with a latent dynamics and uses contactomorphisms to adapt while pre- serving its key features. To assess uncertainty, GCF employs an ensemble of contactomorphisms to ensure convergence to the data support. GCF has been successfully applied to both dynamics reconstruction and control tasks, covering a wide range of problems including spring mesh simulation, quantum dynamics and robotic manipulation.
Our method is implemented using two different latent dy- namics: The stable (7b) and the safe (7c) formulations. Ta- ble 5 reports the reproduction accuracy for all methods across five trials. The results indicate that the different la- tent dynamics do not affect the GCF's ability to learn and reproduce the task. In contrast, EF, NCDS and DHNN fail to achieve satisfactory results. EF struggles to model the wrapping phase of the task, while DHNN lacks robustness to

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
Impact Statement
Chen, Y., Matsubara, T., and Yaguchi, T. Neural symplectic
form: Learning hamiltonian equations on general
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
coordinate systems. In Neural Information Processing Systems (NeurIPS), volume 34, pp. 16659-16670,
2021. URL
htc/sppee_fieessppperr///2/////2 05511991882226772333388888888888 pdf.
Paper.
Acknowledgments
The authors are grateful to Hadi Moll-Mohammadii and Huy Le for the insightful discussions on the core technical challenges and prior work in the field, and for the valuable assistance with the robot setup during the experiments.
Chern, S.-S. Finsler geometry is just Riemannian geometry without the quadratic equation. Notices of the American Mathematical Society, 43(9):959-963 1996. 3, 13
Ciaglia, F. M., Cruz, H., and Marmo, G. Contact manifolds and dissipation, classical and quantum. Annals of Physics, 398: 198:159-179 2018. 21
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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
A. Extended Preliminaries
Hamiltonian and contact Hamiltonian dynamics can be el- egantly analyzed via differential geometry. We begin by introducing the symplectic structure and comparing with the more familiar Riemannian structure. Subsequently, the symplectic geometry is extended to contact geometry, which forms the foundation of our approach. While contact and Riemannian geometries offer distinct perspectives, they are complementary in providing a deeper understanding of the geometry of dynamical systems.
91
A.1. Symplectic and Riemannian Structures
91
Let M be a smooth compact manifold, and let T&M denote the tangent space x .M. The collection of all the tangent spaces identifies the tangent bundle TM- UzeMTaMM A vector field X M TM assigns tangent vector u to each point E M. The set of all the vector fields over TMM is denoted as ((T.). A differential 1-form a TM + I is smooth map field act- ing on vectors of the tangent bundle. For a smooth function f + IR, the 1-form a df generalizes the gradient from Euclidean spaces. Specifically, df measures the vari- ation of f under an infinitesimal displacement on M. This displacement is locally described by starting point and direction such that (zz, v) E T.M. Alternatielly, it can be globally expressed by vector field X. The variation of f along the vector field X is given by (X). This variation is independent of the choice of reference frame. To preserve this invariance, df must transform covariantly with X. Consequently, the 1-form a = df resides in the cotangent bundle T M, the dual space to T.M. The symplectic and Riemannian structures provide two distinct mechanisms for ussociating f-form to a vector field, thereby establishing connections between the tangent and cotangent bundles. By considering the dynamics governed by the vector field and the scalar function defining the 1-form, a relationship between these elements emerges, illustrated Figure 10.
Figure 10. The same scalar function f. associated with the I-form a = df, gives rise to three distinet vector fields under diffcrent geometric structures: Riemannian (top left), symplectic (top right), and contact (bottom). The streamlines of these vector fields are illustrated on a representation of the state manifold. In Riemannian geometry, the streamlines correspond to gradient flow trajectories, following the direction of steepest ascent or descent of f. which acts potential. In symplectic geometry, the streamlines instead tangent to the level curves of f. representing isoenergetic trajectories where f remains constant, thus describing the dynam- ics of conservative systems. In contrast, in contact geometry, a single flow line can traverse different energy levels, with the varia- tion of energy along the flow governed by Equation 8.
In such cases, it defines a Finslerian structure, which is generalization of the Riemannian structure (Chern, 1996).
The Riemannian metric also establishes a correspondence between vector fields and covector fields, thus defining a bijection between the tangent bundle TM and the cotan- gent bundle T*M. Specifically, the metric g associates vector field X, with the differential 1-form df (X) = g(X,. X), VX E I(TM). Since 9 defines an inner product, this connection allows the variation (X) to be interpreted as a measure of alignment between the vector fields X and Xy. A greater alignment corresponds to larger variation of f along X. Consequently, X, is orthogonal to the level sets of f and represents gradient vector field. In this sense, f can be interpreted as a potential function, generating a gradient flow z(t) whose trajectories follow geodesic curves with respect to the metric g.
df through
The Riemannian Metric A Riemannian metric TM x TM + IR is smooth, symmetric, and dositive-defintt bilinear field of maps defined on pairs of vectors the tangent bundle. This enables the introduction of an inner product on the tangent spaces of the manifold, allowing us to measure distances and curve lengths. M, the length w.r. the metric g is I = fhh Vo(i(t), where a(t) E Ma M is the vector tangent to the curve at xtt). The curve minimizing this length between two points ar(o)) and x(41) on M is called a geodesic. Geodesies generalize straight lines in Euclidean space curved spaces, representing the shortest paths in the geometry induced by g. The geometric structure ())(t, (t)) does not need to be symmetric with respect to à to measure curve lengths.
For a smooth curve x(t) [to,
The Symplectic Form differential 2-form w TM x TM + a skew-symmetric, bilinear, and smooth field of maps acting on pairs of tangent vectors. A 2-form is called symplectic if it is both closed (dw 0) and non- degenerate. Unlike the Riemannian metric, the symplectic form lacks the properties required to define an inner product. However, it still establishes a fundamental relation between differential 1-forms and vector fields: Given a 1-form df, the symplectic form w uniquely determines a vector field X,
13

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
that is tangent to the level sets of J. rather than perpendieular as in the Riemannian case. This relation is defined by.
restoring the interpretation of the trajectory as geodesic with respect to the metric 9 (Udrişte, 2000; Udriste & Udriste, 2000). This reparameterization underlies the Mau- pertuis principle, which states that conservative mechanical motion can be recast as geodesic motion in a suitably de- formed geometry.
(x( X),, VX E
(20)
By definition, I remains constant along the flow of Xy, which in turn preserves the symplectic form w, i.c., Cx, = where Lx, denotes the Lie derivative (Silva,
The Maupertuis' principle The trajectories of an Hamil- tonian dynamics on the symplectic manifold (T) are obtained the eritical solutions of the Lagrangian action functional (López & Martínez, 2000),
2001). In this framework, the function f is interpreted as conserved energy, or equivalently,
Hamiltonian H. The symplectic structure thereby endows M with natural geometric framework for formulating Hamiltonian dynam- ies (Tokasi & Pickl, 2022). The pair (M, w) is referred to as a symplectic manifold. Notably, the non-eegeneracy of w implies that M must be even-dimensional.
8 $= $= (tt) =(t) - (((x(t), dr,
(24)
where the covector (1)()), *(t) is denoted simply as at)) to emphasize its dependence on the integration variable t. The resolution of this optimization problem corresponds to a geodesic computation on (M,g) under the arc-length parametrization of Equation (23),
A diffeomorphism ø : (M.w) + (N,w") between two symplectic manifolds is called sympleecoonophhism if it preserves the symplectic form, i.e., d"w' = w (Polterovich, 2012). The symplectic flow z(f), generated by integrating the Hamiltonian vector field X,, can be understood as the action of a one-parameter group of diffeomorphisms ((t) on a point x E (M, w). This is defined as,
sxs)) argminn
V Vo(5(s), ds.
(25)
A modification of the Hamiltonian dynamics on the symplectic manifold (T*R,w) can be represented in the time domain as a change in the action functional (24), while geometrically, it corresponds to an adjustment of the Jacobi metric the Riemannian manifold (R, This geometric perspective, first introduced in classical mechanics in the 18th century, is known as the Moupertuis' principle of least action. In physical terms, it states that the dynamics of a system follow a trajectory that extremizes (typically minimizes) the difference between the kinetic and potential terms over time. Practically, this provides a geometric framework for shaping or controlling the trajectories of a dynamical system by modifying only the underlying Jacobi metric 9 (Udrişte, 2003; López & Martínez, 2000).
x (tt) = (z) [0,2] C - (M,w).
(21)
A.2. Symplectic Bundles of Riemannian Manifolds
The geodesic flow x=() on a Riemannian manifold (R,g) lifts to joint evolution of coordinates (z(t), (()())))())))((((()))))))))))) on the cotangent bundle T (Abraham& & Marsden, 2008). This extended dynamics is governed by an energy function (1, : T - R = 9 (a, a)), which remains constant along the flow. A reparameteriizttion ds dt links the trajectory of the integrated dynamical system at time t on TRR with the length of the corresponding geodesic on R. This framework reveals fundamental connection between geodesic flows and Hamiltonian dynamics in the special case where the Hamiltonian consists solely of a kinetic energy term. The cotangent bundle T-R naturally equipped with a symplectic structure, making a symplectic manifold (T*R,w). w)
A.3. Contact Geometry
While symplectic manifolds provide a geometric framework for modeling the dynamics of conservative systems in clas- sical mechanics, a more general approach is required to describe con-conservative systems. This is addressed by contact manifolds, the odd-dimensional counterparts of sym- plectic manifolds (Geiges, 2001; Bravetti et al., 2017). contact manifold is defined as (M, where M is an odd- dimensional smooth manifold, and 17 is non-degenerate 1-form known as the contact form (Geiges, 2008). The con- tact form satisfies the maximal non-integrability condition, meaning that the top-degree differential form A (dn) + O is nowhere vanishing on M. This form is constructed by tak- ing the exterior product of r with the d-fold wedge product of its exterior derivative drj, i.e.,
The formulation can be further generalized by introducing potential energy function into the Hamiltonian, given by H (z,,a) a)- (a, a) + (xx. In this setting, the geodesic structure underlying the Hamiltonian flow is determined by the Jacobi metric:
g - (H g,
(22)
which rescales the original metric g by position-dependenn conformal factor (Abraham& & Marsden, 2008). The corre- sponding time reparameterization takes the form
(dn) d = dn A .... A dr.
(26)
ds / H VVx) dt,
(23)
d times
14

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
The (2d 1)-form defines a volume form on M, ensur- ing that the hyperplanes defined by ker(n) C TM, which constrain the dynamics the contact manifold, do not form a foliation, i.e., they do not partition the manifold into nnwer-dimensional submanifolds (Geiges, 2001; 2008). Geometrically, this means that the contact distribution im- poses mon-holonomic constraints: it restricts the admissible directions of motion at each point without confining the dy- namies to a fixed submanifold or energy level. This property is crucial for modeling systems where energy can change over time, enabling constraints on energy behavior without enforcing conservation.
specific time-dependent energy profile (1) (Di Cairano et al., 2019). The additional variable defined on R is the Lagrangian action 5, introduced in Equation (24), which represents the geodesic length. In the contact setting, the Hamiltonian function also depends on 8, making the La- grangian action funcctiona implicit:
f. $=
f1 a(t) . -((()) dt.
s=
(29)
This dependence renders the associated Jacobi metric -- dependent, thereby transforming the geodesic computation problem into:
Contact Hamiltonian Dynamies Like symplectic geometry, contact geometry connects scalar functions to vector fields, enabling the description of dynamical systems (Zadra, 2023). Given energy function H M I, the dynamics contact manifold are defined by contact Hamiltonian vector field XH, as follows,
xxs) argmin (()) ds.
(30)
Thus, the length of the geodesic on R directly influences the metric 9. reflecting the non-conservative nature of contact dynamics. The proposed Geometric Contact Flows build on this foundation, modeling the dynamics by harnessing the contact interpretation on the full state manifold while generalizing through the Riemannian perspective on the configuration manifold.
(x) (x, X) -xxxm(()),,, VX ((TM).
(27) Unlike symplectic geometry, where dynamics are confined to energy-pret mnergy-preserving flows along the level sets of the Hamil- tonian, contact geometry allows for an additional component of motion. Specifically, the dynamics on a contact manifold are not restricted the term dny , X), which lies tangent to the level sets of H, but also include transverse com- ponent Lx æ (X). arising from the non-degeneracy of the contact form. Consequently, while in symplectic geometry the symplectic form w is strictly preserved, contact geome- try allows the contact form 1) to be preserved only up to a scaling factor E (Bravetti et al., 2017).
diffeomorphism sp (M, (7, between two con- tact manifolds, a.k.a. contactomorphism, satisfies sp n/' ay, thereby preserving the contact structure up to the scal- ing factor (Zadra, 2023). Analogous to the symplectic case, the integral flow x=f) generated by the contact Hamiltonian vector field X can be viewed as a contactomorphism, i.e.,
<t)) = (z) [0. 71 c R (M.1).
(28)
Contact Flows as Riemannian Geodesics Contact Hamilto- nian flows can also be reinterpreted as geodesics associated with an induced Riemannian metric ÿ. under the reparame- terization given in Equation (23) (Udrişte, 2000) In contrast to the symplectic case, contact dynamics are inherently non- conservative: the Hamiltonian function (t) varies with time, introducing an additional degree of freedom. Let us denote the contact manifold (T*R R. 17) the augmented cotangent bundle of Riemannian manifold (R. .). In this setting, contact dynamics corresponds not to geodesic mo- tion directly on R, but rather on an augmented state-tima manifold R x R. The projection these geodesics onto R. yields family of trajectories, each corresponding to a
15

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
B. Methodology Details
B.2. Integrating Contact Hamiltonian Flow
This section presents the training algorithms for the single contactomorphism and ensemble approaches. We also outline the integration method for contact Hamiltonian dynamics, essential to implementing contactomorphism net- works. Finally, we examine the implications of preserving the contact structure for the physical interpretability of the ambient dynamics.
A contact Hamiltonian dynamic system can be defined using contact Hamiltonian function H on contact manifold n). The contact form 7 establishes connection be- tween H and its associated vector field X via Equation (2), In local coordinates, the vector field X xr is expressed as,
2H 4:
Dp:
X
pi
dg.
0H
Pi
0H ūs
(31)
B.1. Training Algorithms
3 p: dp.
DH
H.
Algorithm 1 Training a Contactomorphism (T)
This vector field generates a flow up satisfying
Input: Dataset: (xx(t) 1 (E(0.ts
bE[1.B]
_ pi( (x))
( (x)
po(x)) = x.
(32)
Output: A trained contactomorphism (T)
1: for to F do
D Iterate over epochs
The flow preserves the contact form v7 up to a scaling fac- tor a E R. such that yo = on). Therefore, the resulting transformation x(t) (() is formally contactomor- phism within (M, n). However, the numerical integration of this flow introduce errors, depending the method used. Contact splitting integrators are a family of numerical algorithms to integrate contact Hamiltonian flows while pre- serving the contact structure (Zadra, 2023). This algorithm applics to Hamiltonian functions that can be expressed as sum of separate terms, with the state {q, P, s} updated in corresponding number of steps. In case, the Hamilto- nian function (11) of the individual contactomorphism (rro, consists of three terms, resulting in the following three-step update process,
2: for b = to B do
Iterate over trajectories
3:
{Z)(t)]e(oo.al = (T') (eejot)) > Map the trajectories into the latent space using (9)
4: (ee(t)) = 1 (0)):
Integrate the latent dynamics according to flow (6)
5:
(Ettoo)) ,F (T) ((Eoo)))
b Map back to the ambient space with (9)
6:
argmin (t), (t), (t), (t)):
Update sp (2)) by minimizing loss (12)
7: end for end for
8:
9/+1 = qi
Algorithm 2 Training the Ensemble (prm (T) InE(L.M)
1.
Pj+1 Pj -Pp;F, + VF5sj)T
(33a)
Input: Dataset: (xx(t))(E)).
(8/1 = 8, - -FJSTT
Output: A trained ensemble {prm (T)
4/42 qj+1
1: for e == 1 to E do
D Iterate over epochs
2.
Pソ+2 = Pj+1 VV++1
(33b)
2: for = 1 to do
>> Iterate over trajectories
88+2 = 8j+1 VJ+IT,
3:
for to No do
9j+3 4j+2 + (Mz+ + DJ42 VM Pf4+2)
4:
(te p, (T)) (t)
D Map the trajectories into the latent space using (14a)
3.
Pj+3 = Pj+2,
5: end for
8j+3 8j+2 + PD (P VMH+2 M5+2) Pj+2T; (33c)
6: {z% (t)ea)) = (t) (0)));
Integrate the latent dynamics according to flow (6)
where Mj =M(P), V, V(g,), and Fj F(g)) This integration method has the key advantage of being analyt- ically invertible. As result, a contactomorphism imple- mented according to this scheme can be efficiently reversed with minimal computational cost.
7:
for j RandomShuffle(L,22
.N do
8:
[x] (t) = Φ. (T) (t) (Eefo.))))
> Map back to the ambient space with (14b)
9: end for
10:
argmin Em (1), (t),,, (t), (t));
P Update the ensemble by minimizing loss (12)
B.3. Approximation Error of the Contactomorphism
11:
end for
We experimentally evaluate the numerical error introduced by our implementation of the mapping between the ambient
12:
end for
16

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
and latent states by leveraging the analytical contact trans- formation conditions established in (Bravetti et al., 2017),
C. Implementation Details
C.1. Importance of Contact Structure Preservation
d& di:
da
The transformation fr (M, - 7") is designed to preserve the contact structure, ensuring that the physical relationships between conjugate variables in the latent base- line dynamics are conserved. The theoretical justification for this is provided in Section 3 and Appendix A B.4. Here, we empirically evaluate its importance by examining the consequences of replacing contactomorphisms by standard diffeomorphisms. This substitution disrupts the underly- ing physical coherence, leading to degraded reconstruction and generalization performance. Figure 11 qualitatively illustrates the impact of replacing contactomorphisms by diffeomorphisms, highlighting distortions in both the recon- struction and generalization of the LEAF 2 character from the LASA dataset. Figure 12 shows a more elaborated com- parison of generalization performance, by considering pre- dicted trajectories initialized from grid of points in the state space. The use of contactomorphisms shows significantly faster convergence toward the data manifold, measured as the percentage of time steps spent near the demonstrations over the trajectory's ' duration. Tables 7 and 8 quantitatively confirm the degradation in reconstruction and generaliza- tion accuracy when contact structure preservation is not enforced.
P: P.P:
+ Ps
(34a)
ds
ds
dq.
dgi
d8 dû: = 0.
p.
(34b)
dp;
dpp
Here, the coordinates (q:, Pt. and (qi, Ps. 8) represent the variables before and after the contact transformation, re- spectively. The partial derivatives correspond to entries in the Jacobian of the contactomorphism. By computing the difference between the left-hand and right-hand sides of the conditions above, we consistently observe a numerical error lower than le- 5, confirming that the implemented transfor- mation closely satisfies the contactomorphism conditions.
B.4. Physical Interpretability
Equipping the latent dynamics with a contactomorphism allows us model a dynamical system directly in ambient coordinates. To predict the ambient trajectory from initial point Xo. we first map it to the latent space via the learned contactomorphism, Zo- = yp, (x)).. We then integrate the latent dynamics to obtain (t) = Pg (Zo), and finally map the result back to the ambient space using the inverse contac- tomorphism, x(t) = up_ (z()). Formally, this process is expressed as a composition of contact Hamiltonian flows, x(1) = p7 (T) o (t) o sp, (T))), i E [0, 41) (35)
Table 7. Reconstruction error via DTWD for the ablation study on the structure of the ambient space latent space mapping
Note that the composition of Hamiltonian paths is itself a Hamiltonian path. The corresponding Hamiltonian function, associated with the Riemannian metric r.. can be derived using (Polterovich, 2012, Proposition I.4.D),
Character Contactomorphism Diffeomorphism
0.44
1.15
S
0.43
0.96
Hm H, + Hy(p, ()
(36)
where the Hamiltonian of the contactomorphism H, is,
Table 8, Generalization measured as average ratio of convergence for the ablation study the structure of the mapping between the ambient space and the latent space
K
TH=H,, + Σ ( 0... (p..oo)...00)) (37)
1=1
Character Contactomorphism
Diffcomorphism
Preserving the contact structure ensures that the predicted ambient dynamics follow contact Hamiltonian form, making them physically interpretable. Specifically, using a contactomorphism instead of a generic diffcomorphism allows for the derivation of an analytical expression for the Hamiltonian function in the ambient space. The damping coefficient, 0Hm/ embedded in the contact form, quamtifies energy dissipation or absorption as described by Equation (8), and governs the interplay between conjugate variables. While GCF morphs the latent contact form n/ and modulates the damping coefficient, it also preserves their relationship, ensuring physical consistency. Figure/ shows that GCF yields a damping coefficient consistent with the real system demonstrating its robustness in modeling conjugate variable interactions.
0.61
0.08
S
0.62
0. 0.12
C.2. Training Details and Weights Ablation
The weights employed in the loss function (12) are w, = 1 and wa 0.01. Training is conducted for 5000 epochs, taking approximately 4 hours on average. Details of the machine used for training are provided in Appendix D.I. The initial learning rate is x 10-3 and is reduced by a factor of 0.9 on plateaus observed for 200 epochs. The loss is clipped at x 10 and the gradient is clipped at 0.1. Optimization is performed using the Adam optimizer with
17

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
1.00
Figure 11. This figure illustrates the reconstruction and generaliza- tion results for the character LEAF_2 when the contactomorphism connecting the ambient and latent spaces is replaced by diffeo- morphism. As a result, the Hamiltonian structure of the ambient dynamics is lost, removing the physical bias that aids in accurately reconstructing the dynamics. Consequently, the trajectories in the ambient space appear more distorted, especially near the target.
0.75
0.50
0.25
0.00
Contactomorphism
Diffeomorphism
default hyperparameters, and 1.2 regularization is applied to the weights with a coefficient of 1 x 10-10, While the ambi- ent weighting factor We focuses on the reconstruction of the demonstrations in the ambient space, the latent weighting factor w. plays a crucial role in regularizing the deformation of the latent space.
Figure 12. The top plots show trajectory predictions from grid points for the character LEAF_2, comparing a model using a sin- gle contactomorphism (left) to one using naive diffeomorphism without physical structure (right). The bottom chart summarizes these trajectories hy representing them points based on the ratio of time steps spent within the data manifold to the total number time steps. Removing the physical structure disrupts the relation- ships between state components in the ambient space, negatively impacting generalization.
During training, detailed in Algorithm 1, the integration of the latent dynamics (t) depends only on the initial ref- erence state (0) and remains independent of the rest of the latent demonstrations (t). The inverse contactomor- phism p. is applied at each point along 2n (t) to recover the corresponding ambient trajectory (t), while the forward contactomorphism yor is used only map the initial states of the demonstrations to the latent space. As result, if train- ing relies solely on the ambient loss term, the latent space receives limited structural guidance. Since the contactomor- phism defines the mapping from ambient to latent space, this can lead to irregularities or lack of smoothness the la- tent representation. In contrast, the latent loss term enforces alignment between the reference trajectory (t) and the in- tegrated dynamics (t). thereby promoting smoother and more coherent latent mapping. This regularization improves the latent space structure, leading to more accurate ambient reconstructions and mitigating unpredictable behavior out- side the data manifold. Ultimately, it supports a more stable and generalizable representation. Figure 13 compares the ambient and latent reconstructions of the demonstrations for three different values of ws, illustrating how small weight can serve as trade-off to enhance both reconstructions. A quantitative analysis of this ablation study is reported in Table 16.
Table 9, Reconstruction error via DTWD for weight loss ablation
Wx
Latent Space Ambient Space
0
0.
5.140.53
0.001
0.50
4.940.43
0.01
0.44
1.940.27
0.1
0.56
1.640.19
1
0.69
0.96
nents representing the inertia, potential energy, and damping coefficient of the Hamiltonian system, respectively. Learn- ing all these components enables the resulting contactomor- phism to effectively model a wide range of target dynam- ics. Table 10 reports reconstruction scores for the character LEAF 2 from the LASA dataset, comparing the performance of the full Hamiltonian with that of simplified versions us- ing fewer parametrized components. The fully parametrized Hamiltonian consistently achieves the best reconstruction accuracy. These functions are implemented using Random Fourier Features Networks (RFFNs), which were selected due to their superior performance in the ablation study re- ported in Table 11. RFFNs provide a strong balance of expressivity and smoothness, making them particularly well- suited for gradient- based training in this context.
C.3. Hamiltonian Function Parametrization
Each individual parametrized flow Proy which collectively defines the contactomorphism 4y, integrates the vector field associated with the Hamiltonian function in Equation (11). The functions Mo, (p), Vo, (q), Fo, (q) are learned compo-
18

---

Geometric Contact Flows: Contactomorphisms for Dynamics and Control
x10 x10
D. Experimental Setup
Uncerande na
D.1. System Configuration and Architectural Settings
Laggangiian
2
The framework runs on a machine equipped with 13th Gen Intel Core i7- 17-13850HX CPUs. Positional data normal- ized to the range [-0.5, 0.5]. Other coordinates adjusted accordingly maintain physical coherence. Additionally, velocity data is normalized to |-1, 1] by scaling the dura- tion of the trajectory. In the experiments, the input to our framework the current system state at lx. while the output is the next state at f This prediction is repeated at each time step to reconstruct the full dynamics. Our approach treats the dynamical system as autonomous, excluding time explicit input. The number of contactomorphisms N used in the ensemble depends on the task, whether it involves reconstructing a dynamical system generalizing demonstrated trajectory for control execution. In the first case (e.g.. the spring mesh and quantum system), a single contactomorphism (N = 1) is used. the second case (e.g.. the handwriting dataset and robotics tests), the ensemble consists of N contactomorphismms, In the latent space, the stable baseline dynamics given by Equation (7b) is used in all experiments. Additionally, for the execution of the robotics tasks, the safe dynamics from Equation (7c) is also employed to leverage its energy-aware aware behavior.
x10-2
x10
Lagrang Action
Uncertainty
5
x10-1
x10-
Lagranpiann Action
Uncertaind
N
Figure 13. Demonstration reconstruction in the ambient space (left) and latent space (right) for different values of the latent weighting factor Wy: W2 0 (top), wN 0.01 (middle), and wa (bottom).
Table Reconstruction error character LEAF_2 DTWD for different parametrization of the contact Hamiltonian
Hamiltonian
Score
Each contactomorphism (r. composed of sequence of K 12 individual parametrized flows irr,, with dura- tion of r 0.2 seconds each. The scalar functions defining each flow, Me, (p). Ve, (q), Fe, (q), are implemented using Random Fourier Features networks with n, = 200 hidden units and a kernel bandwidth of 1. A tanh activation func- tion bounds the output of these scalar functions to the range [-2, 2]. Across experiments, the GCF networks consist of 7200 trainable parameters. The baseline models were evalu- ated with various parameter configurations, ranging from the original author-released versions to implementations with parameter counts comparable to that of GCF, Ultimately, we selected the architecture that yielded the best performance for each method (EF: 2000, NCDS: 2000, DHNN: 4572, HNN: 2286, MLP: 7168).
M(P)p + V(q) + F(g)s
0.44
P p V(q) F(q)s
0.61
P M(P)p V(q)
0.94
P p V(q)
2.14
Table 11. Reconstruction error via DTWD for different learning models used to reconstruct the contact Hamiltonian.
Character
MLP
RBF
RFEN
0.94
0.44 1..8
10.12
0.
0.48
0.43
D.2. Robot setup.
The WRAP-AND-PULLI and the DISHWASHER LOADING robotics tasks employ a 7-DoF Franka Emika Panda robot as the test platform, with ROS2 acting as the middleware to interface either the GCF or the state-of-the-art bench- mark method with the robot. The trajectory predicted by the GCF or the benchmark method is sent to low-level Cartesian Impedance Controller, which is built on top of the Franka torque control system using libfranka functionalities. Via the Franka Control Interface (FCI), motor signals are sent to the robotic manipulator in real-time at rate of 1 KHz. The GCF learning model, trained according to the
19

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
task demonstrations, is loaded by a control node operating at frequency of 10 Hz. This node receives state measure- ments from the robot, normalizes the data, and computes the Lagrangian action before calling the model to predict dy- namics for the subsequent? seconds. At each timestep, the GCF node checks whether the GCF learning model is avail- able for computation. If the model is not already engaged, it processes the latest robot state. The node then updates the Impedance Controller with new frame of the predicted dynamics. Whenever a new trajectory becomes available, the previously executed trajectory is replaced and updated accordingly. The control node used for the state-of-the-art benchmarks shares the same features.
By substituting Equation (39) into (40), we obtain,
(q"à q 6) - Hoe Ja ( (34 (-
(41)
Differentiating this equation with respect to time, we derive,
- (24 (--((a)=)) (0(8, 8,4. q) - -) = 0, (42a)
//4 q 3(q) - (a q - 5) + 2d T a.
p(s, 3, q. a)
8
(42b)
which implies:
A 9 q B(q) -
24 q.
(43)
D.3. Conversion to Canonical Coordinates
This differential equation can be integrated forward to obtain the full time series of Lagrangian actions s, given the initial conditions,
In classical mechanics, the state of physical systems is typi- cally described using positions and velocities {q. q), rather than the canonical coordinates {q, p, 8), required in the con- tact Hamiltonian formulation. Therefore, prior to inferring the network, it is necessary perform conversion and estimate the action Lagrangian value s, which is not directly observable in many applications.
0, so qo T qu.
80-
(44)
Through this process, we obtain a description of the dynam- ics in canonical coordinates {q. s)
Robotic Tasks. The dynamical system under consider- ation consists of the workspace and manipulated objects, represented by the rope. We have access to the end-effector position and velocity, de- noted as {q. q}. To transform these variables into canonical coordinates in real time, we first use the dynamical model of the robot to compute the workspace inertia matrix A. As- suming a Hamiltonian in mechanical form, the canonical momentum obtained as p = Aq. The total energy of the system at given time can be modeled as the sum of the robot kinetic energy, the dissipation term, and the energy stored in the environment, which is represented by the dis- placement of the rope as a result of the work performed by the robot. The total energy is therefore expressed as,
Panda robot and its
Spring Mesh System and Handwriting Datasets. The spring mesh dataset, along with the LASA and Di handwriting datasets, represent the system dynamics using the state variables q, q. To transform these data into canon- ical coordinates, we assume that the dynamical system is characterized by unitary mass, implying i q= p, and gov- erned by a contact Hamiltonian in mechanical form, defined as,
1 p B(a)) + +7(4)s,
H =
(38)
where ((q) is the potential energy, and ((4) denotes the damping coefficient. Depending on its sign, (q)) can ac- count for either energy dissipation (7 > 0) or energy genera- characterized tion (- 0). In this context, the dynamies
H(t) = 1 Aq T 2
f dr.
(45)
by an attraction point, and we model the potential function during demonstrations as linearly decreasing toward zero. In order to compute the Lagrangian action s, the potential energy is parametrized as B(q)) = T - I, where T represents the final time of the trajectory.
where f. represents the external force applied at the end- effector and Eq denotes an infinitesimal robot motion. The integral term fn f. Đq dr is negative during a pulling action and includes both the conservative and conseevsevveeive components of the robot interaction with the environment.
According to the Maupertuis' principle of least action, the variation of is.
The evolution of the Lagrangian action s is governed by the Maupertuis' principle of least action according to the equation,
1999-9 9 3(q)
(39)
$ = q Aq - s+ S.' f. dr.
(46)
In addition, as derived from Noether (1918), the variation of the contact Hamiltonian function (38) regulated by,
This differential equation shows how can be updated at every time instant according to the measured values of q and f.. Through this process, we obtain a description of the dynamics in canonical coordinates (q. 8).
( a T 2
+ + 3(g) + +7()))) = Hoe So (())dt
(40)
20

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Geometric Contact Flows: Contactomorphisms for Dynamics and Control
E. Extended Results
Table 12. Spring Mesh Parameters
E.1. Spring Mesh System
Parameter
Values
Spring networks are meshes of particles connected by springs, effectively modeling deformable surfaces and solid materials (Pfaff et al., 2021). They serve as simple yet versatile proxies for a wide range of physical scenarios in- volving deformable solids and cloth, including mechanical simulations of material deformation such as finite element modeling, as well as robotics tasks and computer graph- ics applications. By benchmarking on a spring mesh, we can examine how Geometric Contact Flows learns large de- formations, wave propagation through a membrane, or the impact of damping. Each pair of connected particles (a, b) exchanges spring forces regulated by the Hooke's 's law,
damping coefficient y [kg/s]
0.1
mass [kg]
stiffness & [N/m]
initial displacement r [m]
Uniform (0.1, 0.35)
Nodia
Node
0.2
Postiomn Pene
Postioo Plane
GCF
DCP
Drthk
1NW
INN
RLP
MLP
EF
NCDS
KCDS
Dynamics
Dyramics
0.3
-d2
1.0
1/6
0.0
-0.2
1.0
Node
Postion
GCT
0.9
DHNN
fab = -- - -1lqa -4i12 12-
lab)
|qqa
9a qb
10ll2
19a, (47)
0.5
HNN
MLP
NCDS
-3.3
Dynamics
-3.0
-02
0.4
1.0
91
where lab is the rest length of the spring. 7 is the damping coefficient, and k is the stiffness. The coupling of multi- ple springs leads to complex large-scale deformations and oscillatory behavior.
Figure 14. Dynamics reconstruction of the first three nodes in the spring mesh experiment. The reference dynamics (dashed gray line) is compared against the predictions from GCF and other baseline models.
In this assessment, we use the 60-dimensional dataset origi- nally introduced by Otness et al. (2021), as part of com- prehensive benchmark for learning physical systems. The system comprises 2D square grid of particles (4 x 4), con- nected by springs, with the top row of particles fixed in place. Springs are placed along both the axis-aligned edges and the diagonals of each grid square. The rest lengths of the springs are chosen such that the uniformly spaced particles remain at equilibrium in their initial configuration. Initial conditions are generated by perturbing the positions of all non-fixed particles. These perturbations are drawn as uniform random vectors within a circle of radius r, en- suring that all sampled initial conditions have zero initial momentum. The training dataset comprises 20 spring-mesh systems, each characterized by a distinct set of initial con- ditions. The system parameters are detailed in Table 12. Each system is simulated over a time horizon of 8 seconds, capturing the complete trajectories of particle displacements and momenta. The resulting dataset offers comprehensive representation of the system's 's dynamical behavior across the explored parameter space. Figure 14 illustrates the os- cillatory behavior of the first three nodes in position space, comparing the true dynamics with the predictions of the GCF approach and baseline models. Quantitative metrics evaluating the accuracy of the dynamics prediction are pre- sented in Table 1. The GCF model achieves a 57% reduction in reconstruction error compared to DHNN, the next-best baseline, owing to the convergence biases integrated into the network architecture.
E.2. Quantum Mechanical System
Contact Hamiltonian formulation of quantum systems. The versatility of the contact Hamiltonian framework ex- tends its applicability beyond classical mechanics, enabling the description of thermodynamic processes (Bravetti. 2019; Simoes et al., 2020) and certain aspects of quantum mechan- ical systems (Woit et al., 2017; Herezeg & Waldron, 2018). Although this formulation cannot account for decoherence or other effects characteristic of open quantum systems, it provides natural extension of the Schrödinger equa- tion for the evolution of pure states in finite-level quantum systems (Ciaglia et al.. 2018). Some authors (Cruz, 2018 8; Bravetti et al., 2017) proposed to use the contact formulation in the context of canonical quantization in position repre- sentation. However, this approach has the drawback of not preserving the norm of the wave function in the Schrödinger equation.
In order to facilitate the application of contact geometry in quantum mechanics, we eliminate the detrimental effects of dissipation on the probability distribution by reformulating the Schrödinger equation in terms of the Madclung repre- sentation. Starting from the standard Schrödinger equation,
OV Ot
ih
AVV
(48)
where h is the reduced Planck constant, V is the wave func-
21

---

Geometric Contact Flows: Contactomorphisms for Dynamics and Control
tion, and the Hamiltonian operator is given by
To account for small fluctuations arising from experimental imperfections, calibration errors, and unavoidable environ- mental disturbances, we introduce a stochastic perturbation to the system's 's evolution. Unlike open quantum system ap- proaches, this modification does not lead to decoherence, en- suring that the system remains in a pure state while incorpo- rating realistic uncertainties. The deterministic Schrödinger equation (48) is therefore extended to,
12
+ (Q), 2m
(49)
expressed in terms of the position and momentum operators, q and p. we rewrite the wave function in polar form as
V
(50)
where the probability density function, p | corre- sponds to the squared amplitude of the wave function, while represents the phase, related to the classical action. Sub- stituting this expression into the Schrödinger equation and separating real and imaginary components, we obtain two valled equations. The first one is the continuity equa- tion,
ih dv diii + ih 8 dW,V,
(57)
where dW, is a Wiener process representing stochastic flue- tuations, and ó determines their magnitude. In this experi- ment, 20 dynamical systems have been generated by inte- grating the Madelung representation of Equation 57 for 8 seconds, i.c.,
Pp 0
+ V (pV 9s) 0.
(51)
0p Vg (pV 98) + + h 2 Agp:
(58a)
ōt
The second one is the quantum Hamilton-Jacobe equation,
Ös
Ot + (V + + Q -
(58b)
as
(V
2m
+V+ =0
(52)
Ot
2m
The parameters of the quantum system were sampled from uniform distributions, as detailed in Table 13. We evaluate the performance of GCF in reconstructing the dynamics of the expected values of the position and momentum operators, q and p. as well as the phase variable s, in comparison to the baselines. Unlike other tests, the HNN and DHNN frameworks cannot be applied to the quantum case, as they operate only in even-dimensional phase spaces and cannot model the additional odd variable s. Stochastic results are summarized in Table 2. GCF reduces reconstruction error by 60% in modeling the system's is evolution. Figure 15 illustrates an example of a single trajectory, comparing the reconstructions produced by the models.
where,
R12 DvVP
Q
(53)
2m
VP
is the quantum potential, with the operator Ag y denoting the Laplacian. The Madelung equations are particularly suitable for adaptation within the contact Hamiltonian framework, as the extension of the Hamiltonian operator (49) the contact Hamiltonian operator,
A
2m +
V()) + 78,
(54)
modifies the quantum Momilton-Jacobb equation (52) to,
0.00
3.00
BCF
Ōs
(V,s) 2m
Expected teettedto Mommentum
0.75
EF
MLP
Ot
+V + Q 7 0.
(55)
Expected
0.50
GCF
0.50
NCDS
Dynamics
-075
M.P
This introduces dissipation into the contact system's 's evolu- tion without altering the properties of the probability distri- bution, which remain governed by the continuity equation (51).
0.00
11.00
NCDS
Dynamics
-0.25
-1.25
200
600
450
600
Time
-0.30
-0.5
Quantum dynamics reconstruction In this test, we focus on reconstructing the dynamics of a single-mode bosonic system, whose Hamiltonian consists of a linear term rep- resenting the free evolution of the mode, a squeezing-le interaction, and a cubic nonlinearity. Expressed in terms of the position and momentum operators, the contact Hamilto- nian is given by.
Agganngin
0.40
GCF
-0.45
MLP
-0.50
NCDS
0.53
Dynamics
200
400
600
Time
Figure 15. Trajectories of position q. momentum P. and La- grangian action for the evolution of a single- phase bosonic sys- tem. The trajectory obtained from integrating the Schrödinger equation (gray dashed line) is compared against the predicted evo- lution from the GCF model and the baselines.
2 + 3
w0p A = 2
696G² - p2² + 77s (56)
22

---

Geometric Contact Flows: Contactomorphisms for Dynamics and Control
Table 13. Quantum Parameters
Parameter
Values
Natural frequency
Uniform(1,22
Interactions coupling term x
Uniform (0.05, 0.1)
Cubic non-linearity coefficient 3 Damping coefficient ?
Uniform (0.05, 0.2)
(a) EF
(b) NCDS
(c) DHNN
(d) GCF
Uniform (0.3, 0.6)
Noise magnitude 5
Uniform (0.01, 0.1)
E.3. Handwriting Datasets
(c) EF
(f) NCDS
DHNN
(h) GCF
1.00
1.00
In motion generation frameworks, learned dynamics are viewed as skills that an actuated system must not only im- itate but also generalize and adapt to new scenarios. The LASA dataset (Lemme et al., 2015), widely used bench- mark, contains 2D handwritten characters but lacks self- crossing trajectories, as it was designed for first-order dy- namics. To better showcase GCF's capabilities, we also include the DigiLeTss dataset (Fabi et al., 2022), which fea- tures more diverse trajectory patterns. Since the demon- strated dynamics converge to a target, we exclude HNN from the baselines, as it is limited to modeling periodic behaviors.
0.75
0.75
0.50
0.25
0.25
0.00
NCDS
DHNN
GCF
0.00
EF
NCDS
DHNN GCF
Time spent on the man- (j) Time spent on the data man- ifold for trajectories generaliz- ifold for trajectories generaliz- ing the character LEAF_2 _2. ing the character EL..F.
Figure 16. Generalization of the EF, NCDS, DHNN, and GCF methods. Predicted trajectories starting from grid points for the character LEAF 2 (16a, 16c, 16d) and for the character ELLE (16c, 16g, 16h). The gray dots (16i, 16j) represent these trajectories as points based on the ratio between time steps spent within the data manifold and the total time steps. The violin and box plots illustrate the stochastic distribution of these ratios. A high ratio indicates quick convergence. NCDS's contractive properties yield good results on LEAF-2, 2, but struggle to converge in ELLE. In contrast, the GCF method demonstrates reliable performance on both characters, achieving the highest average ratio and lowest standard deviation, indicating consistent convergence even for trajectories initialized far from the training dataset.
Dynamics reconstruction. Table 14 compares GCF and baselines on handwriting dynamics reconstruction. It in- cludes four characters from the LASA dataset and four from the DigiLeTTs dataset.
Table 14. Reconstruction error via DTWD on handwriting datasets
Character
EF
NCDS
DHNN
GCF
3
0.43
40.4
0.44 1.65
S
0.45
0.47
0.53
40.07
V
0.42
0.49
0.44 0..3
GCF delivers more reliable results, maintaining consistent performance across both types of dynamics (Figures 16h, 16d). Figures 16i and 16j provide statistical analysis of these results. Each point, representing predicted trajectory, is computed as the ratio between time steps spent within the data manifold (in the position space) and the total time steps. As the DHNN method imposes no constraints outside the data manifold, only small number of trajectories initiated outside the data support converge, indicated by the large proportion of points with ratio of 0. EF guarantees asymp- totic stability, ensuring that trajectories initiated outside the data support eventually converge to target. However, it does not explicitly drive trajectories toward the data man- ifold, leading to a uniform distribution of ratios. NCDS improves the results for the character LEAF 2 due to its guaranteed contractive behavior. However, GCF ensures that all trajectories from both LEAF_2 and ELLE reliably converge to the data manifold. On average, these trajectories spend significantly more time within the data support than outside it, even when initiated far from the data support. Table 15 reports generalization statistics for a set of hand- written characters. GCF stands out for its ability to avoid
G
0.85
0.
0.71
0.68 00.24
L
2. 2.79+0.09 2.22+0.03
0.82+0.22
0.70+0.36
d
3.13 3.08
1.08
0.93
e
2..73
0.71 0.78 0.30
4
1..8
3.1610.10
1.10
0.86
Dynamics generalization. Figure 16 presents generaliza- tion tests for EF, NCDS, DHNN, and GCF using two exam- ple handwritten characters. The position space is divided into a grid, with each point serving as starting location for trajectory predictions, while the remaining state variables are initialized to zero. For DHNN, this means starting with zero velocities; for GCF, both velocities and the Lagrangian action are initialized zero. These tests evaluate how well each model performs when dynamics are initiated outside the data manifold. The trajectories predicted by NCDS for the character LEAF 2 (Figure 16b) visibly converge and remain close to the demonstrated paths. However, NCDS operates only on first-order dynamics and fails to reconstruct second-order trajectories, as seen in Figure 16f. contrast,
23

---

Geometric Contact Flows: Contactomorphisms for Dynamics and Control
uncertain regions and converge the data manifold.
Table 15. Generalization measured as average ratio of convergence
Character
EF
NCDS
DHINN
GCF
0.49
0.18 0.94
1.69
S
0.39
0.22
0.13
10.12
0..1
0.22
0.1840.10
0.62+0.23
(a) Generalization with single (b) Generalization with the en-
G
0.33
0.21 1.
0.68
contactomorphism (I.FAF 2). semble (LEAF 2).
L
0..3
0.17 4..35
0.66
d
1..4
0.17 00.39
0.64
e
0.19 0..4 4.39
0..1
4
0.58
0.51
0.18
0.65+0..15
(c) Generalization with single (d) Generalization with the en- contactomorphism (ELLE).
Ablation Study on Ensemble Contribution The general- ization results achieved by GCF, previously shown in Figure
semble (ELLE).
1.00
1.00
16, benefit from the ensemble of contactomorphisms. Figure
0.75
0.75
17 contrasts these results from the complete GCF framework with those from variant using a single contactomorphism. The ensemble approach, which incorporates uncertainty- aware curves, demonstrates faster convergence to the data manifold. For trajectories initiated from points already close is
0.50
0.50
0.25
0.25
0.00
Single
Ensemble
0,00
Single
Ensemble
(c) Time spent the data (f) Time spent the data manifold for LEAF_2.
man- ifold for ELLE
the data manifold, the difference in convergence less pronounced, reflected in the violin plots (Figs. 17c, 17f), where the distributions of points near 1 are similar. However, for trajectories starting farther away from the data manifold, the effect of the ensemble uncertainty is more evident, contributing to a higher overall aggregated metric. Generalization statistics for set of handwritten characters are presented Table 16.
Figure 17. The ablation study highlights the performance improve- ment in generalization capability achieved through the use of an ensemble of contactomorphisms. Plots 17a and 17b depict trajec- tory predictions from grid points for the character LEAF_2, while plots 17c and 17d present predictions for the character ELLE. The bottom charts (17e and 17f) summarize these trajectories by repre- senting them points hased the ratio of time steps spent within the data manifold to the total number of time steps. The violin plots visualize stochastic distribution these ratios, where higher ratio indicates faster convergence. The use of the ensemble enhances convergence toward data manifold, resulting in a higher ratio and lower standard deviation.
Table 16. Generalization measured as average ratio of convergence
Character
Single
Ensemble
0.61 10.21
0.6940.19
0.62
0.67
V
0.5640.26
0.62:t.23
G
0.64
0.68
E.4. Robot Tests
l
0.61
0.66...12
In robotics, learning interaction tasks from demonstra- tions is more challenging than learning free-space mo- tions (Scherzinger et al., 2019; Le et al., 2021). GCF excels in this context by modeling energy behaviors beyond simple motion and flexibly generalizing learned dynamics when the system encounters unforeseen regions of the state space, such as unexpected positions, velocities, or external forces. This makes GCF particularly well-suited for online robot control in interaction tasks. We apply GCF to two robotics tasks: proof-of-concept PRRAP-AND-PLLA task and more realistic DISHWASHER LOADING task.
d
0.57 +0.21
0.64
e
0.61+0.12
0.67
4
0.57 10.18
0.65
Unsafe Regions on the Handwriting Datasets Figure 18 shows the trajectories and the distances to both the obsta- ele and the data manifold for the handwriting characters LEAF 2 and EL.EE.
24

---

Geometric Contact Flows: Contactomorphisms for Dynamics and Control
x10-1
x1o-1
learned trajectory and successfully grasps the basket, even when disturbances (such as the operator inadvertently dis- placing the robot) cause deviations from the data manifold. This robustness is essential in collaborative settings where humans and robots share the same workspace and may un- intentionally interfere with each other's actions. Moreover, the safe latent dynamics described in Equation (7c) allow the robot respond appropriately when the task cannot be completed due to environmental contingencies. For instance, if an object is poorly positioned and blocks the basket from closing, the robot detects that the required energy exceeds levels encountered during training and stops exerting force, thereby avoiding potentially damaging overexertion. Also this experiment available in the video accompanying this paper.
105
Lagrangian Actios
Obstacle Distance
1.0
Uncorsintyy
0.05
0.5
0.00
0.0
Data Distance
100
steps
x10-2
x10
1.00
Lagrangan Action
Obstacie Distance
0.10
0.75
Uncertainty
1.4
0.05
0.50
0.25
0.00
Data Distance
0.00
200
400
600
Time steps
Figure 18. Obstacle avoidance behavior demonstrated by the LEAF_2 2 (lop) and the ELLE (bottom) characters. Obstacles are represented as red dots.
Space Latent Space
0.0
Lagring an Action
-0.4
Reaching
Reaching
Pulling
-0.6
Programmed Path (1, 3,
-0.8
Operator intervention
Programmed Path (1. 2. 3, 4)
Demonstrations
Operator Intervention
12
500
Time 1500
1000
Figure 19. Position and Lagrangian action trajectories (left and right plots) for the unloaded WR WRAP-AND-PULL task when dis- turbed by the operator's physical intervention.
Pulling
Dish Placement
Pushing
WRAP-AND-PULL Task under Physical Perturbations Figure 19 shows the robot trajectories in the position and Lagrangian action spaces when physically perturbed by an operator who deviates it from its nominal path. The robot's ability to recover the wrapping and pulling behavior even under physical perturbations showcases GCF's robustness. The Lagrangian action s, governed by the Maupertius' prin- ciple (see Appendix D.3), decreases during the pulling task as a result of the work performed by the robot on the en- vironment. However, it increases when the environment, represented by the operator, performs work on the robot. Consequently, the robot gains additional energy to continue progressing through the task.
Pushing
Homing
Figure 20. Snapshots from the collaborative DISHWASHER LOAD- ING task, illustrating key phases: reaching the basket, pulling (opening) the basket, operator placing a dish, pushing (closing) the basket, and the robot returning to home position.
E.5. Scalability Evaluation with Synthetic Dynamics
To evaluate how GCF scales with varying target dimension- alities and model sizes, we conduct experiments on synthetic data generated from contact Hamiltonian systems with dif- ferent numbers of degrees of freedom. The dynamics are governed by single contact Hamiltonian function parame- terized by the target dimensionality d, ensuring consistent comparison across dimensions,
DISHWASHER LOADING Task Household environments represent a key frontier for collaborative robotics, where robots can assist with daily tasks, improving convenience, safety, and quality of life. particularly for the elderly or indi- viduals with limited mobility (Shafiullah et al., 2023). Fig- ure 20 shows snapshots of the main task phases: the robot reaches for and pulls out the dishwasher basket, enabling the human operator to place a dish, then pushes the basket back in and returns to its home position. Thanks to the robustness of the GCF framework, the robot reliably converges to the
d/2
d/2
1 H = == p-ppppp
2 q+
p+
Σ 92: 1 g0 Σ
$2. +s. (59)
The summation terms couple the behavior of the various degrees of freedom, producing unique dynamics for each di-
25

---

Geometric Contact Flows: Contactomorphisms for Dynamics and Control
Table 17. Inference time (seconds) as a function of the number of contactomorphism components (K) and hidden units in the RFF
networks (nz))
K
n
5
10
15
20
200 0.19440.00
0.20420.00g 003
0.216=0.0m 003
0.221+0.000
350 0.19840.000 0.207
0.218
002
0.223440000
500 0..05 0.206=0.002 0.218
0. 0.2244+0.001
mension. This coupling also ensures that the reconstruction task sufficiently challenging.
We experiment with different contactomorphisms structures (Pr, by varying two key parameters: The number K of in- dividual flows Pre, composing fPr. and the number ni, of hidden units, parametrizing each flow via RFF networks. The total number of parameters computed as K x ni X fif, where n, 3 the number of learning functions per indi- vidual flow. Figure 21 shows the reconstruction accuracy of
1.00
0.95
DTWD
0.85
System
0.80
DOFs
DOFs
10 DOFs
12 DOFs
14 DOFs
2500
5000
7500
10000
12500 15000
17500
20000 22500
of Parameters
Figure 21. Reconstruction results achieved by the GCF method, evaluated across varying numbers of parameters used to model contactomorphism, for systems different dimensionalities.
different high-dimensional GCF models, function of the total number of parameters. Reconstruction quality, mea- sured by DTWD, is normalized by the dimensionality value to ensure fair comparisons across systems. Note that GCF's performance remains stable as the problem dimensionality increases, with nigher-dimensional target dynamics ben- efiting from larger architectures, at the cost of increased inference times as reported in Table 17. However, beyond a certain size, performance may decline due to overfitting, likely caused by the limited data available.
26


Reviews:
[{"review_id": "rGWYNKiMtP", "reviewer": "Reviewer_F9hb", "summary": "The paper proposes to learn in the latent contact Hamiltonian space to inject inductive biases and encoding desirable physical properties. Additionally, the paper developed an ensemble method that aims to identify the unseen states and drive the dynamics to avoid these states. Experiments in character writing and robot-object manipulation verified the proposed method outperforms two previous works.\n\n## update after rebuttal\nI appreciate the authors' additional experiments in the rebuttal. The new simulation and robot experiments demonstrate that the proposed method can learn dynamics with more variations (the spring-mesh experiments) and address real-world problems (the robot-dishwasher experiments). I raised the score accordingly.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The evaluation tasks seem like simple trajectory generalization tasks. These tasks are state based and have very limited variations (4 characters and 1 robot trajectory).\n\n> Does the proposed method scale to more complex, real-world problems, for example ... ?\n\nYes, as emphasized in the introduction, the proposed method aims at modelling complex non-conservative dynamical systems. Its applications extend beyond trajectory synthesis for control-related tasks (robotics) to a broader modeling of intricate physical phenomena. The reviewer’s suggestions fall within the natural scope of our framework, and we thus expand our evaluation with other dynamic modeling benchmarks to further demonstrate its capabilities.\n\nTo clarify, our handwriting dataset experiments were conducted on eight characters, not four as pointed out by the reviewer. The results reported in Tables 1 and 2 in the paper represent a subset of our findings, while the complete set is detailed in Tables 6 and 7.\n\nAdded experiments:\n\n- We consider a 60-dimensional [dataset](https://github.com/karlotness/nn-benchmark) describing the dynamics of a 2D square grid of nodes connected by springs. Predicting the dynamics of mesh nodes closely parallels finite element modeling of material deformation. The coupling of multiple springs leads to complex large-scale deformations and oscillations. **GCF reduces reconstruction error by 57% across 20 different dynamic simulations with varying initial conditions**. Experiment parameters and results are available here: [sping-mesh-experiment](https://drive.google.com/file/d/1w4zLbyYc8cx0TV1hpJjqlTPG1YQnxcLW/view?usp=sharing).\n\n- We also use GCF to reconstruct the expected dynamics of a single-mode bosonic system, simulated with a stochastic Schrödinger equation. We generated 20 trajectories by integrating the equation over 8 seconds. **GCF reduces reconstruction error by 60% in modeling the system’s evolution**. Experiment parameters and results are available here: [quantum-experiment](https://drive.google.com/file/d/16ly3ixFi01tDxxZW-jkPfu2_ZBXIfIvM/view?usp=sharing).\n\n> Does the proposed method scale to high-dimensional states, for example, images (videos) or point-clouds' dynamics?\n\nAs demonstrated in the high-dimensional tests in Appendix D.5 and our new spring-mesh experiment, our method scales effectively to high-dimensional states. \nBesides, GCF has the potential to be integrated with state estimation methods that extract system variables (e.g., position and velocity) from high-dimensional representations such as images, videos, and point clouds. \n\nHowever, as this paper introduces an entirely new methodology, we focused on establishing its core capabilities and did not explore such integrations within this work.\n\n> Could the author compare the proposed method with more recent trajectory generation methods, for example, diffusion models?\n\nAs clarified above, our approach models complex non-conservative dynamical systems with physics-informed biases, capturing second-order dynamics, including self-intersections. In contrast, the referenced diffusion model-based methods are path planners, generating motion policies based on first-order Langevin dynamics, which by design cannot capture second-order systems. While self-intersecting trajectories could emerge in diffusion-based approaches due to their stochastic nature, these intersections result from the multimodal distribution of sampled actions, not from the correct modeling of physical laws. Moreover, unlike GCFs, the provided references build on diffusion models without physics-informed bias, preventing them from guaranteeing specific dynamic behaviors.\n\n> Testing on more complex datasets could be more convincing (e.g., image generation, more complex robotic manipulation tasks)\n\nFollowing the reviewer's suggestion, we tested GCF in a dishwasher-loading task. The robot handles disturbances while pulling out the basket and closing it. If obstructed, it detects excessive force and stops. Snapshots are available here: [dishwasher-experiment](https://drive.google.com/file/d/1o-jpCiV6rE21_5mSLWgK5gWgEOZd3P26/view?usp=sharing), [dishwasher-variants](https://drive.google.com/file/d/1t_bDq2wLEL8TMrh3no6jPOzGAGgax1Hh/view?usp=sharing).\n\n> No proofs in the paper. In terms of the idea that ensembling contactomorphisms, it does not only reflect the data support, it could also indicate the randomness of the data.\n\nIn our framework, comparing dynamics predictions from different contactomorphisms helps determine whether GCF can infer a principled dynamics from sufficient training data or should prioritize convergence to the data manifold when information is lacking. Regarding the reviewer's mention of \"randomness of data,\" it is unclear whether this refers to noise, multimodality, or another aspect. The cooking example does not clarify further. We kindly ask the reviewer to confirm that our interpretation of their feedback is correct, or to elaborate it otherwise.", "claims_and_evidence": "The claims in the paper are supported by experiments.", "methods_and_evaluation_criteria": "The evaluation tasks (Handwriting Datasets, Robotic Task) seem like simple trajectory generalization tasks. These tasks are state based and have very limited variations (4 characters and 1 robot trajectory). Testing on more complex datasets could be more convincing (e.g., image generation, more complex robotic manipulation tasks).", "theoretical_claims": "No proofs in the paper. In terms of the idea that ensembling contactomorphisms, it does not only reflect the data support, it could also indicate the randomness of the data. In some scenarios randomness may be preferred, e.g., asking a household robot to mix food ingredients for cooking.", "experimental_designs_or_analyses": "These two experiments in the paper are too simple. These experiments are state based, and have no variations. Consider image-based, or language-based, or the tasks involving agent-env interaction (Artari game or DeepMind control suite, https://github.com/openai/gym).", "relation_to_broader_scientific_literature": "Learning dynamics is important in the learning world models, and could benefit the learning community in general. However, the proposed method demonstrated limited dynamic learning ability in two simple tasks.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strength:\nTheoretically valid: adding inductive bias and learning physically preserving properties are meaningful for dynamics learning.\nWeakness:\nThe experiments in the paper are too simple.", "other_comments_or_suggestions": "The acronyms of the method should be GCF, but there are multiple places typed as GFC.", "questions_for_authors": "Does the proposed method scale to more complex, real-world problems, for example, learning rigid body interactions, or learning dynamics of one / two-link pendulum?\nDoes the proposed method scale to high-dimensional states, for example, images (videos) or point-clouds' dynamics?\nCould the author compare the proposed method with more recent trajectory generation methods, for example, diffusion models[1] [2]?\n\n[1] Cheng Chi, et al, Diffusion Policy: Visuomotor Policy Learning via Action Diffusion.\n[2] Michael Janner, et al, Planning with Diffusion for Flexible Behavior Synthesis.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QJYYOnIeZ3", "reviewer": "Reviewer_sRBK", "summary": "This paper introduces a geometric contact flows model based on Riemannian and contact geometry, which introduces a robust and interpretable inductive bias over the previous MLP based methods. Furthermore, the authors propose a novel framework to learn latent dynamics of contactomorphisms and generalization mechanism based on Riemannian geodesics, which also improves the model robustness. Experiments show superior performance over baseline methods on multiple tasks such as reconstructing handwriting dynamics and robotic interactions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The application of the system can generally applied to contact related tasks such as robots-object interaction and trajectory synthesis. No further broader impacts identified.\n\nWe clarify that the contact Hamiltonian biases in our framework extend beyond interaction tasks in control-based approaches. They represent fundamental physical principles applicable to modeling dissipative mechanical systems, thermodynamic processes, and quantum dynamics. To highlight this broader impact, we introduce two additional physical reconstruction experiments (spring mesh and quantum system), as detailed in our response to Reviewer F9hb.\n\n> I wonder how is the baseline models perform if the model is equipped with a stronger priors, particularly those captured from differentiable simulators.\n\nThe comparison with a simple MLP in our methodology serves to motivate our approach rather than act as the main baseline. As detailed below, the baselines used in our results are equipped with strong priors.\n\nThe referenced differentiable simulators learn policies for physically meaningful behaviors by leveraging physical simulation during training to evaluate policy performance. Since they introduce physical biases during training, there is no guarantee that the learned policies will remain physically consistent when generalized to new scenarios. In contrast, our approach (and the baselines we consider) embeds biases directly within the network structure itself.\n\n> Why are these two baselines chosen and are there other more recent works comparable?\n\nThe selected baselines are well-known for incorporating biases in learning dynamical systems, introducing features that our approach successfully recovers and generalizes:\n- Encoding desirable properties (e.g., periodicity or target convergence) in the dynamics through diffeomorphisms (EF).\n- Embedding physical relationships between the components of the system's state through Hamiltonian dynamics (DHNN).\n\nOur approach extends EF's idea of transforming latent dynamics using diffeomorphisms by considering second-order dynamics and by introducing a (more general) contact Hamiltonian structure in the diffeomorphisms to preserve conjugate pair relationships. DHNN achieves second-order modeling by embedding pure Hamiltonian dynamics in the network structure, but it lacks the ability to enforce desirable properties in the learned dynamics. The comparison with these baselines in our experiments highlights the importance of both biases. A more recent work (2024) aligned with our philosophy is [NCDS](https://openreview.net/forum?id=Q5N3P0SMRr), which extends the properties of a learned latent contractive system to the ambient space using structure-preserving transformations. To strengthen our baseline comparison, we included this approach in the [experiments](https://drive.google.com/file/d/1RcE8tb_gQxbQ3e0ERz-BUPIzk9Lk9uq-/view?usp=sharing).\n\n> how is the network designed and the geometry priors fused?\n\nThe network $\\varphi_r$ (Eq. 6), is implemented as a sequence of chained transformations $\\varphi_{r_k}$ (Eq. 7). Each of these transformations consists of three steps (Eq. 24), which updates the initial state by integrating the vector field associated to the contact Hamiltonian $H_{r_k}$ (Eq. 8). This Hamiltonian is composed of three learning functions $M(p), V(q), F(q)$, parametrized by RFFNs. Therefore, the network $\\varphi_r$, which integrates the dynamics of a sequence of Hamiltonians, characterizes a contact flow.\n\n> I also do not find enough experiments ablating the development of the modules inside the neural networks\n\nWe address the reviewer's suggestion by introducing three additional ablations:\n\n- Is the contact structure truly necessary? We assess its importance by examining the issues that arise when replacing contactomorphisms with naive diffeomorphisms, implemented similarly to EF. The disruption of physical coherence manifests in poor reconstruction and generalization performance: [contact-structure-ablation](https://drive.google.com/file/d/1OV8Wonn9_ITfwHKmRxJMrbULw0geT2Cq/view?usp=sharing)\n\n- How does the Hamiltonian function (Eq. 8) or its parametrization affect GCF performance? We test variations of the Hamiltonian function and compare different architectures for parameterizing the learning functions. The Hamiltonian that incorporates all functions achieves the best reconstruction results, while RFFN proves to be the best parametrization choice: [learning-functions-ablation](https://drive.google.com/file/d/1RiYCiGLknc4DHntgnqejI5Vx3ZJr9zRz/view?usp=sharing)\n\n- Why does the loss function have this specific form? We examine the effect of the second loss term (Eq. 9) by varying its scaling factor and analyzing the resulting performance differences. The study finds an ideal range where improved coherence of the latent space enhances reconstruction in the ambient space: [loss-term-ablation](https://drive.google.com/file/d/1hDVZR168YKjaUdgO12ALfZ-NbIfkconA/view?usp=sharing)", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes, methods are evaluated on LASA and DigiLeT datasets for handwriting trajectory reconstruction. Experiments detailed are included in supplementary materials.", "theoretical_claims": "The paper is not a theoretical paper.", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "The application of the system can generally applied to contact related tasks such as robots-object interaction and trajectory synthesis. No further broader impacts identified.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "1. The author claims that introducing Riemannian geometry as inductive bias shows improved robustness over simple MLP with no additional priors. However, I wonder how is the baseline models perform if the model is equipped with a stronger priors, particularly those captured from differentiable simulators, see some works below [R1 - R3]. While I understand the contact dynamics in these works may not be directly relatable, it would be interesting to see a results of a similar baseline or discussion on these methods.\n\n2. In quantitative experiments the authors compared with EF and DHNN without providing a detailed introduction of these two methods. Why are these two baselines chosen and are there other more recent works comparable?\n\n3. While the section 3 - 4 introduces most concepts from contact geometry, I find limited designs on the learning framework, in particular how is the network designed and the geometry priors fused? I also do not find enough experiments ablating the development of the modules inside the neural networks, make it hard to evaluate its contribution in the ML side.\n\n[R1] SimPoE: Simulated Character Control for 3D Human Pose Estimation\n[R2] Residual Force Control for Agile Human Behavior Imitation and Extended Motion Synthesis\n[R3] DeepSimHO: Stable Pose Estimation for Hand-Object Interaction via Physics Simulation", "other_comments_or_suggestions": "NA", "questions_for_authors": "Please see above weakness", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KJK01p2BAr", "reviewer": "Reviewer_89jF", "summary": "This paper introduces Geometric Contact Flows, a framework that models dynamical systems by incorporating Riemannian and contact geometry as the inductive bias. The learned latent space captures the dynamics by contactomorphically preserving the structure of the ambient space. An additional ensemble approach is proposed to model the system uncertainty. Lastly, the proposed method is evaluated on two handwritten datasets and a real-world rope-wrapping robot experiment.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The proposed system seems able to model trajectories with intersected paths. However, it is unclear whether it is overfitting to a specific trajectory\n\nOur framework reconstructs intersecting paths in position space using the full state of the system to resolve directional ambiguities. Extensive experiments (Figs 8 and 11) confirm that GCF avoids overfitting, successfully reproducing intersecting trajectories beyond the training data support.\n\n> the paper proposes an ensemble approach for uncertainty estimation. However, it is not experimentally verified\n\nNote that all the experiments use our full framework with the ensemble of contactomorphisms. The experimental setup is detailed in Appendix C.1. Additionally, Appendix D.2 presents an ablation study on the handwriting dataset, comparing GCF with the ensemble approach against a single-contactomorphism variant. The ensemble significantly improves generalization performance (Figure 9, Table 9).\n\n> I am still not convinced why contactomorphism is a better choice compared to symplectic geometry\n\nContact geometry naturally models both conservative and non-conservative systems, while symplectic geometry is limited to the former and require modifications to handle dissipation, making its dynamics not purely symplectic. Our approach constructs latent dynamics whose physical properties are fully encoded by contact geometry, so preserving the contact structure in the transformation to the ambient space is sufficient to propagate these properties intact.\n\n> The choice in (4) seems a bit arbitrary to me\n\nThe choice of latent Hamiltonian functions (Eq. 4) is a design decision driven by the properties the user aims to preserve when generalizing in the ambient space. On the data manifold, GCF can recover the demonstrated dynamics regardless of the specific latent Hamiltonian function, while outside this manifold, the latent dynamics structure acts as a physical bias to guide generalization.\n\n> The paper designs the latent dynamics to be contactomorphic to the ambient dynamics. It is theoretically sound but is not verified experimentally\n\nWe can verify this by evaluating the contact transformation equations that map the ambient state $(q, p, s)$ to the latent state $(\\hat{q}, \\hat{p}, \\hat{s})$, as established in [Bravetti et al.](https://arxiv.org/abs/1604.08266):\n\n$$\np_i \\frac{d \\hat{s}}{d s} - p_i \\hat{p}_i \\frac{d \\hat{q}}{d s_i} = - \\frac{d \\hat{s}}{d q_i} + \\hat{p}_i \\frac{d \\hat{q}}{d q_i};\n$$\n\n$$\n\\frac{d \\hat{s}}{d p_i} - \\frac{d \\hat{q}_i}{d p_i} = 0.\n$$\n\nThese partial derivatives are elements of the contactomorphism Jacobian. By evaluating these conditions, we consistently observe an error lower than $1 \\cdot 10^{-5}$, confirming that the transformation is contactomorphic.\n\n> the experimental setup of the handwritten dataset is not clearly stated\n\nIn the handwriting experiment, the input to our framework is the current system state, while the output is the next state. This prediction is repeated at each time step to reconstruct the full dynamics. Our approach treats the dynamical system as autonomous, excluding time as an explicit input.\n\n> the generalization experiment is not well described\n\nIn the generalization experiments, we use models trained to reconstruct a specific dynamical trajectory but initialize the predictions from states far outside the training data distribution. This evaluates model performance on states unseen during training. Specifically, as shown in Fig. 8, we initialize predictions from a grid of points in the position space, while setting the remaining state variables to zero.\n\n> Is the network trained and evaluated on the same trajectory? If so, would it be infeasible to apply to real-world robotics tasks as it takes 4 hours to train the network?\n\nYes, the network is trained and evaluated on the same dynamics. However, reproduction goes beyond imitation, allowing adjustments like obstacle avoidance, and ensuring convergence to the learned dynamics under different initial conditions or external disturbances. In robotics, many tasks are inherently repetitive, allowing a dynamical primitive learned in four hours to be reliably reused across various scenarios, with the robustness of the framework ensuring reliable adaptation and performance.\n\n> evaluating on some dynamic modeling benchmark could better demonstrate the impact of the proposed model\n\nWe incorporated new evaluations on material deformation and quantum dynamics simulation, detailed in our reply to Reviewer F9hb.\n\n> The proposed method is not compared with the state-of-the-art imitation learning methods\n\nWe kindly refer the reviewer to our response to Reviewer sRBK, where we emphasize the rationale behind our baselines choice. Additionally, in response to the reviewer's request, we included two new baselines: NCDS and HNN, the latter in the new experiments.\n\n> Paper can benefit from a flowchart\n\nhttps://drive.google.com/file/d/1xCe6Rh7FU0ZEwVdKfEw16cNSa1MKkS-R/view?usp=sharing", "claims_and_evidence": "The paper proposes a framework for modeling dynamical systems. The experimental results support the idea that the proposed system is able to model trajectories with intersected paths. However, it is unclear whether it is overfitting to a specific trajectory. \nSecondly, the paper proposes an ensemble approach for uncertainty estimation. However, it is not experimentally verified. \nLastly, the system is claimed to be able to perform obstacle avoidance by incorporating the energy term of the obstacle, which is verified qualitatively in the experiment.", "methods_and_evaluation_criteria": "The contactomorphic idea is interesting. However, I am still not convinced why contactomorphism is a better choice compared to symplectic geometry. Additionally, the choice in (4) seems a bit arbitrary to me.", "theoretical_claims": "The paper designs the latent dynamics to be contactomorphic to the ambient dynamics. It is theoretically sound but is not verified experimentally.", "experimental_designs_or_analyses": "The experimental results show the proposed GCF’s capability to learn dynamical trajectories with intersected paths. The proposed method also achieves lower reproduction errors. \n\nHowever, the experimental setup of the handwritten dataset is not clearly stated. The outputs are trajectories integrated from a vector field, but what are the inputs to such a system? Is it an initial value problem with initial position and time as inputs?\n\nSimilarly, the generalization experiment is not well described either. What is the task being performed? Is the task training on two sets of trajectories and evaluating on them?\n\nIs the network trained and evaluated on the same trajectory? If so, would it be infeasible to apply to real-world robotics tasks as it takes 4 hours to train the network? I understand that the paper is positioned as a dynamic modeling framework. In such cases, evaluating on some dynamic modeling benchmark could better demonstrate the impact of the proposed model.", "relation_to_broader_scientific_literature": "The paper proposes a framework for dynamic modeling using contact geometry. It is demonstrated using several trajectory reconstruction tasks in robotics. However, the proposed method is not compared with the state-of-the-art imitation learning methods. As a result, it is unclear where the proposed method stands in the literature of imitation learning. Nevertheless, the proposed model is shown to perform better than the Euclideanizing Flows and Dissipative Hamiltonian Neural Networks.", "essential_references_not_discussed": "The experiments fall into the imitation learning domain. However, it is not compared against state-of-the-art imitation learning methods. Comparing with the state-of-the-art methods in the field can greatly strengthen the paper.", "other_strengths_and_weaknesses": "Please see the comments above.", "other_comments_or_suggestions": "1. The paper can benefit from a flowchart that explains the overall architecture of the system. Currently, the overall flow is not explicitly clear.\n2. There are some typos in the paper. Ex. Ln 17 left and ln 29 right (GFC) should be (GCF)?", "questions_for_authors": "Please see the questions above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: It is an interesting and innovative idea, which is also reflected in the reviewers' consensus (cf., specific reviews below)
Decision: Accept (poster)