Contact Wasserstein Geodesics for Non-Conservative Schrödinger Bridges
Authors: Andrea Testa
Søren Hauberg
Tamim Asfour
Leonel Rozo
Venue: ICLR 2026 Poster (2026)
Keywords: Schrödinger Bridge
Generative Models
Hamiltonian
Contact Hamiltonian
Differential Geometry
Wasserstein metric
Primary area: generative models
TL;DR: We introduce a non-conservative reformulation of the Schrödinger Bridge problem, along with a ResNet parameterization that can efficiently learn its solution.

Abstract: The Schrödinger Bridge provides a principled framework for modeling stochastic processes between distributions; however, existing methods are limited by energy-conservation assumptions, which constrains the bridge's shape preventing it from model varying-energy phenomena. To overcome this, we introduce the non-conservative generalized Schrödinger bridge (NCGSB), a novel, energy-varying reformulation based on contact Hamiltonian mechanics. By allowing energy to change over time, the NCGSB provides a broader class of real-world stochastic processes, capturing richer and more faithful intermediate dynamics. By parameterizing the Wasserstein manifold, we lift the bridge problem to a tractable geodesic computation in a finite-dimensional space. Unlike computationally expensive iterative solutions, our contact Wasserstein geodesic (CWG) is naturally implemented via a ResNet architecture and relies on a non-iterative solver with near-linear complexity. Furthermore, CWG supports guided generation by modulating a task-specific distance metric. We validate our framework on tasks including manifold navigation, molecular dynamics predictions, and image generation, demonstrating its practical benefits and versatility.

Published as a conference paper at ICLR 2026
CONTACT WASSERSTEIN GEODESICS FOR NON-CONSERVATIVE SCHRÖDINGER BRIDGES
Andrea Testa Søren
Seren Hauberg', Tamim Asfour Leonel Rozo
1 Bosch Center for Artificial Intelligence (BCAI). 2 Karlsruhe Institute of Technology (KIT), 3 Technical University of Denmark (DTU), Italian Institute of Artificial Intelligence (AI4I)
Emails: andrea.testa @ de..bosch.co. sohau dtu.dk, asfour @ kit,edu, lconel.rozo@annl aiii.it
ABSTRACT
The Schrödinger Bridge provides principled framework for modeling stochastic processes between distributions; however, existing methods are limited by energy-conservation assumptions, which constrains the bridge's shape preventing it from model varying-eneryy phenomena. To overcome this, we introduce the non-conservative generalized Schrödinger bridge (NCGSB), novel, energy- varying reformulation based contact Hamiltonian mechanics. By allowing energy to change over time, the NCGSB provides a broader class of real-world stochastic processes, capturing richer and more faithful intermediate dynamics. By parameterizing the Wasserstein manifold, we lift the bridge problem to a tractable geodesic computation in fiite-dimensional space. Unlike computa- tionally expensive iterative solutions, our contact Wasserstein geodesic (CWG) is naturally implemented via ResNet architecture and relies solver with near-linear complexity. Furthermore, CWG supports guided genera- tion by modulating a task-specific distance metric. We validate our framework on tasks including manifold navigation, molecular dynamics predictions, and image generation, demonstrating its practical benefits and versatility. Project website: http:://ites.googlee www.le..oom/view/www....
non-iterative
INTRODUCTION
Inferring the stochastic process that most likely generates a set of sparse observations is fundamental challenge, e.g., in cellular dynamics (Yeo al., 2021; Zhang et al., 2024; Moon et al., 2019), meteorology (Franzke et al., 2015), and economics (Kazakevičius et al., 2021; Huang et al., 2024). Here, the target is not merely the distribu- tions of observed data, but rather the underlying dynamics economic phe-
of cell populations, weather patterns, nomena, enabling reconstruction of missing intermediate states and predicting the systems' future evolution.
1.0 Figure 1: Probability paths under energy- conserving (-), energy- energy-decreasing (-), and energy-inercasing conditions (-) (de- tails in App. F.2). Our NCGSB enables en- ergy variation, increasing modeling flexibil- ity in applications where distributions at in- termediate time steps are of interest.
0.0
0.5
The Schrüdinger Bridge (SB, Schrödinger (1931)) is a powerful mathematical framework address this. SB seeks the most likely stochastic path between marginals (i.e., observations), while being close a reference pro- cess, typically Brownian motion. This offers general stochastic optimal control perspective that encompasses both Optimal Transport (OT, Vargas et al. (2021)) and generative approaches such as diffusion mod- els (Ho et al., 2020; Chen al., 2024), which can interpreted as optimal bridges with a Gaussian initial marginal (Bortoli et al., 2021). Unfortunately, current SB solvers operate infinite proba- bility space and rely on iterative of the reference dynamics. This leads to complex and costly optimizations, limiting adoption.
backward stochastic simulations or progressive refinement
Solutions provided by the SB preserve the distribution'ss 's energy throughout the full stochastic path. Here, energy is understood as combination of kinetic energy, which reflects how fast samples move across the probability manifold, and potential energy from the underlying landscape. This energy-prest preservation principle constrains the shape of the bridge and excludes stochastic paths with

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varying energy profiles, such dissipative behaviors commonly encountered in real-world physical systems, e.g. storms gradually losing intensity in weather forecasting.
This paper overcomes both the energy-preserven limitation and the iterative optimization schemes of existing SB approaches by formulating a novel geometric generalization of the SB to model non-conservative systems and by proposing near-linear time solver. Specifically, we build on the geometric perspective of the SB, which casts it as a flow governed by Hamiltonian dynamics on the Wasserstein probability space (Sec. 3). Extending the Hamiltonian system to a contact Hamiltonian (Zadra, 2023), we propose a more general energy-varying formulation of the SB prob- lem: The Non-Conservative Generalized SB (NCGSB, Sec. 4). To make computations tractable, we introduce the Contact Wasserstein Geodesic (CWG), which solves the NCGSB problem by casting it as geodesic computation, reshaping its cost functional into a Riemannian metric whose induced distance is minimized. Discretizing the geodesic leads to geodesic segments that match standard residual blocks. We show how this leads to an efficient solver that avoids outer iteration loops and achieves near-linear complexity in both dimensionality and batch size. Additionally, our approach allows for guided generation, steering the learned process via a task-dependent loss, which in our geometric framework corrssonnds to modulating the Riemannian metric (Sec. 5). We demonstrate our approach on benchmarks and tasks such as LiDAR manifold navigation, molecular dynamics predictions, unpaired images matching, image-based reconstruction of systems such as sea-surface temperature and robotic pick-and-place tasks (Sec. 6).
formulation of the Schrödinger Bridge problem that models a wider range of real-world physical stochastic processes by building on geometric flows governed by contact Hamiltonian dynamics; (2) The introduction of the Contact Wasserstein Geodesic (CWG) framework, a general geometric solver compatible with all Schrödinger Bridge variants, enabling fast and scalable computation; (3) a guided generation method for steering the bridge learned by CWG according to new task specifications.
summary, we contribute: (1) A novel non-conservative
2 RELATED WORK
Schrödinger bridges. The SB problem imposes no constraints on the probabilistic path be- yond matching the endpoint marginals, limiting its applicability when intermediate observations exist. It also does not permit including known physical laws governing the system's 's dynamics. The multi--margial Schrödinger Bridge (mmSB) treats intermediate observations (Theodoropoulos et al., 2025) as constraints, which enables reconstruction of continuous dynamics without piecewise approximations. In contrast, the Generalized Schrödinger Bridge (GSB) adds a state cost, allowing potential energy functionals to be minimized along the probability path. This allows to model mean- field interactions (Gaitonde et al., 2021; Ruthotto et al., 2020), conservative forces (Philippidis et al., 1979; Noč et al., 2020), or geometric priors (Chen & Lipman, 2024; Liu et al., 2018).
Non-conservative Schrödinger bridge formulations. The SB problem assumes constant energy, preventing it from modeling consevsvervtee systems. The momentum SB augments the state space with a velocity term (Theodoropoulos et al., 2025; Chen et al., 2023), allowing damping to be incorporated (Blessing al., 2025; Sterling et al., 2025). However, this doubles the state space and increases computational cost. Other extensions replace the Brownian reference process with the Uhllebee (OU) process (Orland, 2025: Zhang & Stumpf, 2025), introducing a non- conservative prior for the target dynamics. The OU process defines a mean distribution with a curl-free component that drives convergence and divergence-free component that induces rotation. Yet, this formulation lacks a mechanism for energy dissipation in the rotational dynamics. We introduce a more general energy-varying framework in which dissipation naturally emerges across all components of the dynamics, while only requiring a scalar augmentation of the state space.
Schrödinger bridge solvers. Matching- based iterative approaches (Bortoli et al., 2024; Shi al., 2023; Gushchin et al., 2024) have gained popularity by improving the scalability and robustness of traditional Iterative Proportional Fitting (IPF) algorithms (Kullback, 1968; Léonard, 2013) through Markovian projections, thereby avoiding the need for full trajectory storage. However, for GSB, re- strictive assumptions like Gaussian probability paths limit expressivity (Liu et al., 2024; Tang et al., 2025), while for mmSB, global trajectory consistency is compromised due to the piecewise nature of the approach and its sensitivity to the initial choice of the reference process (Shen et al., 2025). To overcome these limitations, the stochastic dynamics can be learned indirectly by leveraging the ana-
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lytical optimality conditions of the SB problem (Vargas et al., 2021; Chen et al., 2022). This has been shown to scale more effectively to both the GSB (Liu et al., 2022; Buzun et al., 2025) and the mmSB problem (Theodoropoulos et al., 2025; Hong et al., 2025; Chen et al., 2023), due to the symmetries inherent in these optimality conditions (App. A). However, these methods are limited by a classical SB solver bottleneck: the computational overhead of their iterative nature, which alternates between forward-backin passes or repeated dynamics integration and reference updates, We propose a cheaper non-iterative solver that scales nearly linearly with both dimensionality and sample size.
Schrödinger bridge guided generation. The SB framework naturally extends to conditional generation, where the marginals and the transition path depend on additional parameters or objectives (Shi et al., 2022). Model guidance techniques (Song et al., 2023; Guo et al., 2024) introduce a guidance term into the stochastic process, typically derived from the gradient of a loss function. This approach steers the flow locally and resembles a gradient-based form of optimal control. Alternatively, Raja et al. (2025) employs a global optimal control perspective. Their approach generates full trajectories and chooses the one minimizing task-specific action functional. However, their method is deterministic and yields a single optimal path rather than a posterior distribution over paths. Unlike previous guidance methods, we propose a hybrid approach to guided generation within the NCGSB framework. By embedding a task-specific loss into the potential energy of the NCGSB, we reshape its Riemannian metric that the resulting geodesic reflects the guidance objective, allowing the learned dynamics to align with the desired outcome.
3 PRELIMINARIES
We introduce the necessary notation throughout the following sections, and summarize it in App. B.
The Wasserstein manifold. Before formally introducing the SB problem, we define its domain. Let (MM) denote the space of smooth, positive density functions supported on manifold M. Each element p E (M) is a function p(x) . + R+ satisfying S. p(x) da = 1. The density dynamics is represented by a time-dependent family { C (M). The infinitesimal variation of the density at time is the time derivative 0,p' (z), which lies in the tangent space Tp (MM). The collection of all such tangent spaces forms the tangent bundle (M+ When equipped with the Wasserstein metric, P+M becomes Riemannian manifold (Ambrosio et al., 2005). The corresponding metric tensor is defined as,
g Ma (Orp', 0,p')) = S. 0.0'(a(( (z) px dz.
(1)
where (-Apt)t ) is the inverse of the weighted Laplacian operator Apt = -V Vz) (Chow et al., 2020). inducing inner product on TP+ (Mt and distance dw, (Pas Pb) for Pai E pt (M). The minimum-length curve p' connecting the two distributions Pa, Pb. is called a geodesic.
Multi-marginal generalized Schrödinger bridge (mmGSB). Given two endpoint densities Pa: Ps E (MM), the SB problem (Schrödinger, 1931; Schrödinger, 1932) seeks the most probable interpolating density path p'. This minimizes the Kullback-Leibler divergence w.r.t. a reference pro- cess Piu typically Brownian motion. The SB problem is equivalent to a stochastic optimal control setting (Dai Pra, 1991), which minimizes the cost required transport a set of diffusing particles from an initial distribution Pa to a target distribution Pb- This dynamic reformulation of the OT problem (Benamou & Brenier, 2000; Chen et al., 2014) has solutions corresponding to geodesics on the Wasserstein manifold p+ (MM). These trajectories are straight, since the classical SB problem assumes that particles dynamics are unaffected by external potential functions. This assumption, however, limits our ability to model complex real-world physical systems.
Intermediate observations represented by marginal distributions {pm ) at specific time steps {Lm M m= can also be incorporated as additional constraints (Chen et al., 2023; Tang et al., 2025).
This leads to the mmGSB problem.
min J( ,-) = ( U(+)) (x) dx dt;
(2a)
s.t. O,p((x) + V. (p' of (2)) = =
(2b)
p0 Pa: p¹ pbs pm Pm, Vm E 1,..., M.
(2c)
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Here, the density evolution p' is governed by the Fokker- Fokker-Planck equation (2b), which generalizes Brownian motion by incorporating a deterministic drift term of alongside a stochastic diffusion term scaled by E. This drift of acts the control variable and ensures that the probability path satisfies the boundary conditions in equation (2c). The objective functional includes the kinetic energy associated with the drift to quantifies the deviation from the (uncontrolled) reference stochastic process and the potential energy U to favor low-potential pathways.
Wasserstein Hamiltonian flows and geodesics. A convenient solution to the mmGSB problem (2) is to specify analytical optimality conditions (Sec. 2). These take the form of a Wasserstein Hamil- tonian Flow (Chow et al., 2020), which describes a probability distribution evolving according to Hamiltonian dynamics. This evolution lies on planes tangent to P (M), specifically on the cotan- gent bundle T*P+ (M), the dual of TP+ (M), and it is governed by the derivatives of a sealar Hamiltonian function H. However, their integration remains computationally expensive (Buzun et al., 2025; Wu al., 2025), so we propose a geometric reformulation that results in a significant simplification. To this end, we introduce Proposition I, a standard result from differential geometry (App. C.2), which is instrumental in lifting these equations to geodesics on pt (M).
Proposition 1. Let the optimality conditions of the mmGSB problem (2) be expressed in Hamilto- nian form, yielding the optimal bridge o² (x). Then, p² (x) can be viewed as a geodesic connecting the marginals in equation 2c w.r.t the modified Riemannian metric g., known as the Jacobi met- ric (Abraham & Marsden, 2008).
To access the Jacobi metric and determine the corresponding geodesic, we first derive the Hamilto- nian optimality conditions of the mmGSB problem (2) using Lagrange multipliers (Cui al., 2024). This introduces a potential function St (z), whose gradient defines the drift via v'(x) = V 5²,'
The potential enforces the dynamic constraint (2b) within the cost functional (2a), whose first vari- ation yields the Hamiltonian optimality conditions.
0.p' = H.H = v. (+)))
(3a)
= -,H -
+ (x))) +U(x).
(3b)
with the corresponding Hamiltonian, S²p K(p''s'') + F(p'), defined as the sum of a kinetic energy IC and a potential energy F --- dependent on the potential function U and the Fisher information J. A detailed derivation of these dynamics and the full Hamiltonian function are given in App. D.I. To handle boundary conditions (2c), the Hamiltonian dynamics (3) are typically integrated backward in time, where the solution each intermediate point (p'". SSt)) serves as the initial condition for the next segment (Theodoropoulos et al., 2025). Equation (3a) links the potential function St (z) to the infinitesimal density variation a,p' TP+ (M) via the weighted Laplacian operator Apt introduced through the Wasserstein metric (1). This connection establishes correspondence between the tangent bundle TP+ and the cotangent bundle Txpt (M). where 5t (x) naturally resides, and where the Hamiltonian dynamics of (0',S*) unfold.
By Proposition 1, the Hamiltonian dynamics (3) corresponds to a geodesic flow on the underlying Wasserstein manifold p+ (MM, which minimizes the Jacobi metric
= (H FFgg The original metric 9¹ (1) accounts only for the kinetic energy of the transport map IC by measuring distances between distributions. In contrast, the Jacobi metric fJ also includes the potential energy F, which is maximized attain values F H. Consequently, computing the geodesic between marginals under this metric is equivalent to solving the mmGSB problem (2).
4 THE NON-CONSERVATIVV GENERALIZER SCHRÖDINGER BRIDGE
Non-conservative formulation. The solution to the GSB problem (2) assumes a constant energy function Ht and restricts the drift o' to depend solely on the potential energy F. This limits the model's 's flexibility in representing dynamics that cannot be described by conservative potential, which reduces its ability to capture real-world processes involving energy dissipation and external interactions. To overcome this, we introduce the non-conservative generalized Schrüdinger bridge (NCGSB), which allows for time-varying energy systems. To do so, we reformulate the cost func- tional J as the time integral of new scalar state z representing the Lagrangian action, whose evolution depends recursively on itself. The NCGSB problem is formulated as follows,
[' 0,2² dt: min J(o ,0) = Jo
(4a)
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s.t. 0,²¹= = f ( (((())))))++()) / (1(x) do - 722;
(4b)
0.p' (x) + V (z(()) = = (x));
(4c)
p Pa. p pb. p' Pm. Vm E ...... M,
(4d)
where 7 E IR is a damping factor. The objective in equation 4a is no longer to minimize a static quantity, but rather timevarying state z'. Its dynamics (4b) depend explicitly its current value. This recursive structure endows the system with a form of memory, as its evolution is influenced by the entire trajectory, implicitly encoded in z'. Because non-conservative forces are path-dependent, augmenting the system's 's state space with the scalar =' allows their effects to be modeled, enabling the system's 's energy to vary over time. The sign and magnitude of y determine the direction and rate of this variation. By relaxing the implicit energy conse-coonsrovttion constraint of the GSB problem, our approach enhances the model's flexibility and improves the quality of the resulting optimal solution.
Guided NCGSB. NCGSB (4) can be extended to the guided generation setting by introducing a guiding function f. which steers the generative process toward desired conditions at any chosen time (Song et al., 2023; Guo et al., 2024). For given time t.. the guidance is expressed as = [(ah), with x²= ~ p'r. To enforce this form, the bridge p² steered according to pa(yly) = Z e'(x))=lv-(((
where Z is normalization constant and x'> denotes a sample from the predicted guided distribution p²a (xrl)). By Bayes' rule, the dynamics of the guided bridge (x(wl)) To perform a guided generation that enforces the constraint y = f(x²*) while preserving the underlying data manifold, we incorporate y into the Lagrangian action constraint (4b) as (see App. D.3 for details),
acquire an additional guidance term via the drift e². determined by l1y - f=="))
0,²²= = [n UUx)
f(tt))) 11l (U)
p (z(x) dz 7z'.
(5)
Wasserstein contact Hamiltonian flows and geodesics. Analogously to mmGSB (2), understand- ing the dynamics of the optimality conditions in NCGSB (4) is essential for reformulating it as a geodesic computation. As detailed in App. D.2, we propose to leverage the contact Hamiltonian for- malism (Kholodenko, 2013), an extension of classical Hamiltonian mechanics to non-conservative systems (App. C. to model the dynamics of the NCGSB optimality conditions as Wasserstein con- tact Hamiltonian flows. This generalizes Prop. 1, since the contact Hamiltonian dynamics defines a geodesic but on the extended space P- (M) x (Udrişte, 2000; Testa et al., 2025). The contact Hamiltonian optimality conditions are,
0,p'(x) = 0.) = V (0' (1(x))
(6a)
(z)²(z) (-pH(() (*)))_H() - -V= + 2 ) +U(x)
1
- -7)'(x) - so
log
(6b)
H() / (1 + +(()) ('()) dx + 1
0,2' =
2
sy(log p' (x) 11)) (x) dx - 722"
(6c)
The corresponding contact Hamiltonian function defined as, H(' St 2²,2 K(pp',''))+ +FFp)( B(p²) +yz". This differs from its conservative counterpart in two ways. First, its explicit dependence on allows the total energy to vary over time. Second, the potential energy augmented by an entropy term, B(p²) Jum e(log (xx) - 1)p²(x) (x) da. producing an additional diffusion in the dynamics, As previously mentioned, for guided generation, an additional potential energy term llv [(ff)]] can be introduced here to steer the flow. Geometrically, the dynamics of (o', St =²) can be interpreted as a flow on the cotangent bundle of the Wasserstein manifold, augmented by the scalar state 2t, That is, the dynamics unfold on the space T**+(M) x R.
The contact Hamiltonian flow evolving on the extended phase space T-P+ (M) x R. and interpo- lating between the marginal densities, induces a geodesic on the augmented manifold pt (M) R. This geodesic minimizes a Jacobi metric ÔJ = (H - F B) g W. which generalizes the classical Wasserstein metric by incorporating the potential energy of the contact Hamiltonian function. Com-
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puting the geodesic under the Jacobi metric IJ corresponds to the NCGSB problem (4). Unlike the conservative case, the contact Hamiltonian H is no longer constant along the flow, allowing the total additional degree of freedom that can be leveraged to
energy to vary over time. This introduces shape the system's energy along the path over P (M). This is the reason why the geodesic (p', H)) is defined on the extended space P (M) R.
CONTACT WASSERSTEI GEODESICS (CWG)
ResNet resembles a discrete geodesic. Our objective is to compute geodesic p² on P (M), induced by the contact Hamiltonian dynamics, that is constrained to pass through set of observed marginals (Pas (/mm/p Pb) (i.e., discretized distributions along the probability path). These constraints naturally lead to a discretized parameterization of p², where the overall density transformation is modeled as a composition of maps, each connecting a pair of consecutive observations. A ResNet is ideally suited for this problem, as its sequential block structure directly mirrors this piecewise, compositional nature of the approximated geodesic. Let À be a fixed reference measure on P+ (M) (e.g., f. standard Gaussian uniform distribution). We define (K + 1)-block ResNet as follows,
T for K = Tax .... Toi Too;
with parameters ( E O. The process begins by sampling an initial batch of points x" that is pushed forward through the first block to obtain r² Tgp (x*). Then, the parameters 8 are optimized such that the resulting pushforward reference measure approximates the initial marginal Pe = Pa. Thereafter, each subsequent block & pushes forward the sample via The full pushforward map induces,
gle+1 Tgets (x**), Itt ~ 1
Po
1e (Tp--)) ) p' (x*+1)) det V Vet[VTTotti (
(8)
Starting from the reference measure A, the ResNet parameters {0 discrete probability transitions (0.p't = which in turn specify the discrete family of densities 1ink Ik=o. Geometrically, the discretizations (depg" Pg fe K provided by the ResNet, approximate (p',Osp')) E TP+(M). which can be seen as inducing mapping from TP+ (M) onto the parameter space @ (Fig. 5). As stated in Proposition 2, the existence of such map endows the finite-dimensional space Σ, where the parameterized densities P reside, with the geometric properties of P (M). This lifting enables faster and tractable computations for SB problems.
1k=0 define a sequence of
Proposition 2. Approximate the evolution of the density p² E (M) by a series of K smooth parametrized pushforwards Tgt, with 6 belonging to finite- siiite-dimessoonal space O. If each push- forward Tøx is an immersion Tgx with Riemannian structure via the pullback of the Wasserstein metric g2, Consequently, the contact Hamiltonian dynamics on T*p+ (M) x RR can be represented the reduced phase space ** x E. with the associated geodesic on P+ (M) x R projected onto Σ x R (see App. E.1).
TPP (M), then the parameter space Q can be endowed
to transform the geodesic computation from the (MM to a geodesic on a finite-dimensional parameterized space Σ such that the resulting geodesic flow on Σ x IR evolves under the pullback of the Jacobi metrie T7 4* N2, where the scalar factor 4 H H Stx 2*x) F(p²x) B(p*)) ))), ), encodes the kinetic energy. Specifying the time evolution of H** determines unique parameterized bridge on Σ. This formulation enables tractable computation of geodesic flows to solve the NCGSB problem. Although different parameterizations (0* {0 may define distinct coordinate systems on Σ, the geodesics solutions remain equivalent and share the same energy (Syrota et al.. 2025).
Proposition 2
allows
Visualization of the ResNet trans- formation. Two successive pushforwards Po ft-1 + P/ - Pg (++1 on P+MM) are shown as local updates O.pt'. O.pp fet1 Each update is parameterized by ,0, gh+z e e, defining local coordinates on Tpt coordinate system is not unique.
Figure 2:
TyP (M)
Tya P(M)
P1
0.P0
a,Po
p
tangent spaces.
(M()
+
(*(M)
TP+(M). (M). This
ghil
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Geodesic computation. The contact Wasserstein geodesic (CWG) corresoonds to the discrete path {pg 1K 1k=0- that approximates the NCGSB solution. This trained to reconstruct the avail- able marginals while minimizing the geodesic energy under the pullback Jacobi metric To of T g,, The initial and final marginals, Pa and ps. are enforced the path endpoints, corre- sponding to the ResNet outputs at times to and fx- When available, intermediate marginals pM must appear at time points matching the ResNet discretization for the condition to be enforced.
CWG training happens in two stages (Alg. in App. E.2): (1) we optimize the first ResNet block to match the initial marginal Pa. and (2) we find the optimal path by minimizing the loss,
8= = (Pg k Po)
divs (Po (pot Pm)
th dlw, (0'0 Po 1)
Terminal marginal
Intermediate marginals
Energy minimization
Here dw, denotes the Wasserstein-2 distance between probability distributions. In practice, this distance is approximated using empirical estimators based on samples drawn from the distributions. Convergence of the algorithm guaranteed in App. E.3. Its complexity
(N K (Tab + D(LW + log N))), scaling linearly in dimension D and nearly linearly in batch size (see App. E.5), rather than exponentially quadratically (Hong et al., 2025). Unlike Chen et al. (2023); Shen et al. (2025), our CWG avoids costly iteration loops and is only weakly affected by the number of marginals.
Guided contact Wasserstein geodesics. In the conditional setting, the Lagrangian action dy- namics (4b) in NCGSB (4) is augmented as in equation 5. Here, the scaling factor 4 = H(p",, S*h 2(x) F(p*)) B()**) of the pullback Jacobi metric is augmented with the guid- ance term lly - f(x*) )|? to enforce the constraint = f(x**) at time i, of the generative process. Under the ResNet parameterization, the desired distribution approximated by z'*, ss x time step lk,, and the Jacobi metric modified as Øj (Φ'* k + |yy [(a*=)] WW, with x'*,, ~ Po. le. This penalizes geodesics crossing undesired regions at ls, The loss for the guided optimization is
e- Pb) pb)++ Σ d'vs (Po (Potm, pm) Σ
di (0 Ps) f +11-- de (0' Po -
(10)
where the modified distance d measures deviations between the generated distribution Pg for and the intermediate marginal P. at te. while incorporating the penalty for samples 2, that violate the guidance constraint y = f(zz). ). c.f. App. F.I. In practice, this loss is optimized through fine-tuning procedure applied to model initially trained without any guidance.
Proof of concept. We demonstrate our framework generation setting using the Two-Moons and Checkerboard benchmarks (Holderrieth & Erives, 2025). These lack intermediate marginals {pm)) and the initial distribution Pa coincides with the reference distribution X. Hence, only the second step of Alg. is needed. Figure shows that our method successfully generates the target distributions, and steers the generation to samples confined to subset of the target space (here, the upper half). This guided behavior is achieved via the term lly f(xt=) I13 with t. = 1,, f measuring 2D sample positions, and y defining the admissible region.
2D distribution-matchingg task and guided
RESULTS
We benchmark our approach against four established baselines summarized in Table details of the experimental setups are provided in App. G.I.
Further
Method
GSB
mmSB
Energy variation
Image Gen.
Guided Gen.
DSBM (Shi et al., 2023)
X
X
X
SB-Flow (Bortoli al., 2024)
X
x
x
GSBM (Liu et al., 2024)
/
X
X
SBIRR (Shen al., 2025)
X
/
X
DM-SB (Chen et al., 2023)
X
/
X
CWG (ours)
/
Table Comparison our CWG with basclines designed to address various SB variants and types of problems.
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Figure 3: Two-Moons (top) and Checkerboard (bot- tom) benchmarks with guided variants (right).
Figure 4: LiDAR Manifold Navigation: CWG before and after guidance (top), CWG vs GSBM (bottom).
LiDAR manifold navigation. First, we tackle a standard GSB task: computing a bridge evolving on a geometric manifold. We use the LiDAR scan of Mount Rainier (Open Topography, 2025) as the reference surface, and we aim to connect two marginals while remaining on the manifold and favor- ing low-altitude regions. These conditions are encoded into the potential function U (see App. G.2). In this experiment, we do not model a physical system but instead compute an optimal transport map between the marginals under conservative setting. Unlike our approach, DSBM and GSBM iteratively fit a deterministic path between the marginals, falling short on representing a posterior distribution. As a result, there is no guarantee that Gaussian path remains the manifold (Fig. 4). This leads to substantially higher-energy paths, as reflected by the Optimality metric in Table 2, and to inaccurate modeling of the target distribution, indicated by the Feasibility metric (both metrics as defined in the baseline Liu et al. (2024)). In contrast, our approach finds finds lower-energy solutions that respect the marginal constraints and converges significantly faster. Furthermore, our method uniquely supports guided generation, illustrated here by steering the probabilistic path to the right side of the mount (Fig. 4; see App. G.2 for quantitative results).
Table Optimality (67a) (4). Feasibility (67b) (L),
Table Wasserstein error at validation (L) training time (It) (L) in Single Cell Sequencing.
and training time (tt) (4) LiDAR Navigation.
Metrie
CWG (ours)
GSBM
DSBM
Metric
CWG (ours)
DM-SB
SBIRR
Optimality
1.40+0.02
2.18+0.02
4.16
dw, (x
1.11 10.06
2.25
1.92+0.02
Feasibility
0.06+0.01
0.83+
0.97
dw, (x*a
0.33
1.64_..03
1.86+0.02
(S)
280
1570+50
1340
tt (s)
710
38120
1740
Single cell sequencing. Next, we reconstruct stem cell differentiation dynamics from series of isolated cellular snapshots. We use the Embryoid Body (EB) dataset from Moon et al. (2019), which tracks cell state progression across five developmental stages [to, t1, t2, f3. tz]. Cell differentiation is fundamentally non-conservative biological process (Zeevaert et al., 2020: Kinney et al., 2014) and the ability model energy-varying bridges is essential. To evaluate generalization in regions with no available data, we split the dataset into a training set [fo, 12, fa] and a validation set ta]. Ac- cordingly, the former contains the distributions {PasPma Pb), while the latter contains {Pmi Pma): }. The geometry of the training distributions is encoded in the potential function U, which penalizes paths that stray from the observed data manifold. Minimizing U ensures the learned bridge remains close to the data manifold, enabling effective generalization. The combination of the data manifold guidance and an energy-varyinne bridge allows our approach to outperform other mmSB baselines in both reconstruction accuracy and computation time. Quantitative results are reported in Table 3, with additional details and an ablation study on the importance of energy variation provided in App. G.3.
Image generation. We also demonstrate our framework's 's applicability to image generation tasks. Given sequences of images capturing the time evolution of physical phenomena, the model aims, from a single input image, to predict the most likely terminal state of the system, along with realistic intermediate frames at unseen time steps. Specifically, we use the NOAA OISST v2 High Resolution Dataset (Huang et al., 2021), which provides daily surface temperature averages over multiple years, and the BridgeData V2 (Walke et al., 2023) dataset, containing image snapshots of robotic manipulation tasks. Our NCGSB problem also be applied to unpaired image-matching tasks
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E
Table 4: FID scores at validation steps (L) and training time (It) ( in Sea Prediction (2020-2024).
GSBM CWG lours)
Metric
CWG (ours)
GSBM
DSBM
SB-Flow
FID (r
121+o
161+s
242+10
177
FID r23)
160
186
236
190
(60s)
17+1
1227
318+15
8345
DSBM
260
scmn Ivaliaatonn
240
LWG (oun))
CSBM
220
DSBM
Reference S8-Flow
180
160
140
120
2020-2074
2015-2019
2010-2014
2009-2009
2000-2004
1995-1909
1990-1934 1934
1985-1989 1989
1981-1983
1995
1990.
1983.
1961
Wars
Predictions generated CWG (ours, top), GSBM (second), DSBM (third), SB-Flow (fourth). The red row displays the corresponding ref- erence samples.
Figure 5:
Figure 6: FID scores at validation time steps for four methods, evaluated over 9 tests (1981-2024). Our CWG scores significantly lower than baselines.
using general image datasets that do not describe dynamical systems, such MNIST, EMNIST, and Flickr-Faces-HE HQ (FFHQ) (Karras et al., 2019). For the high-resolution unpaired image-matching task on the FFHQ dataset, the SB is computed in latent space after encoding the images with the pre-trained ALAE encoder (Pidhorskyi et al., 2020). For all the other datasets, we compute the directly in the ambient image space. To ensure that the generated frames remain faithful to underlying data distribution, we introduce a potential function U, that penalizes deviations from the learned data manifold. Following the baseline (Liu et al., 2024), for samples x' p', Uxx)) is defined the reconstruction error, obtained via VAE (Song & Itti, 2025). Details the energy behavior and extended results are provided in Apps. G.4, G.5, G.6, and G.7.
For the temperature prediction task, cluster data from 1981-2024 into five intervals. Us- ing heatmaps from January, May, and September (i.e., [to t2, tal). our method predicts the tempera- ture profiles of March and July (i.e., (11, Our CWG produces cleaner, more accurate predictions than the baselines (Fig. 5). Since our framework operates efficiently in probability space and is not constrained by energy conservation, it achieves these results with order of magnitude less com- putation (Table 4 for 2020-2024: Fig. for all years), In the robotic task reconstruction, our model
to C2 t
Is
t7
ts
t6
t7
GSBM CWG (ours)
Figure CWG outputs before (top) vs. after guidance (place the item left).
DSBM
Metric CWG
GSBM DSBM SB-Flow
FID
1941
40+2
150
7341
Reference SB-Flow
tt (h)
0.5
25.3
7.6_
1.4
Table 5: FID score (4) and training time (t)) (1)
in the robotic task reconstruction.
Metric Standard Guidance
Centroid 35.84111
22.3+22
Figure 7: Reconstructions from CWG (top). GSBM (mid- dle), and DSBM (bottom). Red shows the reference.
FID
19.52
23.77
Table Item centroid position (px) and before vs. after guidance.
generates realistic intermediate frames connecting the initial and final states of robot's reaching motion (Fig. 7), and demonstrates consistently robust performance, outperforming baselines in im- age quality (Table 8). Also, Fig. 8 shows guided generation, where model successfully steers the

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Published as a conference paper at ICLR 2026
placing motion task toward target location the left side of the table. This is achieved with only a minimal drop in image quality, maintaining a clear advantage over competing methods (Table 6).
Regarding the unpaired image transfer experiments, transfer (Table 38, App. G.7) and validate the soided-bridge capabilities through a Gaussian deblurring task. We then present the FFHQ transfer experiment, in which the SB maps images from a starting adult distribution Pa to its closest analogue in a target children distribution Pb- As shown in Table 7, while GSBM and DSBM produce images visually similar to the input (Optimality), they fail to satisfy the boundary conditions of (2) and achieve poor Feasibility. In contrast, our CWG model consistently generates images within the children distribution pb. with average predicted ages below 18, as illustrated in Fig. 10.
first evaluate the MNIST+0-EMMNIT
Table 7: Metrics for the FFIQQ transfer experiment: Training time (tt) (L), Optimality (72a) (.). measuring the geodesic distance to assess the transport cost between the two marginals (Po-Ps), and Feasibility (72b) (L), indicating how well the marginals are preserved, ie., how closely the bridge endpoint aligns with pa- Metric 18 (L) indicates confidence that the final predicted images satisfy the boundary constraints.
Metric
CWG
GSBM
DSBM
SB-Flow
Optimality
218.142.
206.2+2.9
198.042.7
237.442.2
Feasibility
40.52
6.839 753
7.780+0.80
21.748+0 498
18 (p-value)
2.1 x 10 10-9
6.6 x 10
6.1 10
1.1 x 10
tt (s)
930+30
26504:30
2530
1490
Input
CWG (ours)
CSBM
DSBM
SB- Flow
Figure 9: Adult + Child image generation on the FFHQ transfer experiment.
Extimated a0e
GSBM
50
DSBM
S8-Flow
40-
30.
30
30-
30-
18 years Lo
10
18 years
10
18 years
10
16 years
Insut
5
Inout
Input
Ingit
4
Stop
Stop
Stap
Stap
Figure 10: Average age predictions each generation time step for the images shown in Figure 9, comparing the baseline models with three CWG variants: energy mnergy-ooneervnn, energy-increasing, and energy-decreasing.
7 CONCLUSION
Our work is motivated by the need to model intermediate time steps of Schrödinger bridges (SBs), arising from the underlying dynamics of the observed physical system. As standard SBs conserve energy across time, they cannot meaningfully encode such dynamics. To overcome this, we in- troduced the non-conservative generalized Schrödinger bridge (NCGSB). which extends the usual Hamiltonian to its non-conservative counterpart, the contact Hamiltonian, allowing energy to vary. Through a geometric perspective, we show that the NCGSB is equivalent to geodesics on contact Wasserstein manifolds. This link leads to a non-iterative and near-linear time algorithm for com- puting the non-conservative bridge, which can practically be realized by a ResNet-like construction, easing its implementation. We show that these theoretical contributions lead to SB framework that not only more expressive but also significantly faster than existing approaches, as validated by the significant improvements achieved across range of diverse tasks.
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Grace Hui Ting Yeo, Sachit D Saksena, and David K Gifford. Generative modeling of single-cell time series with PRESCIENT enables prediction of cell trajectories with interven- tions. Nature communications, 12(1):3222, 2021. URL https: wtrpii.org/01........ 441467-0211 014677221-2351
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Federico Zadra. Topics in contact Hamiltonian systems: analytical and numerical perspectives. PhD thesis, University of Groningen, 2023. 20
Alireza Zaeemzadeh, Nazanin Rahnavard, and Mubarak Shah. Norm-preservation: Why residual networks become extremely deep? IEEE Transactions on Pattern Analysis and Machine Intel- ligence (T-PAMI), (3(11):3980-3990 2020. URL https: / htt//iioorrrg /00. 1109/ 1109/TPAMI. 2020. 2990339. 27
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https: //tp:////rxivvv 2550 16644. 2
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A EXTENDED STATE-OF-THE-ARE ON SCHRÖDINGER BRIDGE SOLVERS
Existing methodologies for addressing the Schrödinger Bridge problem can be broadly divided into two main categories, depending on their solution strategy: those that directly fit the stochastic dy- namics on the probabilistic manifold, and those that leverage the analytic optimality conditions of the problem to solve it. We review them in the sequel.
Schrödinger Bridge Solvers via Dynamics Parametrization. Traditional SB solvers often use Iterative Proportional Fitting (IPF) (Kullback, 1968; Léonard, 2013), which alternates forward and backward updates to successively match the initial and terminal marginals. However, IPF is computationally expensive, as it stores full trajectories, and it suffers from error accumulation, nu- merical instability, and reliance on strong priors (Vargas al., 2021; Gushchin al., 2024). Recent matching-barte based approaches (Bortoli et al., 2024; Shi et al., 2023; Gushchin et al., 2024; Peluchetti, 2023) improve scalability and robustness by learning time-reversed drifts via Markovian projections, circumventing the need for full trajectory storage and mitigating the IPF discretization errors. While Liu et al. (2024) extended this idea the GSB setting, their assumption of Gaussian probability paths limits the model's 's expressivity. To overcome this, Tang et al. (2025) proposed branched dynamics using Gaussian mixtures. This enables more flexible path structures but at the expense of higher computational cost. For mmSB, iterative reference refinement with piecewise SB inter- polation (Shen al., 2025) suffers from inconsistencies in global trajectory construction due to its piecewise nature and shows high sensitivity to the choice of initial reference process. Alternatively, Tong et al. (2020) proposed a continuous normalizing flow for deterministic interpolation, removing noise from the reference process but preventing the construction of a true probabilistic bridge.
Schrödinger Bridge Solvers via Optimality Conditions. The optimality conditions of the Hamiltonian phase space, driven by dual
SB problem take the form of dynamical equations potential functions (Chow et al., 2020). These conditions allow the exact dynamics to be recovered via integration and provide flexible framework for generalization through modifications of the Hamiltonian. Furthermore, the state-dependent nature of the Hamiltonian framework offers a natu- ral way to obtain Markovian approximations of stochastic process. Vargas al. (2021) and Chen et al. (2022) leveraged this view and solved the SB problem using control- and likelihood-based approaches, both employing iterative
barkwrd updates on the Hamiltonian dynamics. Liu et al. (2022) extended this idea to the GSB setting. although without convergence guarantees. How- ever, a critical bottleneck of these and derived methods that a tractable integration of optimality conditions relies on iterative updates of reference process. For example, Buzun et al. (2025) improved stability by directly modeling the dual potential and minimizing residuals of the Hamilto- nian conditions, yet iterative updates incur significant computational overhead and may destabilize training due to the dependence on self-generated samples (Bertrand et al., 2024). This issue persists in mmSB settings (Theodoropoulos et al., 2025; Hong et al., 2025). Even when belief propagation used to reduce time complexity (Chen et al., 2023), scaling to high dimensions remains poor. Therefore, while leveraging the optimality conditions offers clear advantages, it remains essential to develop computationally efficient, non-iterative algorithms with favorable scaling properties.
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B NOTATION
Table 8: Summary mathematical notation
Notation
Description
Context
Probability Distributions and Manifolds
M
Underlying data manifold.
Sec.
(+M)
The space of smooth, positive probability density functions sup- ported on M (the Wasserstein manifold).
Sec. 3
p(z), (x())
A probability density function and its time-dependent path. The initial and final endpoint marginal distributions.
Sec.
PuiPo LMM {Pml 1mmm=1
Sec. 3
A set of intermediate marginal constraints at times (fm 1.M
Eq. (2)
A fixed reference measure (e.g., standard Gaussian).
Sec. 5
Transport Dynamics
o'(x)
The drift term (control variable) in the Fokker-Plaane equation. The external potential function (state cost).
Eq. (2)
U(x)
Eq. (2)
c
Scaling factor for the stochastic diffusion term in the Lokker-Plande equation (related to entropy regularization).
Eq. (2)
The objective functional (cost) to be minimized both in the mmGSB problem and in the NCGSB problem.
Eq. (2)
(*xx)
The potential function/Lagrange multiplier that acts as a canonical momentum, with ,,. Sz) = v (xr).
Eq. (3)
2'
The augmented scalar state representing the Lagrangian action.
Eq. (5)
7
The damping factor the Non-Conservative GSB formulation.
Eq. (5)
Hamiltonian variables
The Hamiltonian function, defined the sum of kinetic (K) and potential (F) energy: (p',5% = K(pp', C St) + F(p'). The contact Hamiltonian function, defined (p' K(p',S') F(p') + B(p²) + 72'_
Eq. (3)
(
Eq. (6)
K.
Kinetic energy of the transport map.
Sec. 3
F
Potential energy, F= - =U Fisher information term arising from entropy regularization, defined as T JM V, log p² (2) dr.
Sec. 3
Sec.
Entropy term, defined as [M æ(log -1))) (x)) dz, produc- ing additional diffusion in the contact Hamiltonian dynamics.
Sec.
Guidance Terms 18 f(x*))]
Guidance loss terms, penalizing the difference between the detected features and the desired values y Feature function applied to the sample x-x at time t.. It can indicate class, describe a property of the sample, or quantify the sample's ss value depending on the task.
Eq. (5)
f(x*))
Eq. (5)
Geometric Terms (.(-.))
The Wasserstein metric tensor on P PMM).. The Wasserstein distance between and Pb- The weighted Laplacian operator, A or -V (p'Va).
Eq. (1)
dws (Pa.Po) Apt
Sec. 3
Eq. (1)
gs
The Jacobi metric, gJ (H F)g W/2 which minimized by the optimal bridge geodesic for the mmGSB problem.
Sec.
9.
The Jacobi metric, ÿ. = (H F W2, which minimized by the optimal bridge geodesic for the NCGSB problem.
Sec. 5
ResNet parameterization
TioKyK
ResNet model.
Eq. (7)
Top
A parametrized pushforward map (one block of the ResNet).
Eq. (7)
gk
The parameters associated with the pushforward map Tge.
Eq. (7)
Φb
Scaling term for the Wasserstein distance, defined as ttx
Sec. 5
H(ph, ,h,,,()) -F(p*t) -(p²*).
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C EXTENDED PRELIMINARIES ON DIFFERENTIAL GEOMETRY
C.I HAMILTONIAN AND CONTACT HAMILTONIAN DYNAMICS
Hamiltonian and contact Hamiltonian dynamics are governed by specific energy constraints that can be analyzed via differential geometry as flows on specialized manifolds. Hamiltonian dynamics energy-conin mnergy-conserving and evolves on symplectic manifold. Contact Hamiltonian dynamics is more general, allowing for variable energy levels, and takes place on contact manifold. Their formal definitions and key differences are discussed next.
Symplectic and Contact Structures. Let M be smooth compact manifold, and let T&M denote the tangent space at z E M. The collection all the tangent spaces identifies the tangent bundle TM U M. A vector field X M M assigns tangent vector each point & M. The set of all the vector fields over TM is denoted as I(T (M. A differential 1-form a TM - I smooth map field acting vectors of the tangent bundle. For smooth function f M R, df generalizes the gradient from Euclidean spaces. Specifically. df measures the
the 1-form variation of f under an infinitesimal displacement on M. This displacement is locally described starting point 2 and direction such that (ær, v) E T.M. Alternatively, it can be globally expressed by vector field X. The variation of along the vector field X given by df (X). This variation 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 M, the dual space to T.M. The symplectic and contact structures provide two distinct mechanisms for associating a 1-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, as illustrated in Fig. 11.
92
91
92
q1
Figure 11: The same sealar function f. associated with the 1-form a df. gives rise to two distinct vector fields under the symplectic (left) and contact (right) geometric structures. The streamlines of these vector fields illustrated representation of the state manifold. In symplectic geometry. the streamlines the level curves of representing isoenergetic trajectories where f remains constant, thus describing the dynamics conservative systems. In contrast, in contact geometry, a single flow line can traverse different energy levels.
tangent
Symplectic Geometry. A differential 2-form TM IR is skew-symmetric, bilinear, and smooth field of maps acting on pairs of tangent vectors. A 2-form is called symplectic is both closed (dw 0) and non-degenerate. The symplectic form lacks the properties required to define an inner product. However, it still establishes a fundamental relation between differential I-forms and vector fields: Given 1-form df, the symplectic form w uniquely determines a vector field Xy that is tangent to the level sets of f. This relation is defined by.
df(X)= = -XX,, AX E(TM).
(11)
By definition, f remains constant along the flow of X,, which in turn preserves the symplectic form w, i.e., Lxj = 0 where Lx, denotes the Lie derivative (Silva, 2001). In this framework, the function ff is interpreted as a conserved energy, or equivalently, as a Hamiltonian H. The symplectic structure thereby endows M with natural geometric framework for formulating Hamiltonian dynamics (Tokasi & Pickl, 2022). The pair (M,w) is referred to as a symplectic manifold. Notably, the nondeegenyyycy of implies that M must be even-dimensional.
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Contact Geometry. While symplectic manifolds provide geometric framework for modeling the dynamics of conservative systems in classical mechanics, a more general approach is required to describe con-cesservvtiee systems. This is addressed by contact manifolds, the -idddinnnsional counterparts of symplectic manifolds (Geiges, 2001; Bravetti et al., 2017). A contact manifold is de- fined as (M, 17), where M is an odd-dimensional smooth manifold, and rj is non-degenerate 1-form known as the contact form (Geiges, 2008). The contact form satisfies the maximal non-integrability condition, meaning that the top-degree differential form 1y A (dn)² d + is nowhere vanishing on M. This form is constructed by taking the exterior product of v with the d-fold wedge product of its exterior derivative dij, i.e.,
(dn)) dh db A ....A. dn.
(12)
d times
The (2d + 1)-form defines a volume form on M, ensuring that the hyperplanes ker(n) C T.M, constraining the dynamics the contact manifold, do not form a foliation, i.e., they do not partition the manifold into lower-dimensional submanifolds (Geiges, 2001; 2008). Geometrically, this means that the contact distribution imposes non-holonomic constraints: it restricts the admissible directions of motion at each point without confining the dynamics to fixed submanifold or energy level. This property crucial for modeling systems where energy can change over time, enabling constraints on energy behavior without enforcing conservation.
Like symplectic geometry, contact geometry connects scalar functions to vector fields, enabling the description of dynamical systems (Zadra, 2023). Given an energy function H .M + I, the dynamics on a contact manifold are defined by a contact Hamiltonian vector field XH, as follows,
n()) (X) dn((X H1 X) Lx (X)), AA ((TM).
(13)
Unlike symplectic geometry, where dynamics are confined to energy-preserving flows along the level sets of the Hamiltonian, contact geometry allows for an additional component of motion. Specifically, the dynamics on contact manifold are not restricted to the term dn(Xu, X)) which lies tangent to the level sets of H, but also include a transverse component Lx, (X), arising from the non-degeneracy of the contact form. Consequently, while in symplectic geometry the symplectic form w is strictly preserved, contact geometry allows the contact form 1) to be preserved only up to a scaling factor a IR (Bravetti et al., 2017).
C.2 RIEMANNIAN AND JACOBI METRICS
The Jacobi metric is a rescaled version of Riemannian metric g that allows Hamiltonian dy- namics on the cotangent bundle T be represented as geodesics on the Riemannian manifold (M,g). The construction is detailed below.
The Riemannian Metric. Let M be a smooth compact manifold. A Riemannian metric g : TM x TM + R is a smooth, symmetric, and positive-definite bilinear field of maps defindd on pairs of vectors in 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. For smooth curve x=(1) [to, 4]] - .M, the length w.r.t. the metric g is l f (()) where +(t) E Tats) is the vector tangent to the curve at x(t). The curve minimizing this length between two points s=(o) and x(tz) on M called a geodesic. Geodesics generalize straight lines in Euclidean space to curved spaces, representing the shortest paths in the geometry induced by g.
The Jacobi Metric. The geodesic flow xxt) on a Riemannian manifold (M,g) lifts to the joint evolution of coordinates (z(t), (e(x(t),((t)) on the cotangent bundle T*M (Abra- governed by an energy function
ham Marsden, 2008). This extended dynamics (z, T*M + R = g- (a, which remains constant along the flow. A reparame- terization ds dt links the trajectory of the integrated dynamical system at time on M with the length of the corresponding geodesic on M. This framework reveals a 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 TM is naturally equipped with symplectic structure, making it symplectic manifold (T*R, w).
This formulation can be further generalized by introducing a potential energy function into the Hamiltonian, given by (xx,a) 9 (', F(x). In this setting, the geodesic structure un-
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derlying the Hamiltonian flow is determined by the Jacobi metric,
9. = (H - F(x))
(14)
which rescales the original metric by a 2008). The corresponding time reparameterization takes the form ds = VH - F(x) dt. restoring the interpretation of the trajectory as geodesic with respect to the metrie gJ (Udrişte, 2000).
conformal factor (Abraham & Marsden,
D INSIGHTS ON THE SCHRÖDINGER BRIDGE
D.1 HAMILTONIAN STRUCTURE OF THE GENERALIZED SCHRRDINGER BRIDGE
This part presents the derivation of the Hamiltonian structure of the mmGSB problem (2), intro- duced in section 3, obtained via the method of Lagrange multipliers. To transform the constrained optimization problem into an unconstrained one, we incorporate the Fokker-Planck equation, scaled by the Lagrange multiplier St, into the original running cost L (i.e., the Lagrangian), as follows,
J p' = 1:
St)dt;; dt;
(15)
C(o', ('.' = / (1))(()) + +(()) (U()) (x)
M
+ { (x) (0,p' (x)
+ V. (z) =Ap'(z) de.
(16)
Pokker-Planck equation
The optimality conditions resulting from the extremization of the cost functional in equa- tion (15) follow from the Euler-Lagrange equations, generalized to the setting of classical field theory (Blohmann, 2024). In this framework, the arguments of the Lagrangian are viewed as smooth fields defined over space and time. By setting to zero the variations of L with respect to these fields, we obtain the stationarity conditions for J. For generic field vf (z). the corresponding Fuler-Lagraage equation takes the form,
in equation (16),
d = C + (,0, C) + V
+ A. = 0.
(17)
Applying equation (17) to equation (16) for the fields v², p², and St, we obtain the following system of optimality conditions, d.L = v'
=0 = vx))) ) (z))
(18a)
1 d, ~ (1)( - - =
(18b)
dgL = (.)(()) + V.
- (zp
(18c)
Substituting the expression for the optimal velocity from equation (18a) into equations (18b) and (18c), we obtain the following Hamiltonian system,
a,p² (x) = 0- = - ( (d())
(19a)
(19b)
with the corresponding Hamiltonian function,
1 H(p's5)) = 2 / IV, (z))) S. (x)) +e S ('(r) dr. (20)
The Hamiltonian system (19) can be reformulated in a linear and decoupled form via the coordinate transformation from the original variables (p',St) to an alternative canonical set (*)))) This transformation, known Hopf-Cole coordinate transformation (Léger Li, 2021: Chow et al., 2020), is derived from the generating function F.
F(p, $) = p(r) (log -1),
(21)
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and preserve the symplectic form, ie.. dF Sdp Sdp. Following the standard generating-
function rules (Chapter 18.3 Johns (2005)),
(x = 0g F(), (² = a.F((,)
(22)
the new coordinates are explicitly given by,
(xx = p²(x;) (x)) = (x) log (x))
(23a)
(23b)
The Hopf-Cole transformation is well established in the literature, not only for simplifying the mathematical form of the equations, but also for enabling the design of efficient numerical inte- gration schemes (Léger & Li, 2021). In our context, is particularly advantageous for obtaining Hamiltonian with separable kinetic and potential energy components. The Hamiltonian system (19)
then becomes.
(xp'(x)) + Vx V 0,
(24a)
(,5" (z) + (zx)) 0rp(()) = 1
1
+ V
(24b)
- e² A. log log (I) U(x).
Expanding the squared norm equation (24b) and substituting 0,p² (z) from equation (24a) yields,
(,5* (())) V (z)
- (z) - V log, (2))1
-VV,S(=VV log (z)) (A)
(25)
ea log (z) Us)).
Using the identity,
(xx V
= +.
(26)
equation (25) simplifies to,
a,5 = -x)= -V -
e² --IV. log ('))) eaD log (x) (x) ((:).
(27)
This form reveals the emergence of the Fisher information, defined as,
T(p'(x)) = / I (x) de, (p'()(x))) = , log p' - V log (x=))
(28a)
(28b)
Thus, equations (24a) and (27) admit the linear decoupled Hamiltonian formulation described in equation (3),
(.)(()) 0 H.H(() V (+))))
0,55(z) = --,l((-- - -
V. S (z) + 1
2 (=)))
governed by the Hamiltonian function,
1 H(O',55 = [ IV. / (l)) de - (T(((t)))
(29)
Kinctic energy KC
Potential energy FF
In this formulation, the Fisher information contributes the potential energy and encodes the effect of stochastic diffusion. Minimizing the Fisher information term promotes smoothness in the density and steers the Hamiltonian flow toward the target distribution pt, providing greater robustness
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due to the regularizing effect of diffusion. This mechanism has been studied in the literature and employed in control applications for its regularization properties (Chen et al., 2025).
D.2 CONTACT HAMILTONIAN STRUCTURE OF THE NON- CONSERVATIVE GSB
Here derive the contact Hamiltonian formulation of the NCGSB problem, introduced in equa- tion (4) and discussed section 4. The structure of the derivation closely mirrors that of the GSB in Appendix D.I, with key distinction: the Lagrangian L now depends explicitly on the accumu- lated action 2¹. Specifically, the augmented cost functional and Lagrangian are given by,
J( , L/¹ C(v',p' ,',''' ,',22) dt:
(30)
p' 2 = S.d. (210((0))++ o² (x) 72
+ I (z)) (Oip' + v (z)) =x) (z)))))) dr,
(31)
where the scalar y acts as a damping factor. Since L depends explicitly on the evolving action e', the problem lies outside the scope of classical variational calculus. Instead, it fits within the framework of non-conservative variational principles, where the cost functional J evolves dynamically with the system. This is naturally addressed by the Herglotz variational principle. which extends the Euler- Lagrange equations to systems with dissipative effects. The optimality conditions obtained from the variations of L. namely, the Herglotz-type Faler-La equations, for generic field argument vi in this case take the form,
d L = Jx (0a, () + Va
(0s, - O. 0g C = 0.
(32)
Applying equation (32) for the fields of, p², and St, and recovering the dynamics of 2 from the optimization problem (4), we obtain the following system of optimality conditions,
d&L = d'(x)''(x) - ('(z)V =0 = v'(z)= (x))
(33a)
1 dnL _ - (.))
( - - EA, (x) + +(x)) (S))))))()))) =0,
=0,
(33b)
dsL = 0.p² (z) + V. (p'()))()))) - (x) =0, 0,
(33c)
+U()) (x) dr - 72 = 0.
0,2²
(33d)
Compared to the optimality conditions for the GSB problem presented in equations (18), the set (33) includes additional term, ->)'(x), in equation (33b), which accounts for the dissipation term.
By substituting the expression for the optimal velocity from equation (33a) into equations (33c), (33b), and equation (33d), we obtain the system of contact Hamiltonian dynamies,
a (x) (.)) - V. (z())) + =
(34a)
(x)11 (x5)(x) (.) (())0, = - E - (x)++ (x)) (34b)
- - 0,2' = (0()0) - H() H()=
[V (
(x() (z 72',
(34c)
with the associated contact Hamiltonian function given by,
St / M
2t S U())p'z)) te / 5'(x) (x) dr
(35)
In this case well, it is beneficial to derive a decoupled and linearized representation of the dynamics. To this end, we perform a coordinate transformation from the original variables (p', St 2*) to an alternative canonical set (Øt, (+ 2²), while preserving the contact structure,
dz Sdp dā Sdp.
(36)
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This is achieved via contact transformation generated by generating function F, defined S, - p(x)$ + (log) p(x) 1) + 2t.
(37)
which induces the following coordinate change according to the contact generating rules in Struckmeier & Redelbach (2008),
(((x) = 0zF(),
(38a)
(x) = 0,F(-), 3' = F()) - Sp.
(38b)
(38c)
As result, the transformed variables are given by
(x()) = px(x)
(39a)
5 e log (xx,, 2' = + Ep' (x)
(39b)
(z) (log p² (z) 1).
(39c)
By substituting the new coordinates (39) into the dynamical system (34), we obtain the following transformed contact Hamiltonian dynamics,
0,p'(x) = =ggH()) = V. (0 (x))).
a,8 -()))(()-(((00))1)))
2 - -7) - log (x),,
1v= + 1 (000)((()))++(U
0,20 = ( H()= / (1 1 (x) + U(m)) ('(x) de + 1 216' 2 ey(log pf (x) 1)p²(a dx - -22"
2
with the associated contact Hamiltonian function given by.
1 H S0,,S,EE =²) 2 J S (xc))
Non-conservntive potential
Kinetic energy K.
(40)
U(r) (x) de - 2 ((')) / or 10)0)(1000 (x) dr.
M
Potential energy F
Entropy B
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D.3 GUIDED SCHRÓDINGER BRIDGE
We consider the bridge p', computed between the terminal marginals Pa and pb, using any variant of the SB problem (e.g., GSB, mmSB, NCGSB). This process can be modified to enforce desired conditions at any chosen time ia defined as,
y y= (a'";, x² p²:,
(41)
while preserving the underlying data manifold. Conditioning in this way modifies the probability flow of (Guo et al., 2024) as,
1 p'()e 11l 110-5(x'>11
(42)
where is a normalization constant, and x² denotes sample from the predicted prescribed distri- bution p'>(x-y) This weight biases the generation toward samples that satisfy the desired property y. a dynamical setting, we perform this conditioning by incorporating a control term G into the Fokker-Planck dynamics,
(x) (d))
(43)
A naive choice such G (zx Var f(x²) often drives the dynamics off the data manifold, produc- ing unrealistic samples far from it and the target distribution Pb. Instead, Bayes' rule provides the correct structure of the guidance term. The gradient of the conditional liglliield decomposes as,
Ue log (a" 1 ()) Vue log (x") + Vat log log x²)
(44)
estimated by G
Substituting the conditional form from equation (42) yields,
(x) = Vze log e -1a- -11a-1(=">)11 = V I1y {1²
(45)
Specifically, in the NCGSB framework (4), this guidance is incorporated directly into the drift of by adding the penalty |yy f(x*'))] to the Lagrangian in the action dynamics constraint (4b),
p') L'' 0,2² dt,
min v
0,²² / (1 U()) s.t.
1(0-)) d'(x) de
(46)
a,p' + V. (p' (1))) ='),
= Pa: p¹ Pb: pmm= = Pm: Vm E (1...,,, M).
Here, the inclusion of |1l - f(x*> 112 in the Lagrangian produces the desired - V. fly )( correction the drift, while preserving the Schrödinger bridge structure and constraints.
E RESNET PARAMETERIZATION FOR DISCRETE GEODESICS
E.1 GEOMETRIC INTERPRETATION
As stated in the main paper, our objective is to compute a geodesic p¹ on P+MM), induced by the contact Hamiltonian dynamics, that is constrained to pass through set of observed marginals {Pa, Pm.Po) (i.c., discretized distributions along the probability path). These constraints naturally lead to discretized parameterization of p', where the overall density transformation is modeled as composition of maps, each connecting a pair of consecutive observations. In this context, a ResNet architecture is ideally suited for this problem, as its sequential block structure directly mirrors this piecewise, compositional nature of the approximated geodesic.
Geometrically, each parameterized pushforward defines a vector O,p' in the tangent space of P ft-1 representing its change rate. The pair "p111 (0'=') thus corresponds to point on the tangent bundle TP+ (M), with parameters 0 E O representing one of the possible coordinate charts for
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this update. As such, the parameter space ② forms finite dimensional subspace of TP+ (M) (see Fig. 5). The block transformation defines a smooth immersion Tg. è - TP* (M) with full-rank Jacobian VzTga, ensuring the pullback of the Wasserstein metric g to θ, denoted T 9 is well-defined and induces a Riemannian structure. This Riemannian metric identifies with its dual O' via the standard aangent-ootangen isomorphism (do Carmo, 1992). Consequently, the contact Hamiltonian dynamics on T*P+ (M) x R can be equivalently expressed in the reduced phase space O' & (Wu al., 2025). At the same time, the Wasserstein manifold is approximated by the fiite-dimensional submanifold Pø (M), whose tangent space is TPg (M) = O.
E.2 TRAINING ALGORITHM
Algorithm Training the Contact Wasserstein Geodesic (CWG) Framework
Input: Dataset: samples from marginals Fa, {i,f) Lb, (6.) ~ Pby Fm, (6.8) ~ {pm Jm= Output: A trained ResNet "
Part I: Initialization to Match the Initial Marginal
1: for = to E do for j to 13 do x² ~ X x= [xa(x*)
b Epoch loop Batch loop
2:
3:
b Sample from reference distribution Apply initial block b Match initial marginal
4:
5:
mingo (2'0 (2'0, La (i.j)
6:
end for
7:
end for
Part II: Geodesic Optimization
8:
for I E do
9:
for j B do
10:
x² ~ X (x ,..,, I".] Tiorge (x*)
Sample from reference distribution
11:
Full ResNet transformation
12:
mingg00 {((x")) k=o: Fo, (is) I.,, (153)
b Minimize geodesic loss (equation (9))
13:
end for
14:
end for
E.3 STABILITY CONSIDERATIONS
The proposed solver discretizes the geodesic flow on the Wasserstein manifold and can be inter- preted as a single-shooting direct method for solving the optimal control problem determining the optimal set of parameters {0k" - Specifically, the ResNet-based parametrization Tio yk. K produces discrete sequence of intermediate batches {-/ Jk=0 converting the continuous optimal control problem into nonlinear programming problem. This problem is then solved using standard static optimization methods, with gradients computed through backpropagation. The loss function in equation 9,
8 == dev LK pb) +
Σ
(Po pm)
4 ders (ptt Pg P
Terminal marginal
m==1
Intermediate marginals
Energy minimization
involving the computation of the Wasserstein distance, which can be explicitly rewritten as,
N
Ta ||
Tu
fum
Vi
2
(47)
1.1
1=
Terminal marginal
Intermediate marginals
Energy minimization
where म. denotes the optimal transport plan between samples and the data batches, optimized in order to minimize the transport cost and preserve the constraint, Σn тa = 1, ensuring full mass conservation between batches. Importantly, the optimization of the transport plan ma is
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subproblem carried out for fixed set of parameters {0 k= This subproblem is linear program whose solution depends Lipschitz-continuuss on {0th
k== i...
K
T (10 )E=) x (()))))) - L
10 M
100 1k=
(48)
where L, is scalar constant. The two differences quantify, respectively, the change in the transport plan induced by a perturbation in the parameter set, and the perturbation in the parameter set itself.
The computation of the gradient of the loss in equation (47), which is responsible for updating the set of parameters according to,
10 (0) ) =2 Vol.
(49)
leads to the appearance of three types of terms:
VH which is Lipschitz in {0*) k= because the transport plan is the solution of a con- vex optimization problem whose dependence on the parameters is Lipschitz-contimuus:;
Jk-o Vollz,- - 1/2 where x²" depends nonlinearly on [0 due to the nonlinear ResNet parametrization T(o=) ) used for expressivity. This implies that multiple different param- eter sets may minimize this term. However, around local optimal point the problem of minimizing the Euclidean distance is locally convex;
K Vodle, which is nonlinear function of (arg Jk=0- If th is least piece-wise smooth, it is possible identify a local minimum for {xy) ) J==0 and therefore a corresponding local minimum for {0*) depending on the initialization of the parameters. Even in the most complex scenario considered in this paper, the energy-varying SB case, the required smoothness assumptions are satisfied. Specifically, for g H (p**, S ) +U(*** T(p**) B(p*),, ). H is a smooth function depending either via at least piecewise smooth learning models (Apps. G.4, G.5, G.7) or via a LAND metric (Arvanitidis et al., 2018) with smooth kernel (Apps. G.2, G.3), and both I and B are smooth functions.
hyperparameter, U is implemented
(م does not destabilize the training and con- verges toward a local minimum, which geometrically corresponds to one of the possible coordinate charts that can describe the tangent bundle TP+ (M) (see Appendix E. for the geometrical in- terpretation). The stability of the gradient for long time horizons is moreover guaranteed by the ResNet architecture itself, where the skip connections prevent detrimental effects such as exploding or vanishing gradients (Zaeemzadeh et al., 2020).
Therefore, each update 1 -
E.4 GEODESIC PARAMETERIZATION: TIME AND ARC-LENGTII PERSPECTIVES
The proposed ResNet architecture T T(ax), > discretizes the geodesic p¹ on P (M) into a piecewise- linear path segmented at nodes (pb k= geodesic. By default, the discretization obtained by minimizing the curve energy,
thereby defining specific sampling of points along the
K
L (Pa, Pb) = Σ Rum
(50)
corresponds to uniform are-length discretization s, where each segment has approximately the tv same length, i.e., d2, (p': Pg and the time discretization f, which describes the progression along the geodesic with respect to the time variable, is given by ds / Kkphy dt, where Kphy denotes the kinetic energy of the system whose dynamics is interpreted as geodesic. In the meery--consevve case, Kenhy remains constant, and the time and lengthh discretizations differ only by constant rescaling of the geodesic length. =k=o remain unchanged, uniformly distributed in space and in time. In contrast, when modeling energy-varying system whose associated trajectory
C. The relationship between the arc-length discretization
Nevertheless, the discretization nodes {p²* K
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projects onto geodesic, the curve energy takes the form,
Lig (Pas pb) Σ Thi dww, (Pz Pa di ");
(51)
where the scaling factor Φ4 - H S ,"), - F(p²*) )- - B(p**) accounts for the use of the Jacobi metric. This scaling factor also determines the length of each segment, which is inversely proportional to it, i.e., di (p'. Pg 1)
C Consequently, the arc-length discretization 4%
dwg (pg Pi () - As is non-uniform, while the time discretization A - -V remains uniform, as the nodes assumed to be sampled at constant time intervals. The origin of this discrepancy lies in the kinetic energy term relating the arc-length and time increments, As = /Kppy which no longer constant in the energy-varying case.
This reasoning provides insight into the relationship between the energy term 4'x used in the geodesic minimization and the physical kinetic energy of the modeled system. Specifieally, since the geodesic minimization problem yields.
dwyy th Po 1x-1) C
(52)
T
while, as established above,
1 = As² = Kghy = Kphyy C
dy (00 P
(53)
obtain 4hx Kppy This result shows that the energy employed in the geodesic computation inversely proportional to the physical kinetic energy, scaled by the constant k. This relationship arises from the coupling between the arc-length and time discretization of the system's 's trajectory.
E.5 TIME COMPLEXITY AND PRACTICAL CONSIDERATION
At each iteration of the geodesic optimization of Alg. 1, we sample N points x E .M of dimension D from the reference distribution À and pass them through the ResNet. Assuming its architecture consists of K blocks, each being an MLP of L layers and hidden dimension W, the computational cost of this forward pass O(N D K L W)) The loss function é in equation (9) requires M K evaluations of the Wasserstein distance between sample batches, and K evaluations of the factor d'x For the Wasserstein distance, we employ the geomloss library (Feydy, 2020). which uses the Sinkhorn algorithm with time complexity (N Tas).. where To is the number of Sinkhorn iterations until convergence (Feydy, 2020). To evaluate T², we apply its complete definition,
[ ([((6')) 2: (2ogg(225) - 1)] Pe dæ + I=2
t' = = +
(54)
batch of samples. As discussed in Sec. 3. the Fisher information I [(p)) ) in the potential energy originates from the entropy regularization in the SB formulation and does not require explicit com- putation. Its effect implicitly captured by the regulay-re Wasserstein distance. Therefore, evaluating 6'x reduces to computing the scalar functions H and U, along with estimating the en- tropy term log(2p)²) pb"" which is the computational bottleneck. This term can be estimated using k-NN entropy estimator with time complexity O(D N log N) (Borelli et al., 2022). Considering all components, and given that K 2 M, the overall time complexity becomes,
(M))++))))())))))))) (K d log N) = (TV
N))).
(55)
Our method demonstrates highly favorable scaling properties, offering a significant advantage over existing approaches. Notably, its computational complexity scales linearly with data dimensionality D and nearly linearly with the batch size N. Furthermore, our model's 's performance is only weakly influenced by the number of marginals and it cireumvents the expensive outer iteration loops. This stands in stark contrast to existing methods, as elaborated next:
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Hong et al. (2025) scales quadratically in N and exponentially in D, with time complexity,
(TTier DKN exp(a(D + +)))
(56)
where Ther denotes the number of Belief Propagation iterations required for convergence, and a is a parameter determined by the order of the Gaussian process and structural
sumptions on their representation basis.
Chen et al. (2023) proposed method with time complexity.
((
(57)
where Tiar denotes the number of Bregman iterations needed for convergence. The quadratic dependence on 2D arises from the state space being doubled by incorporating velocity, which comes the cost of computational efficiency.
Shen al. (2025) employed score-matching SB solver (Vargas et al., 2021) for the mmSB problem with computational complexity. O M2(2D)),
(58)
where Tier again denotes the number of Bregman iterations needed for convergence, with reported values between 2000 and 4000. Although the method is nearly linear in both N and D, it depends explicitly on the number of intermediate marginals and suffers from additional outer-loop iterations.
As highlighted in this analysis, CWG method advances the state of the art remarkably efficient computational approach: it scales linearly all relevant terms and does not require itera- tions. The baselines (Shi et al., 2023; Bortoli et al., 2024; Liu et al., 2024) do not explicitly report their computational efficiency, so the comparison with our method addressed empirically in the experiments.
F IMPLEMENTATION DETAILS
F.1 EMPIRICAL APPROXIMATION OF THE WASSERSTEIN-2 DISTANCE
Let Pe and Pa be two probability distributions from which we draw batches of samples (Ie, ~ M Pe and {xa.5 Jj=1 ~ Pa. respectively. The Wasserstein-2 distance, denoted by dw., Pa). measures the minimal cost of transporting mass between these two distributions. To approximate this distance empirically, we first construct a cost matrix C' E N.Nx.. XM, where each entry,
C = || l2
(59)
represents the squared Euclidean distance between sample He, and sample zaj. A transport plan is then defined as a matrix æ E N that assigns how much mass to move from each Fe, to each Edj. minimizing the total transport cost weighted by C. To avoid degenerate solutions where all mass is concentrated on few points, entropy regularization introduced, encouraging smoother and more distributed transport plans. For this computation, we employ the SamplesLos function from the geomloss library (Feydy, 2020), with parameters resumed in Table 9.
In the conditional generation setting, the probability flow p'xz y) is conditioned f(**x, ), with x' ~ Pa. to ensure that the generated samples satisfy y at time step tz. Therefore, when comparing the prescribed distribution Ps with marginal Pm.. modified Wasserstein-2 distance d' (Ps: Pmm) incorporating the feature penalty is used. This distance defined via the cost matrix
the feature y
Ci
|| Zm. lly f(mm,))1
(60)
which penalizes transport plans assigning to samples Im. inconsistent with the conditioning feature y.
F.2
TEASER IMAGE
The system illustrated in Figure depicts the time parameterization of three different Schrödinger bridges connecting two 1D Gaussian distributions with identical variance but different means. The
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energy-conserving bridge (-) corresponds to the solution of the mmGSB problem (2) in the specific case of U = and without intermediate marginals pm. The computed bridge can be interpreted as piecewise geodesic on the Wasserstein manifold Σ endowed with the pullback Wasserstein metric T" ggy
In contrast, the two energy-varying curves (-, -) instead correspond to the solutions of the NCGSB problem (4) in the case of no intermediate marginals and U = 0. These trajectories are piecewise geodesics with respect the pullback Jacobi metric TT (H())) F(gg B(pg)) )) T"g Here. the potential energy term FF -U duces to the Fisher information contribution, yielding the equivalent expression Tp = (H())) [( - B(()))
re-
g Minimizing the geodesic energy with respect to this metric yields trajectories that tend to reduce the scaling factor ( (H T(pb') B(()))) As result, these paths preferentially traverse low-energy regions of 2 while concentrating the discretization nodes {pg 1 in the higher-energy portions of the curve, since long segments between nodes in such regions are penalized by the large value of the scaling factor. Specifically, the geodesic com- putation exhibits the following behaviors:
The minimization of the Fisher information 1 (p'g) ). promoting smoother and more regular trajectories by penalizing rapid variations in the density. This ensures stable and coherent dynamics along the path.
The maximization of the entropy term B((g)), upper-bounded by the Hamiltonian H since the metric Tu must remain positive. Consequently, higher Hamiltonian values correspond higher-entropy paths.
The concentration of the discretization nodes in regions of higher energy (i.e., where H(pg") ) is larger). As the evolution of the Hamiltonian can be prescribed, selecting de- creasing or increasing law results in a stochastic trajectory that slows down or speeds up over time, respectively, as discussed Appendix E.4.
These behaviors can be observed in the two energy-varying probability paths illustrated in Figure 1. In the red curve (-), the Hamiltonian function, ranging from 1.5 to 0.5, is defined as,
p(x*x) u(x*x)
1t 0.5 + u(x*k) - u(x'))
(61)
where (x(**)) denotes the mean of the samples x²* ~ p8+- The Hamiltonian is thus decreasing, producing dynamics that evolves slowly at the beginning and accelerates toward the end of the time interval. The opposite behavior is observed in the purple curve (-), where the Hamiltonian increases from 0.01 to 1.01,
H 1.01 p(x*K) - 1((x**) p(x u(xto)
(62))
In this case, the dynamics gradually slows down over time. Moreover, the lower overall energy level results in reduced entropy, as evidenced by the narrower Gaussian distribution observed the midpoint of the bridge in the purple curve compared to the red one.
G EXTENDED RESULTS
G.1 EXPERIMENTAL SETUP
The LiDAR Manifold Navigation and Cell Sequencing experiments were conducted on a machine equipped with 13th Gen Intel Core TM 17-13850HX CPUs. The Image Generation experiment was run on system with an NVIDIA GeForce RTX5090 GPU (32GB VRAM, CUDA12.9, driver ver- sion 575.64.03. (Table 3), Sea Temperature Prediction (Table 4 and Appendix G.4), Robot Task Reconstruction (Table 8). FFHQ Transfer (Table 7), and SONIST-t-EENNSTS (Table 38) experiments are based on a total of ten evaluations obtained from training runs with different initial conditions. The tables report the mean and standard deviation of these distributions.
Results for the LiDAR Manifold Navigation (Table 2). Single Cell Sequencing
The ResNet architecture varies depending on the task. For the LiDAR Manifold Navigation, the Cell Sequencing and the Unpaired Transfer experiments, each block is an MIP that processes the
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output of the previous block and generates update, which is added to the input with step size T: 2' 6- x' + T block(z). Details of this architecture are provided in Table 11. In contrast, for the Image Generation experiment, the imput consists of images, and each block is implemented as 2D U-Net. Details of this architecture are provided in Table 12. The total number of CWG parameters used across the different experiments, alongside the baselines, is summarized in Table 13.
Parameter
Lidar
Cell
Sea
Robot
FFHQ
Mnist
Parameter
Value
samples
1000
1000
200
100
1000
300
Entropy
0.05
Weight wb
10
10
100
10
1
1
Euclidean norm order
2
Weight Wm
10
100
Scaling 0.7
Weight wy
1
1
1
1
Table 9: SampleLess parameters.
Table 10: Loss function (9) parameters for the experiments.
Component
Image ResNet
Component
ResNet
Sea
Robot
Mnist
Number of blocks K
8
6
Lidar
Cell
FFHQ
Input channels
3
1
Number of blocks K
20
5
6
Output channels
3
1
Layers per block
Layers per block
2
2
Layer hidden size
30
128
1024
Downsampling blocks
2 (32 and 64 channels) 2 (32 and 64 channels)
Step size 7
0.1
0.1
0.1
Upsampling blocks
Input dimension d
3
512
Step size
1
Input dimension d
4096 12288
784
Table 11: Configuration the ResNet
Table 12: Configuration of the Image ResNet.
During training, the weights {wb, Wm: Wy balancing the loss terms in equation (9),
M
K
ê = wp dlw, (ptx pb) + Wm
Σ
dews (Pg 1tmm .Pm) + Weg Σ th drv, (p% 00
(x-1).
m==1
along with the number of samples N used for Wasserstein distance estimation, are experiment- specific and summarized in Table 10. The GPU memory consumption for the two image generation experiments, comparing our CWG method with the baseline approaches, is reported in Table 14. Due to the explicit handling of the full probability distribution (albeit discretized form) within the ResNet architecture, CWG exhibits particularly high memory requirements. In contrast, methods such as GSBM and DSBM model only the drift component and subsequently integrate the dynamics. While this makes them significantly more demanding in terms of computation time, they are more memory-efficient than CWG.
Experiment
DSBM
SB-Flow
GSBM
SBIRR
DM-SB
CWG
LiDAR Navigation
0.14 M
0.33 M
0.017 M
Cell Sequencing
M 2.5
0.017 M
Sea Temperature
79.2 M
11 M
271 M
5.2 M
Robot Task
79.2 M
11 M
271 M
7.3 M
Unpaired Transfer
5.7 M
6.6 M
5.7 M
15.7 M
to-niisstm
4.6 M
3.3 M
4.6 M
5.2 M
Table Number parameters millions, M) of the models reported in Table
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Methodology
CWG
GSBM
DSBM
Sea Temperature (MB)
25200 + 200
9600 = 300
9000 + 200
Robot Task (MB)
16100 + 200
12500 = 300
10200 + 200
Table 14: Comparison of GPU memory consumption across the methods evaluated in the Image Generation periment. CWG shows decrease memory usage for the Robotic Task Reconstruction experiments, whereas the other methods exhibit an increase due to the reduced batch size used in this training (Table 10).
G.2 LIDAR MANIFOLD NAVIGATION
The LiDAR dataset (OpenTopographyy 2025) consists of point clouds contained in the domain [-5, 5|8 C R The objective of the experiment is construct a bridge across the data manifold for connecting two distributions while avoiding regions of high elevation and remaining closely aligned with the manifold structure. The initial distribution Pa. composed by a mixture of 4 Gaussian distributions. the target distribution is composed of 2 Gaussians on the two sides of the mountain. The manifold shape is incorporated in the problem through the potential function U, inherited from the baseline (Liu et al., 2024),
J (z) dx J/m (Umanitola()) + Unoight(1)) pf (x) dx,
(63)
Umamitold (x) = Wimmitold (1b(()) x:l?, Uneigha (x) = Wheight | (x))1
Here, (x) denotes the projection of point x onto an approximate tangent plane, estimated from its p nearest neighbors on the data manifold. The notation (x())(x) refers to the z-coordinate of (x)) iee, the height of the fitted plane. The weights Wmanifold and Wheight control the relative importance of the two terms in the potential function. We now detail the construction of ((xx). Let N₂ = (ui....,z4%, x denote the set of p nearest neighbors of x E IR in the dataset. To approximate the local tangent plane, employ moving least-squares (MLS) procedure (Levin, 1998). Specifically, the plane parameters (a, b, c) are obtained by solving,
2 min Σ wo, 2n)) (ax) (ax!(a) + bxr 1()) + x ) a.b,c P
(64)
where the superscripts indicate coordinates and the weights are defined as,
w(x, a!) = exp |
(65)
with y being a scaling parameter. Given the fitted plane, the projection operator (xx) is defined as,
(x)) = x +
|||
n
1]"
(66)
where n denotes the plane's normal vector. Differentiation through v naturally restricts gradients to this tangent plane, thereby ensuring that optimization of the state cost U evolves within the geometry of the data manifold. The values of the parameters used in the computation of the projection operator (xx) are provided in Table 15.
The quantitative results reported in Table 2 aim to characterize the two key aspects of the Schrödinger Bridge problem, formulated in the stochastic optimization problems (2) and (4). The Optimality metric reports the length of the stochastic bridge, computed as the square root of the cost functional J. quantifies the ability the approach to find low transport-cost solutions connecting the marginals Pa and Ps- In contrast, the Feasibility metric is computed as the Wasser- stein distance between the marginals and the nodes of the discretized bridge approximating them. This metric measures how well the boundary constraints in the optimization problems are satisfied. Formally, these metrics are defined as.
Optimality = VJ;
(67a)
Feasibility dw3 (ob' pa) + dw, (o'" ,b).
(67b)
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In the guided generation setting, the feature function f, defines follows,
((a) = (w (a) - b,) + ReL by); Th ~ Ph
(68)
used penalize samples from the terminal marginal distribution Ps on the left side of the moun- tain. The parameters used in this experiment are listed in Table 16, and the results of guidance fine-tuning are reported in Table 17. The training time (tt) metric shows that guiding the generation requires only 10.7% of the original training time.
Table 16: Parameters for the feature function f (68), used for guided generation in the LiDAR Manifold Navigation experiment.
Table Potential function parameters for the LiDAR Manifold Navigation experiment.
Parameter
Value
Parameter
Value
Weight manifold Wmannfoldd Weight height hhighht
Weight (z) we 1 Bias (x) by
1
Spatial scaling 7 # neighbor points p
0.1
Weight (v) Wy Bias (y) by
20
Table Bridge energy (0) with penalty f the LiDAR Manifold Navigation task, reported for our CWG method before and after guidance fine mue-tunnn..
Metric
before
after
J
12.31 10.18
2.49+0..3
tt (8)
+20+
310
G.3 SINGLE CELL SEQUENCING
The Embryoid Body (EB) stem cell differentiation dataset (Moon et al., 2019) captures cell state progression across five developmental stages [to. t1, 12, t3. t1 over a 27-day period. Snapshots were collected at five discrete time intervals: to [0, 3). t [6, 9), t2 E [12, 15). t3 E [18, 21], and 14 e 27]. These stages involve significant structural changes, with cells moving and reorganizing within increasingly stiff tissue while consuming and releasing mechanical energy (Zeevaert et al., 2020; Kinney et al., 2014). Consequently, the resulting dynamics exhibit energy dissipation and are better described by the NCGSB framework than by mergyyconservvng models. In this experiment, evaluate the framework's s ability to generalize to regions with available data by dividing the dataset into a training set [to, t2, t1. and a validation set [t1, ts]. The geometry of the data manifold incorporated into the NCGSB problem through potential function U, defined as,
N
N₂
U(x') = N1
# lay -
(x)
x..
(69)
N
where 2² pf denotes sample from the posterior distribution, and x4 ~ {Par Pms pb} are samples from the marginal distributions. N1 and N2 indicate the number of samples taken from p¹ and (Pa, Pm: Pb), respectively, while 7 represents a spatial scaling parameter. The exponential term acts as a kernel-like weight that reduces the influence in the inverse summation ΣM1 of the data points far from the bridge. Globally, the potential function U measures the distance of the bridge p' from the available data {Pa. Pm: Po} (the training set) and its minimization guides the construction of bridge that stays close to the known manifold while generalizing effectively to regions without data (the validation set). While other approaches leveraging this dataset (Tong et al., 2020; Shen et al., 2025) first embed the data into a 100-dimensional feature space using principal component analysis (PCA) and then restrict the analysis to the first five dimensions, this procedure excessively linearizes and flattens the data manifold, making navigation trivial and eliminating the need for intermediate marginals (Shen al., 2025). To better preserve the manifold's geometry, instead apply the PHATE algorithm (Moon et al., 2019) to the 100-dimensional representation, producing a 5-dimensional nonlinear embedding that more faithfully captures the original structure.
In the results reported in Tables 18 and 19, corresponding to the cell dynamics reconstruction shown in Fig. 12, the scalar Hamiltonian function H in the pullback Jacobi metric, T و3 (H
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F(p') -B('') T used in the CWG approach, is linearly varied from initial value H 0.82 to a final value H along the trajectory defined by the first principal dimension D1, i.c..
H ,1 u(=),)
Ho H + H st pt
(70)
This law is treated hyperparameter of the methodology- All ten experiments reported ducted under this Hamiltonian behavior. This increasing trend, associated with the reduction of physical energy (see Appendix E.4), aligns with the underlying biological process and has been shown to produce better results than the mnergy-conserve (see Table 21 and Figure 13a). detailed in Appendix F.2, the behavior of H plays a crucial role in determining how quickly the nodes of the stochastic path depart from the initial reference distribution and approach the subsequent reference marginal. In Figure 13b, the dataset exhibits decreasing Wasserstein distance dwy, between successive cell snapshots, reflecting that cell differen- tiation progresses more rapidly at early stages (Zeevaert et al., 2020). Training with the Hamiltonian behavior defined in equation (70) enables the model to capture this trend more accurately.
case, where H = 1 is kept constant
Ref.
CWG (ours)
DM-SB
SBIRR
1901
BeI
Din
Figure 12: Reconstructions from the models in the Single Cell Sequencing experiment.
Table 18: Wasserstein error (L) for all the time points in Single Cell Sequencing.
Table 19: Maximum Mean Discrepancy (x (x10-3) (L) for the time points in Single Cell Sequencing.
Metric
CWG (ours)
DM-SB
SBIRR
Metric
CWG (nurs)
DM-SB
SBIRR
dy.
0.10+o.0z
0.
0.59
0.86 0.02
MMD x
0.940.08
0.3
1040.11
dy x
1.11+0.06
2.25
1.92 0.
MMD xc
7.0+
10.1 10.33
9.9.
dyw, (2:²2
0.16+0..01
1.17 10.01
1.05 1.002
MMD
1.240.11
2.940.18
2.540.17
dw;
(3
0.33
1.64
1.86
MMD
3.5
6.7 0.3
dw, (x**)
0.11
1.03+0.01
1.38 00002
MMD
x
0.2
1.54
5.94
Table 21: Wasserstein error at validation (L) and training
Table 20: Single Cell Sequencing Parameters.
time (tt) (L) in the ablation study comparing the energy- varying (e-v) and energy-conserving (e-c) versions of CWG on the Single Cell Sequencing task.
Parameter
Value
Spatial scaling
0.3
Metric CWG (e-v)
CWG (e-c)
# bridge samples N1
1000
dw, (x
1.11
1.31
# marginal samples N₂
3000
dw, (x
0.33
0.49+0.03
tt (s)
710
710
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CWG (e-v)
CWG (e-c)
to
Dislanes dw..w.
Den
C2
f3
(--4
Dmm
Dir
(a) Reconstruction provided by the CWG method, in the energy- varying (e-v) and
(b) Geodesic distances.
(e-c) cases.
Figure 13: Visual reconstruction (left) and geodesic distances of cell snapshots the reference dataset (-). CWG (c-v) (-), and CWG (c-c) (-) (right). Snapshots are equally spaced time, but cell diffcrentiation progresses more rapidly at the beginning. as indicated by the larger early distances in the reference. The energy profile in CWG (c-v) more accurately captures this temporal dynamies.
G.4 SEA TEMPERATURE PREDICTION
The NOAA OISST v2 High Resolution Dataset (Huang et al., 2021) is a long-term Climate Data Record that integrates observations from multiple platforms (satellites, ships, buoys, and Argo floats) into a global gridded product. For this experiment, we use daily averages of sea surface temperature in the Gulf of Mexico between 1981 and 2024, represented as 64 64 single-channel images. We cluster the measurements over five-year periods and select five representative months to define five time frames: January (ln), March (41), May (l2), July (ta), and September (14). Each month corresponds to a distribution of images, denoted as {Pa (January), Pm:: (March), Pma (May),Pms (July), Ps (September)).
sample from one of these distributions is a heatmap of the Gulf'ss temperature for specific day in the corresponding month of the specified five-year period. The goal this test is to evaluate our method's ability to interpolate across missing time frames, generating realistic temperature maps for months without data. To this end, we partition the dataset into a training set (tu, t2, ta} and a validation set (t1, ts), and assess the quality of predictions the held-out months. To encourage generalization beyond the training data, we introduce a potential function U that penalizes deviations from the learned data manifold. Building on the approach of Song & Itti (2025), where generative models are evaluated by measur- ing the distance between their outputs and a geometric manifold of real images learned by VAE, we adopt similar strategy. Specifically, we use a state-of- the-r-r-tttttttt VAE architecture, with parameters listed in Table 22, to learn the manifold of the training images in our dataset. The potential function Uar"), for samples x² ~ p², then defined as the squared distance between bridge sample x' and its VAB-projected reconstruction j VAE(x"): U(*)) I|x =11 Results for each five-year periods are presented below.
In these results, the scalar Hamiltonian function 4 in the pullback Jacobi metric, T5 (H F(p') B(p')) Tý Wa is linearly varied from initial value H 1.36 to final value H' 1.0. This law is treated as a hyperparameter of the methodology, and all ten experiments reported in the following tables were conducted under this Hamiltonian behavior. In the geodesic computation, the solution path seeks to minimize T7 Ti- The functional F(p') measures the distance with respect the data manifold, i.c.. it evaluates whether the samples from the bridge remain coherent with the images observed during training, and it is minimized along the trajectory. Since H is prescribed, the entropy term B(p') is effectively maximized up to the limit set by H as the metric cannot become negative. Hence, the assigned Hamiltonian energy 4t determines the admissible entropy level of the bridge. In this experiment, a decreasing trend of H corresponding to an increase in physical energy (see Appendix E.4), was observed to be beneficial for modeling the warmer months, which appear to be more distant from the rest of the dataset in terms of Wasserstein distances compared to the colder months. A higher final energy H and thus higher final entropy B, was observed to be beneficial for modeling the warmer months. This effect can be associated with the increased thermodynamic entropy of such cases, leading to more diverse samples and larger
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variability across the data manifold. By prescribing higher energy level, the model is encouraged to capture this diversification, exploring regions of the data manifold that in other contexts (Arvanitidis et al., 2018) regarded uncertain and are typically avoided.
Table 22: Architecture of the VAn used in the Sea Temperature Prediction experiment. All Conv2D and ConTTraaspose2D layers kernel size stride and padding 1, followed by ReLU activations (except the last layer, which uses Sigmoid).
Stage
Layer (channels)
Output size
Input
Single-channel image
x 64 x 64
Conv2D (1-+32)
32 x 32 x 32
Encoder
Conv2D (32 +64)
64 x 16 x 16
Conv2D (64-+128)
128 x 8 x 8
Flatten
8192
Latent space
Linear /
Linear log o²
Linear reshape ConvT2D (128-+64)
128 8
Decoder
64 x 16 x 16
ConvT2D (64-322)
32 32
32
Conv (32-+1), Sigmoid
1 X 64 x 64
Table 23: FID scores at training and validation steps (1), and training time (It) (L) in Sea (2020-2024).
Metric
CWG
GSBM
DSBM
SB-Flow
FID(2 1
41.5641..0
FID(2") f1
121.47+5.61
160.68+1.54
242.26 +9.91
176.76+
51.5145.52
54.47=5.88
56.51
49.08
FID(x
159.53
185.5447.11
235.83
189.53
F
61,48.16.79
59.14_5.52
58, 58.39
59.96
(S)
1030
7360043300
19100
4950
Table 24: FID scores training and validation steps (L) in Sea Temperature (2015-2019).
Metric
CWG
GSBM
DSBM
SB-Flow
FID(2
42.96
FID f
130.33...994
145.97 16.12
220.
18.61 18.661
174.8445.47
FID (2:²2
58.72 1:446
62.13
65.47 16.16
60.88 15.11
FID Fa
135.25
142.82
228.
14.2
163.53
63.0216.14
59.73+5.86
61.29
6.05
59.0445.76
Table 25; FID scores training and validation steps (L) Sea Temperature (2010-20144
Metric
CWG
GSBM
DSBM
SB-Flow
45.19
FID 51
132.47+6.58
168.9346.22
255.36+100
171.03 771.03+6.35
60.57.1.08
59.79-5.98
61.89_a. 6.
58. 26.5.
140.84+6.01
144.60+6 83
235.08 08.9.07
179.81 +7.74
47.77
FID
54.92+5.23
51.63
50.27
51.3845.15
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Table 26: FID scores training and validation steps (tt) (4) Sea Temperature
Metric
CWG
GSBM
DSBM
SB-Flow
F
45.1942.27
FiD(x
172.69
195.83+8.04
260.97 10.92
194.62 26.93
FID(x
56.08
59.57=5.87
62.03
60.87=5.41
FID
x
126.87 16.94
152.26 63
243.58
176.32
FID FID(z"a
66.4346.58
63.17-6.34
64.86-6.63
64.37=5.59
Table 27: FID scores training and validation steps (L) Sea Temperature (2000-2000)
Metric
CWG
GSBM
DSBM
SB-Flow
FID (z
45.19
FID
140.4847.92
172.9648.34
250.86
184.58
FID(x
56. 73.33
56.97
59.64_6.11
59.53
FID (
142.1847.00
179.42 16.98
265.78 78 165.78110.65
10.67
195.34 17.62
FID x4
68. 68.29
65.91 6.68
64.73
64.23
Table 28: FID scores at training and validation steps (4) Sea Temperature (1995-1999).
Metric
CWG
GSBM DSBM
SB-Flow
45.19
FID
f
138.9247.95
176.1447.0r
257 167.26
174.53
FID
60.1946.01
63.48 6.12
66.07 66.7
6.557
62.95 16.06
154.37+s.14
193.28 18.47
266.80 11.334
191.4647.80 17.93
69.57
66.83 6.69
66.18 6.94
65.62
Table 29: FID scores at training and validation steps (L) in Sea Temperature (1990-1994).
Metric
CWG
GSBM
DSBM
SB-Flow
FID(2
45.19
FID 1
145.5947.22
184.3747.46
258.97
179.43
FID(2">
72.38+6642
78.69
82.13=7.01
75.75-6.27
FID(r"5
160.08
181.67+8.19
236.
136.46+11.59
187.26+8.00 2.
FID(x**
73.42
70.68
69.83_7.00
69.47 +6.98
Table 30: FID scores at training and validation steps (L) in Sea Temperature (1985-1989)
Metric
CWG
GSBM
DSBM
SB-Flow
45. 49.
FID 51
151.27
188.79
188.7947.68
262.47 11.7+11.44
194.87 +7.61
FID(2*2
66.8916.31
69.19 6.47
71.23 7.08
67.83 16.20
(
157.5848.77
160.37
258.96411.78
179.36+8
(14(
66.024741
62.87=7.16
61.59-7.34
63.37=7.19
Table 31: FID scores training and validation steps (L) in Sea Temperature (1981-1983).
Metric
CWG
GSBM
DSBM
SB-Flow
FID 45.19
FID (r
166.08
188.79 48..79
252.59 12.03
194.22
8.59
69.29 16.65
71.68 6.74
73.87 7.26
70.75 17.36
168.73 48.448
185.57
260.47
111.92
198.63
1.845
FID(2
77.8317.66
74.97
74.29
74.87
77.09
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G.5 ROBOTIC TASK RECONSTRUCTION
BridgeData V2 (Walke et al., 2023) is large and diverse dataset of robotic manipulation behaviors, designed to advance research scalable robot learning. this experiment, our goal is to reconstruct the full video of a robot performing manipulation tasks while training the network only on images from the beginning and end of the sequence, interpreted as samples from the endpoint distributions Pa and Pb- No intermediate marginals Pm are used. Following the Sea Temperature Prediction experiment, we introduce a potential function that penalizes deviations from the learned data manifold. This manifold is learned from the initial and final frames of the videos, and the penalty encourages plausible intermediate frames consistent with these distributions. We employ a state- of-the-art art VAE architecture, with parameters listed in Table 33, to model the image manifold. The potential function U(x²) for samples x² ~ p² is defined as the squared distance between bridge sample x' and its VAE reconstruction ze VAE(="): U(=²)
|| ==112. Fig. 7 presents snapshots of the reconstructions produced by our CWG method compared to the baselines. this experiment, the Hamiltonian function H is held constant, as varying yields no apparent benefit.
In the guided generation setting, we define the feature function f as
f(x)) ReL.U.....) be),
(71)
where c/th) denotes the (z)-coordinate pixel position of the centroid corresponding the target location of the item placed by the robot, extracted from the image Th sampled from the reference marginal Pb. We impose a penalty on this image (parameters of this function are available Table 34) so that samples corresponding to placements in undesired locations are discouraged, while those leading to desirable targets are favored.
The centroid extraction is performed by applying morphological opening and closing operations from the OpenCV library to remove noise and refine object boundaries, followed by color-based masking to isolate the object of interest.
to
t1
t2
t3
t4
t5
t6
t7
GSBM CWG (ours)
DSBM
Reference SB-Flow
Figure 14: Reconstructions from CWG (top), GSBM (second), DSBM (third), and SB-Flow (bottom) in the
Robot Task Reconstruction experiment. Red row shows the reference.
Table 32:
FID score (L) and training time (u)) (L) in Robotic Task Reconstruction.
Metric
CWG (onrs)
GSBM
DSBM
SB-Flow
FID
40.23+1.9g 18.83
95
149.78+0.81
73.4644.52
tt (s)
1090
91000
4900 2740042500
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Table 33: Architecture of the Von used in the Robot Task Reconstruction experiment. All Conv2D and ConvTranse Transpose2D layers use kemel size stride 2, and padding 1, followed by ReLU activations (except the last layer, which uses Sigmoid).
Stage
Layer (channels)
Output size
Input
Single-channel image
3 x 64 x 64
Conv2l) (3->32)
32 x 32 x 32
Encoder
Conv2l) (32 (64)
64 x x 16
Conv2D (64-+128)
128 8 x 8 8192
Flatten
Latent space
Linear + 1
Linear + log o²
Linear + reshape
128 x 8 x
Decoder
ConvT2D (128-+64)
64 X 16 x 16
ConvT2D (64-322)
32 x 32 x 32
Conv (32-1 -3), Sigmoid
3 x 64 x 64
Table 34: Parameters for the feature function / (71), used for guided generation in the Robot Task Reconstruc- tion experiment.
Parameter
Value
Weight we Bias be
1
30
G.6 FFHQ TRANSFER
The Flickr-Faces- HQ (FFHQ) dataset is high-resolution (1024 x 1024) collection of human faces that exhibits a remarkably wide range of visual variations (Karras et al., 2019). We closely follow the experimental protocol adopted by Gushchin et al. (2024), using their publicly available datasets from https: //ggttitttt /ht. cwm/ http//ggshchiiicc i Lights.. We split the dataset into training (first 60K images) and testing (last 10K images) subsets. Each subset further partitioned into age groups, specifically adults and children, corresponding to the source distribution Pa and the target distribu- tion Pb. respectively. The objective of this experiment is to generate a realistic child image from a given adult image, representing the same individual at younger age. For each image, we employ the traitra ALAE encoder (Pidhorskyi et al., 2020) to extract 1i2--immnsional latent vector. The SB solvers are trained directly these latent representations. During inference, we encode given image into the latent space, apply the learned bridge using the SB models, and then decode its outcome to obtain the transformed image. Because the probabilistic bridge is already defined within the ALAE latent space, which captures the FFHQ data manifold, the CWG method does not rely the potential function U to quantify deviations from image manifold. In contrast, the baseline GSBM approach retains a potential energy term U that penalizes excessive entropy and overly concentrated probability densities p², distinguishing from DSBM, which does not include any potential term.
Qualitative examples of the images generated by our CWG method and the baseline models are presented in Figures 15-17. A more compact, side-by-side qualitative comparison is provided in Figure 9, while quantitative results are reported in Table 7. The Feasibility metric indicates whether the computed bridges respect the imposed constraints, that is, whether they match the target marginal distribution. It is computed by evaluating the FID score between the terminal distribution generated by the models, p and the target distribution, Ps. defined in equation (72b). The starting distri- bution Pa corresponds to the source domain from which the input samples are drawn, and therefore it is naturally satisfied and does not require matching. The Optimality metric measures the geodesic length of the Schrödinger bridges computed by the models, in order to assess the transport cost that characterizes each solution. It is computed by sampling from the probabilistic path 2 p** in the latent space. The pre-trained ALAE decoder is then used to map these latent vectors to images x't, is which are subsequently encoded into feature representations z using a pre-trained Inception-v3
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network (the same used to compute the FID score). The curve length is then obtained from equation (72a).
K
Optimality
dw3 (#
th Frea ");
(72a)
A=11
Feasibility =FD Pb).
(72b)
Hence, the length of the geodesic computed from the bridge in the latent space is evaluated according the Wasserstein metric, pulled back from the feature space to the image space, and from the im- age space to the latent space. Although the baseline models achieve better Optimality scores (lower values), this improvement comes the cost of failing satisfy the marginal constraints, as reflected by their poorer Feasibility scores. Since constraint satisfaction is of critical importance, our CWG method exhibits superior overall behavior. This conclusion is further supported by Figure 19, where three facial age estimation networks, h1 classifier), and hı däbbilash&&/age-gnndeee waagggg-ennde--rddiiiiee ated faces at each generation step. The different curves in the plot represent the predicted ages for the images shown in Figures 15-17. While the CWG model exhibits more decisive transformation across time steps, the baseline curves remain comparatively flat, often producing outputs that fail to convincingly resemble children images. The age prediction models (h1, h2, ha) provide stochastic metrics that quantify their predictive uncertainty, which
moods/fccil--ag
h2 (nateraw/vit-age- were used to estimate the ages of the gener-
estimate their variance o² under the assumption of uniform error distribution. The average predicted age and the corresponding variance for the plots represented in Figure 19 are computed as,
(a( of
ba(z'))
o
Page
Dage
(73)
03
As shown in Figure 10, the energy of the bridge fitted in the latent space affects the age trends of the predictions. Decreasing the energy encourages smaller distances between the discretized nodes of the bridge at its beginning, resulting in images that are closer in age at early steps and gradually diverge toward later steps. Conversely, increasing the energy produces the opposite behavior. Im- portantly, in all cases, the Feasibility preserved, as the target distribution and mean remain well below the 18- yeay-orodld threshold. This contrasts with the baseline models, which in average does not respect this constraint.
Table 35: FID scores (1) measuring the distance to the initial adult distribution p. and the target child distribu- tion ps for samples generated different time steps using the CWG method the FFHQ transfer.
Metric
Pa
Pb
FID
3.143
33.953+0.930
FID
7.596+0.749
27.413+0.879
FID 2a
25. 75.73541.893
7.16410.806
FID 20
30.63640 8/13
4.375
FID f6
36.001 856
4.331640..60
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Input
Figure 15; Adult Child image generation in the FFHQ transfer using the CWG method.
Input
Figure 16: Adult Child image generation FFHQ transfer using the GSBM method,
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Input
Figure 17: Adult Child image generation FFHQ transfer using the DSBM method.
Input.
Figure 18: Adult
Child image generation FFHQ transfer using the SB-Flow method.
60-
70
60
60
70
Input
50
Input
40
40
40
Input
3.
Input
20
20
20
10
10
Input
Input
Input
Input
Step
Step
Step
(a) CWG
(b) GSBM
(c) DSBM
Figure 19: Age prediction for the samples generated in FFHQ transfer experiment.
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G.7 MNIST-TO-EMNIST
The MNIST and EMNIST datasets are low-resolution (28 x 28) handwritten character datasets that serve as long-standdee stand-sgnn benchmarks for generative modeling, representation learning, and probabilis- tic transport (Cohen et al., 2017). In this work, we adopt MNIST as the source distribution Pa and EMNIST (letters) as the target ps to study unpaired image-to-image translation. We restrict the anal- ysis to ten digits and ten letters. The two datasets are visually related but statistically distinct, making the MNIST-tO- HOM-NNS- transport problem sufficiently non-trivial while still permitting clear quali- tative and quantitative evaluation. Our goal is to produce terminal samples that match the EMNIST distribution while ensuring that the Schrödinger Bridge generates meaningful intermediate states that remain on the data manifold rather than collapsing into noise or artifacts.
For this purpose, we introduce a potential function U that penalizes deviations from the learned data manifold. Building on the approach of Song & Itti (2025), where generative models are evaluated by measuring the distance between their outputs and a geometric manifold of real images learned by VAE, we adopt similar strategy. Specifically, we use a state-of tfrth-rrtt VAE architecture, with parameters listed in Table 37, to learn a manifold composed of images from both marginal distributions Pa. and pb- The potential function (x*)), for samples x' p", is then defined as the squared distance between a bridge sample x' and its VAE-projected reconstruction 3² = VAB(a"): (xx')= | 1-1 No intermediate samples Pm used in this experiment.
The results are shown visually in Figure 20 and quantitatively in Table 38. As observed in the other experiments, DSBM fails to regularize the stochastic path between the two distributions, and SB- Flow searches for an interpolation that first removes features of the source digit not present in the target letter, rather than producing intermediate characters that remain close the data manifold at every time step. GSBM produces only deterministic trajectory along the data manifold, which it then uses as the mean of transient Gaussian distribution. However, because the stochastic com- ponent is not properly constrained, these Gaussian samples can drift off the manifold, leading to unrealistic intermediate states. This drift appears as increased noise in the midpoint samples and results in higher FID scores. In contrast, the CWG method produces intermediate characters that re- main closest to the source and target distributions at every discretized time step. Notably, our method also demonstrates strong computational efficiency, achieving training times significantly lower than those of the baselines.
To evaluate the guided version of the CWG method, we propose a Gaussian deblurring test, well- established inverse problem commonly used in image restoration (Feng et al., 2025; Wang et al., 2026). In this setup, reference image from the EMNIST dataset is processed with Gaussian kernel and provided as the target for restoration. In this test, the guided loss function, [|f(x**) yll. uses the blurry image as the target y, and the feature function f is defined as,
(( K ganssion (o, ② =x*,
(74)
where K Kaaussian( (a,, r) is a Gaussian kernel operator of variance a and size r, applied to the sample If generated the last time step of the bridge. The specific kernel parameters are detailed in Table 36. This loss measures the similarity between the blurred version of the generated image and the blurred reference, ensuring compatibility between the two. The restoration results are shown visually in Figure 21 and quantitatively in Table 39.
Table 36: Parameters for the feature function f (74), used for guided generation in the Mnist-to-Emnist exper- iment.
Parameter
Value
Kernel variance
Kernel size
9
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Table 37: Architecture of the ConvVAE used in the Vrannspose2D layers use ReLU activations unless otherwise specified.
experiment. All Conv2D and Con-
Stage
Layer (channels)
Output size
Input
Single-channel image
1 28 28
Conv2D (1 -32, k4, s2, pl)
32 14 14
Conv2D (32->64, k4, s2, pl)
64 X 7 x 7
Encoder
Conv2D (64->128, k3, pl)
128 x 7
Flatten
128 7 7 = 6272
Latent space
Linear + μ
2 2
Linear + log o²
Linear - reshape
128 X 7
Conv 142 k4, s2, pl)
64 X 14 x 14
Decoder
Conv (64-+32, k4, s2, pl) Conv (32-+1, pl,
32 28 28
1 x 28 x 28
Table 38: FID scores computed w.r..
EMNIST distribution (J) and training (it) the MNIST-to-
EMNIST experiment.
Metric
CWG
GSBM
DSBM
SB-Flow
FID FID
fa
17.25
86.80
102.94 17.04
79.35 4.86
(
83.38
114.53
228.30 728.36413.28
106.28 106.28.5.46
96.29
128.69 17.86
195.38 112.24
121.80 16.32
56.42+5.63
89. 10.75.mm 16.25
163. 163.69 111.34
107.29.1477 14.74
(1
29.73
58. 18..1817.17
108.53 11..1111.99
14..1712.17
(1D(
11.42
11.75+10.36
11.69
11.29+0.27
tt (s)
570+30
30100 1000
8750+300
2200
Input
7
4
5
9
7
4
5
9
7
4
5
9
7
4
5
9
f
7
6
4
9
7
7
t
5
9
ta
T
5
P
M
7
S
T
2
S
P
7
M
R
NNO
N
S
R
T
h
S
la
(a) CWG (ours)
GSBM
DSBM
(d) SB-Flow
Figure 20: Snapshots from the bridges computed in the MNIST-o- experiment.
Clean
Degraded
Restored
R
R
Metric
Score
FID
12. 12.72
LPIPS
0.28+0.02
T
T
PSNR
12.48
SSIM
0.37+0.04
tt (s)
320
S
S
Table 39; Quantitative metrics for the Gaus- sian deblurring experiment: FID score (L) between the clean and reconstructed sam- ples, LPIPS (L), PSNR (t), SSIM 1), train- ing time (tt) (L).
Figure 21: Gaussian deblurring test for the Mnist-to-Emnn -m-nmssstt experiment. Results for the guided CWG method.
44


Reviews:
[{"review_id": "NOmaD2R4mB", "reviewer": "Reviewer_7xEZ", "summary": "This paper addresses the Generalized Schrödinger Bridge (GSB) problem by relaxing certain energy-conserving constraints and instead applying a Hamiltonian mechanism to enable efficient training and inference. Furthermore, by adjusting the time-varying energy term, it allows for guided generation. The proposed method, called CWG, is implemented in the form of a ResNet, which effectively reduces training time. The authors also validate guided generation using interpolation when intermediate marginals are provided partially.", "questions": "See Weaknesses section.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Powerful Theoretical Basis: This study generalizes the conventional GSB framework from the perspective of Hamiltonian dynamics by introducing an energy-varying term. The theoretical analysis presented in the appendix provides a strong justification for the proposed methodology and effectively supports its validity.\n\n- Compact Model Design: The proposed method is implemented using a ResNet architecture, which represents the most essential structure for expressing the Schrödinger bridge. This choice is appropriate, as additional modules can be incorporated when the task complexity increases.\n\n- Efficient Training Speed: Compared to existing methods such as GSBM and DSBM, the proposed approach demonstrates superior performance and faster training. In particular, its advantage in training efficiency further enhances the novelty of this work.", "weaknesses": "#### Major Weaknesses\n- Weak Presentation: Some important details necessary to fully understand the study may be missed unless the appendix is read carefully. Moreover, the core ideas of the proposed method appear too late in the manuscript. In my opinion, it would be better to move the Related Work section toward the later part of the paper and highlight the main contributions more prominently.\n\n- Insufficient Emphasis on Mathematical Notation: The manuscript includes all key equations and attempts to distinguish important terms using various colors. However, the chosen colors are not sufficiently distinguishable, and the equations are interspersed throughout the text, which somewhat reduces readability. To emphasize key terms more effectively, it would be helpful to use a separate table or structured summary highlighting the core notations.\n\n- Inconsistent Explanation of Intermediate Marginals: In Equation (9), the loss function appears to require intermediate marginal distributions. However, in the actual experiments, only a subset of them—or none at all—seems to be used. This inconsistency is not clearly emphasized. It would strengthen the manuscript to consolidate and highlight these explanations right after Equation (9), rather than scattering them across different sections.\n\n- Novelty of Guided Generation: The authors emphasize guided generation as the main contribution of their work. In my view, guidance-based generation should ideally enable generation through indirect guidance rather than direct guidance toward an intended mode. In the current formulation and experiments, however, it seems that only interpolation between intermediate distributions during the optimal transport process is evaluated.\n\n#### Minor Weaknesses\n\n- In Appendix D.2, the notation for the intermediate distribution is inconsistent. It should be corrected from $\\rho_n$ to $\\rho_m$.\n- In Appendix D.3, the time complexity analysis could be strengthened by comparing it with other existing methods. Such a comparison would help to better emphasize the novelty of the proposed approach.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "Ad0qzQWzG6", "reviewer": "Reviewer_GjBx", "summary": "This paper proposes the Non-Conservative Generalized Schrödinger Bridge (NCGSB) and the Contact Wasserstein Geodesic (CWG) framework to model stochastic processes with energy variation, overcoming the energy-conservation limitation of classical Schrödinger Bridges. By leveraging contact Hamiltonian mechanics, the authors extend the Wasserstein geometry to handle non-conservative dynamics and introduce a ResNet-based non-iterative solver with near-linear complexity. The approach also supports guided generation through metric modulation. Experiments on LiDAR manifold navigation, single-cell sequencing, and image generation demonstrate efficiency and accuracy improvements over strong baselines.", "questions": "- Could you clarify the numerical stability of the proposed solver, especially under long time horizons or many intermediate marginals?\n- How sensitive is the method to the choice of potential function $U(x)$?\n- Is the contact Hamiltonian augmentation equivalent to introducing an auxiliary energy channel in the latent space? If so, can this be viewed as a generalization of underdamped diffusion bridges?\n\n- Could the authors provide an intuitive geometric visualization of how the energy-varying geodesic differs from a standard Wasserstein geodesic?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The extension of SBs to non-conservative systems via contact geometry is quite interesting.\n- The proposed method demonstrates faster convergence and better accuracy across diverse tasks.\n- The paper is well-organized presentation with clear motivation and theoretical grounding.", "weaknesses": "- Limited large-scale validation:\nAlthough diverse, most experiments remain moderate in scale. Evaluation on large-scale datasets or high-dimensional physical systems (e.g., long protein trajectories or large image benchmarks) would better demonstrate scalability claims.\n- Ablation studies and interpretability:\nThe role of the contact energy term, and its impact on stability or generalization, is not extensively analyzed. A detailed ablation on the energy variation factor and metric modulation would strengthen the empirical evidence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "PArjSSEuAv", "reviewer": "Reviewer_TgbM", "summary": "This paper proposes the nonconservative generalized Schrödinger bridge (NCGSB), which covers a mathematical generalization of SB for nonconservative systems, increasing model flexibility. The noticeable changes include applying decaying energy with a time-varying state, representing the (stochastic) Lagrangian action. Furthermore, this Lagrangian formulation can also be applied to guided generation using a guidance function. The authors used a ResNet to model successive pushforwards for the discretized geodesic, enabling high-dimensional applicability, including image generation.", "questions": "* Could you please provide the FID scores for MNIST-to-EMNIST SB training?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "* The manuscript is clearly written and straightforward.\n* The authors introduce a new generalization with purposeful intent and physical motivation, which could draw interest from audiences in many fields.\n* The overall approach is sound and appropriate.\n* The proposed method can be applied to various domains and is scalable.", "weaknesses": "* Lacking comparison results in the image domain. The overall volume and benchmark comparison for image-to-image translation can be considered insufficient. The authors are encouraged to review scalable SB methods for images from 2024 and 2025.\n* I could not precisely follow how the proposed system ensures the uniqueness of the solution. Unlike in conservative systems, the rate of energy decay can be varied in nonconservative systems, so should there be a hyperparameter (or an assumption of 1) to ensure uniqueness? Also, the guidance function is another degree of freedom and seems to require careful design by practitioners based on the specific problem.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)