HOTA: Hamiltonian framework for Optimal Transport Advection
Authors: Nazar Buzun
Daniil Shlenskii
Maksim Bobrin
Dmitry V. Dylov
Venue: ICLR 2026 Poster (2026)
Keywords: Optimal transport
optimal control
generalized Schrödinger bridge
diffusion models
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: A new method based on Hamilton–Jacobi–Bellman equation that solves generalized Schrödinger bridge problem.

Abstract: Optimal transport (OT) has become a natural framework for guiding the probability flows. Yet, the majority of recent generative models assume trivial geometry (e.g., Euclidean) and rely on strong density-estimation assumptions, yielding trajectories that do not respect the true principles of optimality in the underlying manifold. We present Hamiltonian Optimal Transport Advection (HOTA), a Hamilton–Jacobi–Bellman based method that tackles the dual dynamical OT problem explicitly through Kantorovich potentials, enabling efficient and scalable trajectory optimization. Our approach effectively evades the need for explicit density modeling, performing even when the cost functionals are non-smooth. Empirically, HOTA outperforms all baselines in standard benchmarks, as well as in custom datasets with non-differentiable costs, both in terms of feasibility and optimality.

Published as a conference paper at ICLR 2026
HOTA: HAMILTONIAN FRAMEWORK FOR OPTIMAL TRANSPORT ADVECTION
Nazar Buzun
Daniil Shlenskii
AXXX, Innopolis Universityf n. bbuzun@seevia.de
AXXX*, Applied AI Institute d. w.shhensski@seeee
Maxim Bobrin
Dmitry V. Dylov
AXXX*, Computational Imaging Lab m. ww..orinnseeeii..
Computational Imaging Lab AXXX
d. dy ll
ABSTRACT
Optimal transport (OT) has become natural framework for guiding the probabil- ity flows. Yet, the majority of recent generative models assume trivial geometry (e.g., Fuclidean) and rely on strong density-estimation assumptions, yielding tra- jectories that do not respect the true principles of optimality in the underlying manifold. We present Hamiltonian Optimal Transport Advection (HOTA), a Hamil- ton-Jacobi-Bellman based method that tackles the dual dynamical OT problem explicitly through Kantorovich potentials, enabling efficient and scalable trajec- tory optimization. Our approach effectively evades the need for explicit density modeling, performing even when the cost functionals are non-smooth. Empirically, HOTA outperforms all baselines in standard benchmarks, as well as in custom datasets with non -iffeerrrfeteiblee costs, both in terms of feasibility and optimality. Code: https: githhb...u.ggg com www.rblza.bb.cccca HOTA. Project page: htttp:://
INTRODUCTION
Static MMonge-Kantorovie optimal transport was originally considered as the main framework for comparing and finding a mostmminiii coupling between distributions [Villani) 2008), while optimality was mainly measured through the boundary marginals. Development of efficient and scalable OT solvers (Cuturi 2013 Peyré al.) (219)) popularized OT across different areas. such as
generative modeling (Makkuva et al. 2020 Korotin et al., 2022 Buzun et al., 2024 Choi & Choi
2024], computational biology (Bunne et al. 2022), graphics (Bonneel Digne 2023), high-energy
physics (Nathan T. Suri 2224)), and reinforcement learning (Klink al., 2022] Asadulaev et al. 2024]
Bobrin al.) 2024) Rupl et al. 2025). However, one crucial limitation of static formulation is its inability produce ntn-straight paths, which completely ignores the underlying geometry of the manifold of the data. classical OT, the underlying geometric structure is solely determined by the choice of cost function (e.g.. Euclidean distance), inherently limiting the capacity for fine- tne-grained
control over the trajectories. We refer to (Montesuma et al. 2024 Pereira & Amini] 2025) for recent
overview of practical applications of OT and to Villani 2008); Santambrogio (2016): Peyré al.
(2019) for a formal treatment.
On the other hand, the dynamical OT paradigm, developed by Benamou Brenier (2000), recasts static OT as time variational problem on the space of probability paths, effectively incorporating time variable and enabling more nuanced control over optimal trajectories (e.g., through velocity, acceleration, length, or energy over the paths). Importantly, such formulation enables one to directly operate on manifolds of non-trivial geometry, whenever the underlying space contains curvature,
AXXXX, Moscow, Russia
1 RRseearch Center of the Artificial Intelligence Institute, Innopolis University, Innopolis, Russia
1 Applied AI Institute, Moscow, Russia
6 *Computational Imaging Lab, Moscow, Russia

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obstacles, or is defined through potentials. This formulation is closely connected to stochastic optimal control (SOC), where trajectories are stochastic yet must still maintain optimality, a problem class known as the generalized Schrödinger bridge (GSB) Liu et al. 2022 Bartosh et al. (2024).
The GSB problem generalizes the classical Schrödinger bridge by incorporating an arbitrary po- tential function (x)) into its cost functional. This potential allows the model to incorporate rich, application-specifee physics and constraints, steering optimal paths away from high-cost regions and towards low-cost valleys. The ability to handle non-differentiable potentials significantly expands its applicability, as many real-world constraints and penalties are inherently discontinuous or non- smooth. In computational biology, simulating molecular pathways with hard-core repulsion and steric clashes is represented by non-smooth potentials (Bunne et al. (022). While the paper uses a static OT formulation, GSB approach could model the entire most probable path a cell takes from health to drug-affected state. Another relevant scenario arises when the potential ((x; t) is defined by a large, non-differentiable system, such as a black-box Large Language Model (LLLM) or a complex neural network where propagating gradients through the entire system is computationally prohibitive or impossible Krishnamoorthy et al. (2023).
A common strategy for GSB involves solving the dual formulation via HamiltonnJacobi-Bellleee (HJB) equations, which provide a flexible and a theoretically grounded framework for deriving optimal trajectories (Liu et al. (2022), Neklyudov et al. (024)). These methods parameterize the cost through a Lagrangian, enforeing optimality via the minimization of kinetic energy or using other path-based penalties. The dual approach is motivated by the intractability of directly minimizing the primal problem, which includes integral cost with, probably, non-smooth potentials U (z)) Such transformation yield a tractable objective that depends only on the value function and its derivatives and permits unbiased estimation from samples. We obtain a density-free objective that naturally incorporates non-smooee con-smooth potentials and enables the use of dynamic programming for efficient multi- step trajectory optimization.
While HJB-based approaches yield theoretically sound solutions, they suffer from critical drawbacks: (1) unstable optimization dynamics, leading to high- variancreeee gradients and poor sample efficiency in high dimensions, and (2) the absence of a strict terminal distribution matching criterion, resulting in inexact couplings. In the current work, we study the Generalized Schrödinger Bridge problem between two measures, where the underlying geometry is defined through potentials. We propose framework that explicitly solves the GSB task, resolves the learning stability problems of the previous approaches, and ensures exact terminal distribution matching through theoretically grounded objective. We conduct extensive empirical evaluations on existing low- dimensional physically inspired benchmarks, as well as in the high-dimensional generative setting, including image-to-image translation. In short, our contributions
new
follows:
Dual reformulation of GSB that binds Kantorovich potentials with HJB value function, turning the HJB constraint into built-in regularizer. This yields stable objective amenable gradient-based optimization in challenging non-Euclidean and non-smooth settings;
We design a specialized training procedure with data replay and target network stabilization, enabling the effective joint learning of the Kantorovich potential and the optimal drift policy;
HOTA attains satte-of-thh-art empirical results diverse set of tasks, demonstrating both better feasibility (exact marginal matching) and optimality (cost along trajectories) compared to current GSB solvers.
2 RELATED WORK
Diffusion Models and Matching Algorithms. Diffusion models have emerged as powerful tools for generative modeling by prescribing the time evolution of marginal distributions. Matching algorithms, such as Action Matching (Ncklyudov et al. (2024) and Flow Matching (Lipman et al. (2023)) learn stochastic differential equations (SDEs) that align with prescribed probability paths (Blessing et al.) 2025). These methods typically assume explicit or implicit intermediate densities of the flow, whereas approach (HOTA) optimizes complete stochastic path from source to target distributions.
Generalized Schrödinger Bridge. The GSB problem extends the classical Schrödinger Bridge (SB) by incorporating a state cost function, which penalizes rewards specific trajectories. Prior methods
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for solving GSB, such as DeepGSB (Liu et al. 2022) and NLSB Koshizuka & Sato (0022) suffer from critical limitations. Particularly, absence of a strict terminal matching criterion leads to inexact couplings between the target distributions. Furthermore, they rely explicit density estimation during their iterative procedure, which introduces additional complexity and instability, especially in high- dimensional spaces. A recent approach GSBM (Liu al. 2024) follows alternating optimization scheme: in the first stage, it learns the drift field U/ while keeping the marginal distributions Pa (x;) fixed, using Flow Matching-style objective. In the second stage, it updates the marginals conditioned on the haundry--cuulled distribution ( | Tu, II), which is defined via the previously learned drift. While GSBM demonstrates strong empirical performance, imposes two limitations: 1) requires the state cost function (1x)) to be differentiable everywhere, and 2) it learns the conditional marginals p(a, wo; 21) with suboptimal distributions on pairs (zig v1), which may lead to a biased p(zz) model. Moreover, as our experiments show, this alternating optimization is computationally expensive, requiring multiple iterative updates between the drift and marginals to converge.
Stochastic Optimal Control. The connection between GSB and stochastic optimal control (SOC) has been explored in prior works (Theodorou et al. (2010) Levine (2018). SOC formulations often relax hard distributional constraints into soft terminal costs, which can introduce bias or require adversarial training (Liu et al. (2022)). Recently introduced Adjoint Matching approach (Domingo Cominng-EEnrich et al., 2024a) and Stochastic Optimal Control matching (SOCM) (Domingo-Ennicch et al. (202bb) address several existing limitations, but still produce highly unstable variance estimations. Our method provides a natural way preserve the feasibility via Kantorovich potential sum.
FROM THE GSB PROBLEM TO THE DUAL OBJECTIVE
The core of our approach begins with the primal formulation of the Generalized Schrödinger Bridge (GSB) problem. We consider a controlled Markov process I₁ and seek the stochastic control policy 2t that minimizes the expected cost of transporting source distribution to target distribution 8:
inf TE V C(1,,22,1)
C(1, 0x) ||0 + Uax),, To a, 3.
The potential term (xxe) usually characterizes the geometry of the space. But in addition, we can also include some physical properties of the flow, e.g., entropic penalty or "mean-field" interaction (Liu et al., 2022). Thus, the optimal trajectories are curved to avoid regions with high values of U(z)). The conditional map ((-(az)) induced by the optimal drift of that minimizes the primal objective (1) is, in general, stochastic. This means that for a given starting point x ~ a, the endpoint æ1 is a random (zr) This formulation, where the transport plan is described by kernel pa-[x;) (x) rather than a deterministic function, directly relates our problem to the framework of weak OT (Gozlan et al., 2014). To handle the terminal constraint 21 ~ 3. we relax by Lagrange multiplier, which takes the form of a Kantorovich potential function g(v) (ref. Definition 2.4 from Gozlan et al. (2014)) This relaxation yields the following saddle-poin problem:
variable distributed according
{ E [[.
inf sup de (ET
C(t, ve) dt d + Fyy-olo(w))
The marginality requirement of the distribution of æ1 is ensured by the potential difference Egg(y) and E., ()))) which tends to infinity otherwise. Under suitable regularity conditions, we can interchange the infimum and supremum. The inner minimization over controls vn for a fixed potential g defines the stochastic c-transform of g:
(x)) = inf E S/ Ctt, Uz) dt g(x1) xo
The function (xr can be interpreted as the optimal cost-to-go from the initial state when the terminal cost is given by -z(1)) This yields the dual problem, which is maximization over the Kantorovich potential function g:
sup {Teza (x)) + Bgyrs (EL1(
This dual formulation is central to our method. Neural networks can effectively solve high- dimensional OT problems by learning the Kantorovich potentials, which maximize the dual objective
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(Korotin et al. (2022), Buzun et al. (2024)). But unlike classical OT, need to minimize the cost throughout the full trajectory 24, t [0, with the objective (3). In order to circumvent this
challenge, will address the dynamic programming principles and define the value function. Let for any 0 S t S and a E Rd, the value function satisfies:
a(1, x) = inff E J." TT. 2(x1)
(5)
such that (x))) x) and the boundary condition at time point / is
Vx R s -g(x).
(6)
To derive a computationally tractable objective, we must now specify the stochastic dynamics governing the state evolution It. We focus on the widely used formulation a diffusion process:
dæ, = vl,, (.)t ot,
where vtt, zz) is the controllable drift and xt) is the diffusion coefficient. In Appendix F provide the complete derivation for a more general class of dynamics, demonstrating the broader applicability of our framework. When the dynamics are given by the diffusion process (7) the
function stt, x) satisfies the Hamilton-Jacohi-Bell
(HJB) equation (Fleeting & Soner 20066. The
solution this equation subsequently enables the evaluation of the infimum in problem By definition, the HJB equation is
-es((t, -= +U()) -U)) in
(rt
(8)
Representation of the Lagrange function as sum of kinetic and potential energy I allows us to find the minimum by control (vz) in explicit form, such that Vi v Iz). Combining the potential optimization objective 4 with the characterization of the c-transform via the HJB equation and its
boundary condition ( we arrive at the dual formulation of the GSB problem. This result is formally stated in the following theorem:
Theorem (Dual GSB problem). Given distributionss o,, B E P(Rd) and stochastic dynamics with vl,,2)) - (l, the GSB problem 7 admits the following formulation:
Ezona Lo" C(t, 21, - Vs)) dt + s(1,x1) Eyno sL.
max ((L,)EL1 (g)
(9)
where (l,,x)) (1,2((0,1] x IRe)) satisfies the HJB PDE
8 4I E [0, 1] and Var E RRd.
The first expression (9) plays the role of discriminator and guarantees matching the target distribution 3, and the second one (8) is responsible for the optimality of trajectories. A detailed proof is provided in Appendix E Additionally, Appendix contains convergence analysis, quantifying the deviation of the learned function s(t, x) from the viscosity solution in terms of the HJB residual and the potential matching loss.
By means of the optimized function s(t, x)) we can generate the OT trajectories using the Euler- Maruyama algorithm:
T++ = - st, + oAW, To a.
(10)
Unlike most other methods, here we do not need to model intermediate density of the a (t E (0, 1)) distribution, which greatly simplifies the learning process, but we need to store the generation history in a replay buffer for more stable HJB optimization high-dimensional spaces.
METHOD
To find stable and balanced solution s(t,z) for the given GSB problem composite approach that combines optimal control (via HJB PDE constraints) and RL techniques (policy-based trajectory optimization). We approximate the value function using parametric model st z). We have to maximize the potential matching functional ( subject to the constraint that so ()) satisfies the HJB PDE. For that divide the time interval [0, into T' time steps and simulate
can follow
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trajectories (to: z ... I.,, 2T =k=1 using initial distribution and Euler-Mann geler-Maruyamaa method (10). Sample also n points ys from the target distribution B and compute the potential matching loss as
pot (sa)
so
(11)
n2
T2
The dual objective in Theorem decomposes into the integral cost and the terminal potential difference. Under the HJB constraint enforced by Lujb. the cost term is already accounted for and minimized for the given terminal condition. Consequently, maximizing Lar under the HJB constraint directly optimizes the full dual objective, allowing us to omit the explicit integral term from the potential matching loss.
The HJB PDE must hold for all & E [0, 1] and x E R. but in practice, for more effective training, the training data should be sampled in the region the flow (trajectories) concentration (according to Liu et al. (2022)) We enforce this by linear interpolation between datasets from and & rough estimation of the flow region and subsequently use the replay buffer B to collect points from the from B or the linear interpolation previously obtained trajectories. Using data samples (t" 1 Tk=1 we compute HJB residual loss as
Osh
+ o² rr( + da Aallat1)
Ligh (sp, 8)
V
112 + U((*))
(12)
Ot
2
2
Os
V
b81 + U(**)
+ 2 tr( [/( + Aalla*11
(13)
n 1=1
Ōt
where sy so _²*) and g s(th, ,²) denotes the target model. The target model is updated
as an Exponential Moving Average (EMA) of the main model parameters: + 7 (1 7) 0. And at is angular acceleration defined as
a d Stp (tth x²)
(14)
dt Vsg (t),
Algorithm 1 HOTA: Hamiltonian framework for Optimal Transport Advection
1: Input: model sp, optimizer s_opt, distributions and B, potential Uxx), diffusion coef. a.
2: Hyperparameters: train steps N, iterpolation sample steps No. temporal discretization T, batch size n, hjb-loss weight Ang. acceleration coef. Xas learning rate Ir, gradients scale EMA coef.
3: Initialize target model 3; replay buffer + 0; gradients scale 4- 1.0
4: for iteration i to N do
5: Sample train data {xb) ~ a; ( Jizn B
6: if í < No then
7:
Sample times {th [ U0, 1)
8:
For S k & n set x.t = =x - 1*) + 1² 1
9:
else
10:
Sample {th x Fh-1 B 1
end if
12:
Generate n trajectories (t). .................. .., 14 1 using current policy e; -vs(t,z)
13:
Add one of the resulting trajectories {th, .h...., th.. xt} to B, where 1 5 5
Compute gradients:
15:
Dhjb Volngj (80, 8, [1k x I1 }k=1)
16:
Spot V VoLpa (80. {at J== far kn 12
17:
Update Parameters:
18:
Compute norms Ghi = [1pp ll and Gpot [1 1 1|2
19:
EMA update of gradients scale ( = TGpa/ Giyb + (1 -)
20:
Sum the gradients
Spot + Any Ç 9bjb
21:
Update model parameters with s_opt(g)
22:
EMA update of target model
23:
end for
5

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The first three datasets-St Stunnel,, Vneck, and GMM-are adopted from Liu et al. (024)). These benchmarks incorporate state cost functions U(x)) that encourage the optimal solution to respect complex geometric constraints. Each dataset is designed to highlight specific capabilities of the evaluated algorithms. Stunnel assesses whether a method can capture drift fields that undergo rapid and localized changes. Vneck evaluates the ability to model drift that compresses and expands the support of marginal distributions. GMM tests whether the method can disambiguate closely situated points and assign them to distinct trajectories. The remaining datasets-Bann BabyMaze, Slit, and Box constructed using similar underlying principles but pose additional Published as a conference paper at ICLR 2026
The angular acceleration with coefficient Xa forces the straightening of the trajectories (optionally). We divide the model into se and 3 as is usually done in Reinforcement Learning (Mnih et al. 2015) to make the optimization problem more similar to regression.
As result, our model is trained on two criteria (Lpor and Lnga) simultancously, and to balance both impacts, we scale the gradients of the hjb-loss and sum them with the pot-loss:
V1 [
VoLLoe(S0) EMA
(iLnnpssa. 8).
(15)
o s)]]
The complete method is implemented as shown Algorithm 1. It effectively combines the theo- retical guarantees of optimal transport with the flexibility of neural network approximations, while maintaining numerical stability through careful gradient management. The adaptive balancing of the potential matching and HJB residual losses ensures stable convergence a solution that satisfies both the optimality conditions and the boundary constraints.
5 EXPERIMENTS
In this section, we evaluate our method on a series of distribution matching tasks with non-trivial geometries. In Section 5.1 compare HOTA with state-of-the-an baselines, demonstrating its superior performance on standard benchmarks including datasets with almost non- differentiable potentials. In Sections 5.2 5.3 we demonstrate the scalability of our approach by showcasing its effectiveness in high-dimensional settings. Finally, in Section 5.5 method.
we ablate key components of our
Evaluation. We assess performance using two main metrics: feasibility and optimality. Feasibility reflects how well the method matches the target distribution, evaluated via Wasserstein distance with squared Euclidean cost (W2 (T B)), where the transport mapping T uses optimized value function sy and simulates 21 by procedure (10). Optimality measures the quality of the resulting mapping. estimated through the integral trajectory cost where follows 7 and 2t - -V,, st,, zz).
Baselines. We use source code from GSBM repository for running it in our experiments. Other results were taken from the original papers NLOT Pooladian et al. (2024), NLSB Koshizuka & Sato (2022). DeepGSB Liu et al. (2022) where datasets were previously introduced.
Runtime. Our method achieves efficient high-dimensional OT computation through JAX's auto- matic differentiation, with minimal overhead (<5%) for computing second-order derivatives. HOTA demonstrates near-linear scalability with both data dimensionality and simulation steps, signifi- cantly outperforming GSBM with speedups of 50- 100x. While this section focuses on comparing the effectiivnnes of these methods, provide analysis of their computational efficiency in Appendix B
Reproducibility. To support reproducibility, we attach full code in the supplementary material. All synthetic datasets can be generated with the provided scripts, and high-dimensional experiments use standard public benchmarks. All experiments conducted on single GPU A100 80GB. Complete experimental details, including hyperparameters and network architectures, are documented in Appendix A
5.1
COMPARATIVE EVALUATION ON LOW-DIMENSIONAL DATA
In this section, we compare our method to previous state- of-the-art approaches on the standard bench- marks including datasets that feature almost dindiifferentialee potential functions. Visualizations of the datasets are provided in Figure
(Pooladian al. 2024))-

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X
W1
-5
Figure I: Evaluation of HOTA method on smooth (top) and non-smooth datasets (bottom): Stunnel, Vneck, GMM, BabyMaze, Slit, Box. Blue regions indicate high values of potential (zx). Distribu- tions (red), 3 (black) and the mapped a (green).
Table 1: Quantitative comparison between baseline methods and our approach, HOTA, Performance evaluated using two criteria: Feasibility (how well the target distribution is covered) and Optimality (efficiency of the learned mapping). N/A cells indicate that original authors of particular method did not include results for those tasks. The mean and the standard deviations of our method are computed across 5 different seeds. Best values are highlighted by bold font (lower is better). Gray values correspond the method's 's divergence (in terms Feasibility).
Feasibility (2 # ( B)
Optimality (integral cost)
Stunnel
Vneck
GMM
Stunnel
Vneck
GMM
NLSB GSBM
30.54
0.02
67.76
207.06
147.85
4202.71
0.03
0.01
4.13
460.88
155.53
229.12
HOTA
0.006 #0.003
0.002
0.19 Ho0s
383.25
115.09
80,44=26
BabyMaze
Slit
Box
BabyMaze
Slit
Box
NLSB
0.013
0.024
N/A
N/A
N/A
NLOT
>
0.013
0.016
N/A
N/A
N/A
GSBM
0.01
0.01
0.02
6.5
4.9
3.8
HOTA
0.004
0.0004=0.000
0.002=0.01
4.87
3.06
2.84
difficulties due to the presence of almost non-differentiable state cost functions. A summary of the quantitative results across all datasets is provided in Table Gray values this table means that the method with corresponding configuration significantly diverges from the target distribution. For this reason, we omit such comparisons with the others by the Optimality metric. Our method, HOTA, consistently outperforms existing approaches in terms of both feasibility and optimality. In particular, HOTA achieves a substantial performance gain on the GMM dataset, which may refer to superior capability in trajectory separation for closely situated points.
In addition to the synthetic 2D benchmarks, we evaluate HOTA on a more complex problem: the transportation of 3D LiDAR point clouds (Appendix D). This setting features irregular potential, Uxx), derived from real-world scanned surfaces. This experiment tests the method's 's robustness and ability handle complex, state-dependent costs in applied scenario.

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Data dim
10
100
500
1000
Gaussian measures a and |
HOTA, W2 Normalized
0.001
0.013
0.025
0.051
GSBM, 2 Normalized
0.99
0.63
0.24
0.21
HOTA, Optimality GSBM, Optimality
3.37
3.16
3.78
3.17
20.9
20.03
20.88
22.1
Mixture measures
(10 Gaussian) a and |
HOTA, W2 Normalized
0.003
0.049
0.064
0.045
GSBM, W2 Normalized
0.097
0.64
0.30
0.30
HOTA, Optimality
2.40
2.76
3.24
3.09
GSBM, Optimality
5.31
6.17
14.59
14.63
(a)
(b)
Figure 2: (a) Visualization of our Sphere dataset for N = with Gaussian (top) and Mixtute of 10 Gaussian (bottom) measures a and B. (b) Performance comparison between our HOTA model and recent state-of-theear GSBM on nigh-dimensional Sphere datasets. Models are evaluated on Feasibility (W2 Normalized) and Optimality. The best value for each metric is bolded.
5.2 SCALABILITY TO HIGH-DIMENSIONAL SPACES
In this section, we test the scalability of our method, demonstrating its stable performance in higher-dimensional settings. For this purpose, we use Sphere datasets parameterized by data dimen- sionality N. The source and target measures projected onto the sphere's ' surface. To enforce geometric constraints, we introduce a potential Uzx) that penalizes deviations from the manifold: Usphere (z) C]I([ < 1), where C2 0 is a dimension-deeeendent constant. The performance of our method across varying data dimensions shown in Figure with three-dimensional case visualization. Since the W2 metric becomes less informative in high dimensions, we report the normalized W2. This is computed as the (T (a), 3) distance divided by W2(a, 3) and substituting distance between independent samples from the target distribution (3, 3'), i.e.,
Gaussian
mixture of Gaussian distributions
Norm W2(T (a), 3) - (2)
(16)
W2(0,
This normalization provides a better measure of convergence relative to the intrinsic spread of the target data. Notably, HOTA demonstrates robust and stable performance as the dimensionality N increases.
5.3
DIMENSILMNSIO OPINION DEPOLARIZATION
polarize dynamics
DeepGSB obj-914.3
GSBM 505.3
HOTA(ours) obj=460.4
larget
DS PCA 1 vs.
Figure 3: Opinion prediction in dimension 1000 and correspondent directional similarities (DS). Polarize dynamics is the opinion distribution formed without control. The target measure B is multivariate Gaussian with mean 0 and standard deviation 4T.
8

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(a) Male (CelobA) + Female (CelebA).
(b) Female (CelebA) + Anime
Figure 4: Examples of learned transport maps on (left) CelebA male->female and (right) CelebA Faces. HOTA shifts domain-speciie attributes (style/texture/colrr) while maintaining content such as identity cues, expression, and pose.
Our model employs the polarization mechanism from Liu et al. (022)) which builds on the framework of the political parties introduced by Gaitonde J. (2021). During each iteration every agent observes a shared random stimulus & E R 1000 drawn independently from the current opinion distribution Pt- The agents then update their positions using dynamic response mechanism. This mechanism drives an agent's movement based on its alignment with other agents regarding a piece of information. an agent's evaluation of the information aligns with another agent's, the interaction attracts them toward that agent's 's direction. Conversely, if their evaluations differ, the interaction causes repulsion. This Formulation captures the psychological tendency for individuals to gravitate toward aligned viewpoints while resisting opposing perspectives, ultimately leading to group polarization. In this case, we consider the stochastic process with polarization drift (plaariee defined in Liu al. (2022))
dx, = v(t, xz xt + follaiize (Iri Pt, & d + odW.
(17)
The control task is to depolarize the final distribution Φ1 and match with the target measure (0,0044) 47). The optimization objective here is the integral kinetic energy (1) where == v /2 and U1x+) = 0. We analyze the distribution angular separations between opinion vectors (direc- tional similarities [Schweighofer S. (2020)) A more uniform angular distribution indicates reduced polarization, while peaked distributions suggest stronger factional divisions. The comparison in terms of the task objective and directional similarities is presented Figure
5.4 IMAGE-TO-IMAGE TRANSLATION
To validate HOTA'S capability on high-dimensional, real-world data, we evaluate it on image-to-image translation tasks. Specifically, we learn transport maps (1) between male and female faces within the CelebA dataset (Liu et al. (015 and (2) between CelebA (female faces) and the Anime dataset, translating from photographic to illustrative domain. These tasks require learning transport map that modifies core semantic attributes (e.g., style, texture, and coloration) across domains while preserving the underlying content and structure of each input image.
Table 2: FID scores for image-to-ime -ommagg experiments.
Task
Model
FID
CycleGAN Zhu et al (017))
12.94
NOT Korotin et (2022
11.96
Male (CelebA)-+Femenn (CelebA) (64x 64)
EUOT Choi & Choi
2024
8.44
HOTA (ours)
6.28
CycleGAN Zhu et al. (011
20.80
NOT Korotin et al. 2202
18.28
Female (CelebA) (lrehA))Annmme Anime (64x64)
ENOT Buzun et al
20044
13.12
HOTA (ours)
11.67
kaggl htt.cm//aattaettsereettaaa////// httpa///aaaalannnnnnneeeeee.eee

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Qualitative results in Figure4show show that the transport learned by HOTA produces samples that closely match the target distribution while preserving key semantic attributes of the input. Quantitatively, measure performance using the Fréchet Inception Distance (FID), which assesses the quality and diversity of translated images relative to the target distribution. As reported Table 2 HOTA achieves competitive FID scores, outperforming strong baselines, including recent Neural Optimal Transport methods (NOT Korotin et al. (2022). ENOT Buzun et al. (2024)).
5.5 ABLATION STUDY
Table g presents comparison the full HOTA model against variants without the replay buffer B that stores simulation history, target EMA model 3(t, z), the adaptive gradient balancing by means of p evaluating as previously feasibility and optimality metrics across Stunnel, Vneck, and GMM datasets. The full HOTA achieves strong metric scores, while removing the buffer severely degrades feasibility in Vneck and GMM and increases costs Stunmel. Disabling gradient balancing harms feasibility in Stunnel and GMM. The results highlight the buffer's 's critical role in maintaining feasibility and the nuanced trade-offs between gradient balancing and transport efficiency across different scenarios.
Table 3: Comparison of HOTA method against variants without the replay buffer B, adaptive gradient balancing and target EMA model. Best values are highlighted by bold font (lower is better). Gray values correspond to the method's divergence (in terms Feasibility).
Feasibility W W (a)) B)
Optimality (integral cost)
Stunnel
Vneck
GMM
Stunnel
Vneck
GMM
HOTA HOTA w/o buffer
0.006
0.002
0.19
383.25
115.09
80.44
0.076
16.47
1.248
706.89
82.49
121.6
HOTA w/o grad. balancing
3.60
0.026
2.64
385.62
109.25
72.77
HOTA w/o EMA model
0.018
0.004
0.65
408.60
109.88
97.30
LIMITATIONS AND FUTURE WORK
While HOTA exhibits strong performance, we observed sensitivity to certain network design choices particlllay the Fourier feature encoding of time, a commonly used technique in models that estimate ODE drifts. Additionally, because the value function in our framework must simultane- ously support optimal control estimation and serve Kantorovich potential, it requires network architecture capable of aggregating rich temporal and spatial information. Despite these limitations, HOTA opens new possibilities for solving optimal transport problems in real-world applications, such as computational biology, robotics, and generative modeling. Future work will focus on extending the theoretical guarantees weaker regularity assumptions and on incorporating more structured neural architectures to further improve scalability and stability.
7 CONCLUSION
In this work, we introduced HOTA, a new OT method based on the Hamilton-lacoiiBellnna (HJB) framework for solving the Generalized Schrödinger Bridge problem. We demonstrated that HOTA consistently outperforms recent state-of-the-a methods on standard benchmarks and scales effectively to high-dimensional settings. Remarkably, it works for non-smooth potentials and with non-differentiable cost functions, yielding robust performance gain in terms of strictly defined concepts of feasibility and optimality.
ACKNOWLEDGMENT
The work was supported by the Ministry of Economic Development of the Russian Federation (agreement No. 139-10-2025-034 dd. 19.06.2025, IGK 000000C1139955440000))
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A ADDITIONAL EXPERIMENTAL DETAILS
Hyperparameters. Table 4 summarizes the hyperparameters used for each dataset presented in the paper. Note that the Sphere datasets, which are parameterized by data dimensionality. share all hyperparameters except for the potential weight, which may take the value 10 for low dimensions and 50 for high ones. In our image-to-image experiments, we replace the MLP backbone with a ResNet architecture and adopt a specialized configuration of Fourier feature embeddings tailored for images.
Table 4: Hyperparameters used for each dataset presented in the paper.
Hyperparameter
Stumnel
Vneck
GMM
Baby
Slit
Box
Sphere
Opinion
LIDAR
Imagee2maage
MLP hidden layers
5522, 612, 1]
Fourier frequencies optimizer
(l.... 20)
{10 (10-f....,
Adam with cosine unnealing (ar 10
initial leaminng rate
Adam 32]
0.99
[0.5, 0.9 100000
training iterations batch size scale EMA, r medel EMA. # control steps
70000
1024
64
0.9
0.99
30
diffussio coef... control weight, Aua
0.3
0.03
0.01
1.0
2.0
0.7
0.5
2.0
0.3
0.4
0.4
1.0
0.5
acc. weight, Aa
0.0001
0.001
0.2
0.05
0.001
0.01
0
0.001
potential weight
25
1000
10
30
700
(10,50)
20
0.1
Network. In all our experiments (except image datasets), we employ a simple MLP, augmented with Fourier feature encoding of the time component. For general time embeddings of the form emb(l) = sin(ʃ - + sp), the time derivative is given by d,emb(() = cos(f 1 p). As the frequency f increases, the magnitude of this derivative also grows, potentially leading numerical instabiltt-esppeciall when the time derivative of the network is explicitly involved in the objective. This issue has been previously discussed Lu Song (2024). To address this, we restrict the fre- quency range to [1, 20) and normalize the resulting Fourier features by dividing by the corresponding frequencies.
Impact of acceleration and control weights. We have evaluated the influence of the balancing coefficient Abjo and the acceleration term Ag |a] in the loss Lnjo- The latter penalizes changes in angular velocity straighten trajectories- our results show that increasing X improves trajectory optimality while introducing a slight bias in matching the target distribution 3, as reflected in the feasibility metric. In the GMM task, due to the specificity of the dataset and the divergence of trajectories in different directions, small penalization of acceleration also improves feasibility. Simultaneously, we optimize the losses Lpot and Laga, scaling the HJB-loss gradients for stability before summing them with the POT-loss gradients, and we investigate the sensitivity of learning to Anjb and its impact on performance in the Stunnel and GMM tasks (Figure 5)
Image-to-image translation. For image-to-image translation tasks parameterize the value function (t, x) using 5-layer ResNet architecture from WGAN-QC (Liu et al. (2019)). The network uses channel progression of [64, 128, 256, 512, 512], with skip connections to facilitate gradient flow. A critical design choice is the integration of the temporal dimension: the time embedding with Fourier features is concatenated the parameters using the Adam with momentum (B1, 32) = (0.5, 0.9). For more stable training, we also used LA loss in HJB objective 12, scaled the kinetic energy term uz by constant 0.01 and simulations in both forward (source to target) and backward (target to source) directions to populate the replay buffer more effectively. The other training hyperparameters are listed in Table
the input of each residual block. We optimize
B SCALABILITY AND RUNTIME
While parametrizing the value function s(t, with neural network requires computing its second- order derivatives, our method incurs minimal overhead in practice. This efficiency stems from two factors: the overhead scales linearly with the number of simulation steps, and JAX's automatic
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(Feasbiity
0.08
(reasibiity)
0.25
0.06
0.20
0.02
0.15
0.0
0.1
0.2
0.3
0.4
0.5
0.1
0.3
400
90
380
Uptiimslity
85
80
0.2
0.4
acceloration
A,
W: (Feaabiityy
NoN
Wo (Faadbillit))
1.0
2.0
1.5
OT cast (Optimalit)
(Optimalty)
90
380
80
360
340
0.5
1.0
2.0
1.5
control weight Are.
control weight Ave
Figure 5: Impact of acceleration coefficient Aa and control weight Anp Left: Stunnel, right: GMM datasets. The results show tradeoff: increasing Xa or Anjo improves optimal transport (OT) cost but reduces feasibility. Larger instability arises for both excessively low and high values of Xijb-
differentiation computes Vs and As in near- identical time to s(t, x) itself (consistently below 5% overhead). This was verified using 4-layer MI.P across dimensions from 102 to 105. The primary speedup is achieved through JAX's scan-based optimization of the simulation loops.
To further illustrate efficiency, we conducted experiments of training iterations at 70,000 and varied the data dimensionality (with simulation steps fixed at 30) well varied the number of simulation steps (with data dimensionality fixed 1000). The results in the Tables 5 below show that HOTA scales well (near-linear) with both increasing dimensionality and an increasing number of simulation steps. HOTA achieves significant speedups over GSBM due to (1) simpler optimization objective and (2) better dimensional scalability. Speedup factors are calculated the ratio GSBM's 's runtime HOTA's runtime. Notably, GSBM also encounters out-of-memon (OOM) errors beyond 5,000 dimensions under 24GB GPU memory constraints, while HOTA maintains stable performance through 9,000 dimensions.
Sphere dataset. We fixed the number
Table 5: Runtime dependence on dimension and simulation steps.
Data dim
1k
5k
9k
HOTA time
4.4 min
7.1 min
9.9 min
12.8 min
15.7 min OOM
GSBM Time
241 min
620 min
1233 min
OOM
Speedup (x)
54.8
87.3
124.5
Simulation steps
10
20
30
40
50
HOTA time
3.4 min
3.7 min
4.4 min
5.8 min
7.1 min
Datasets
2D benchmark
1000D opinion depolarization
GSBM time
30-45 min
1495 min
HOTA (ours) time
6-7 min
26 min
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C HOTA OBJECTIVES CONVERGENCE
In this section we provide an additional empirical analysis of our HOTA convergence. We tracked the evolution of the losses Lujb and Epoi during training on the Stunnel and Slit datasets. The results in Figure show that both losses consistently decrease towards zero, meaning that the learned value function Sp increasingly satisfies the conditions of Theorem T In particular, Lpoe corresponds to (a,(1, - Egs(1, :/) in equation 9 so its convergence indicates that terminal potential s0(1, approximately optimizes the dual GSB problem and enforces marginal matching. At the same time, Lijh is the squared residual of the HJB PDE in Theorem [] and its decay shows that so (t, becomes increasingly accurate solution of the associated optimal control problem. Taken together, these trends empirically support that HOTA is optimizing the dual formulation of Theorem while approximately satisfying the HJB constraint along the learned trajectories, rather than only fitting terminal marginals.
200
5
-100
100
-150
50-
-250
-10-
0
50000
100000
50000
100000
steps
steps
15a
10-
63-
0.0
0.5
60-
45-
-2.0
-2.
75000100000
50000 steps
-5
255
8.6
23
steps
Figure 6: Empirical convergence justification of Lijo and Lpui losses. Top row: Stunnel (smooth U (x)),, bottom row: Slit (non-smooth (z)) datasets.
D EXPERIMENTS WITH LIDAR DATASET
In this section we provide an additional evaluation of HOTA on surfaces observed through LiDAR scans from (Liu et al. 2024) (see Figure 7). We adopt the state cost used in (Liu al. 2024):
U(x) Imanifota (x: + Lneigh((<)) Lmaanitot( ||<(<) =1122
[neigh( =
(18)
where πt (z)) projects onto approximate tangent plane fitted by k-nearest neighbors, and _ (=) (z)
refers to the z-coordinate of 7 (xx). Le.. its height.
Table 7 (left) shows that HOTA achieves the lowest integral trajectory cost, improving upon GSBM and DeepGSB, which indicates more efficient transport on this complex real-world manifold. At the same time, HOTA attains small Wasserstein distance, demonstrating accurate matching of the target distribution, as qualitatively illustrated in Figure Together, these results confirm that HOTA extends effectively beyond synthetic benchmarks irregular LiDiA-derived geometries.
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20000
18000
Optimality (integral cost)
16000
HOTA
6179
GSBM
6209
14000
DeepGSB
7747
Feasibility (W2)
-2
HOTA
0.012
-4
Figure 7: LiDAR dataset: quantitative results (left) and qualitative visualization of HOTA trajectories on the scanned surface (right).
E PROOF OF THEOREM 1 (DUAL FORMULATION OF GSB)
prove, the first step the equivalence between the GSB (stochastic control formulation) and its dual formulation using Kantorovich-styye duality. Remind that we consider the stochastic process Dz with conditions zo ~ a, 21 ~ 3, a control function v(t, 24), and Brownian motion (tt, a, dWW, :. The primary problem of GSB optimization is:
inf E 1 C(t, 21, dt
s.t.
To
B,
(19)
where in the particular L(t, 2;, v) v² + (z). Since the stochastic process x, starts from Tg ~u a, the primal problem is equivalent to:
E [.
inf
C(1, (11,2,,0)d
+ sup (-E[g(æ1)) Es(g(vi)) +Bsl))))))(((()))) 9EL.
(20)
where the supremum over g enforces the constraint 21 B (via Lagrange duality). Rewrite the Lagrangian problem as
g(0)) [ C(t,ze, - -tt + Bolo(w)).
inf (
E
(21)
Assuming strong duality holds under mild regularity conditions (e.g., L convex in u, 9 absolutely continuous, see Villani 2208)), we swap inf and sup:
in V C(t,, Iz, dtt s(2r)) + Bsle(w))
sup (8)
(22)
Moreover, since the optimal control can be chosen independently for each starting point æ, we have
inf V Ldt g(x11) no
inf E L. L olax)]
E
where the infimum on the right is taken over all Markovian controls. This identity follows from the dynamic programming principle and the fact that the value function depends only on the current state. By the definition of c-conjugate transform 3):
Expaa inf E, [ L(t, V1, we - g(x,) = x. =E (x).
(23)
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Thus, the dual problem becomes:
sup (EL((
(Ra(g"(<) Relo())))
(24)
In the second step, find the optimal control solution e" (t, x) by means the dynamic programming principle. Define the value function s(t, 2) that, for any t S 7 1, satisfies:
s(t, -inf IE U. ((2,22: + s(T,
It =
(25)
Applying Ito's formula to s(T, 2r), we obtain that
ds(r, 8, + V8 dr, 2 =
(26)
=
0,s -(0,s
Vs e + 2 2
dr + Vs odW,
(27)
Consider the evolution of the value between times and
(o.s ff: Vs us + 1
dz + [: V odW.
s(T, s(t, -
(28)
Based on the martingale property Ito integrals [E V8 odl br. = x| = 0), holds that
[ O2s V + d2
E[s(T, -x = x] = stt,z) +
(29)
Substitute back into dynamic programming and plug the last expression into the equation 25
S." [(2, x., da + st,,x) [: (0.5 + Vs by + 1 _
st,, = inf E 0
2z
(30)
Cancel stt, 2)) from both sides and divide by (r - t):
B [.. ( [(2z, L2, 02) + 028 Vo" 02 1 = dz
= inf
(31)
v(*.))
Take the limit r to derive the HJB equation for general Lagrangian C
0 = inf L(t, 0.s V v + [r((2
(32)
v
Identify the optimal control for the particular [(t, z, v) = 8² /2 + U(r). The infimum is attained when Vs,, yielding the final result of Theorem
For the HJB equation to have a unique solution (in the viscosity sense), we require coercivity (Theorem (Fleeting & Soner 2006)) of the Hamiltonian for some constants C₁ and C₂
H(x, Vs, V2s
1 v U(x) - x²
- (
(33)
2 C V C2(1 || v2
(34)
The term unique
s. dominates for large values, so in case U(x) is bounded and >> 0, the solution is
F GENERAL HJB
To derive the solution for (xr). one may consider the expectation over general stochastic processes
(not necessarily diffusion or one driven by an SDE). In a general case, we define the value function as:
st,, xx) = infE [.."
TTT v,) dr - g(xg) 84 26
(35)
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with terminal condition s(1, -g(x). Then, (x) s(0, Using the Dynamic Programming
Principle, one can obtain the HJB equation
(t, inf (EELt, 2, + A' s(t,
(36)
where A is the infinitesimal generator of the process:
E|s(t + h, reth) - s(t,x) xe =
(1, lim
(37)
h
and the optimal control satisfies:
v² argmin, {BL(I, 2z, v) + A" (()
(38)
At this step we need to decide on the type of the random process zt. For diffusion (in the main text, we often assume that dæ, dt + a dW,):
1 A's s V
(39)
2
the control iy given by a general function ii = [(t, with e, representing external random- ness, e.g., noise, the HJB equation becomes:
a,, x) = inf {E,, [L(t, f(x, e,)) E, (f(x,,)))) Vas}.
(40)
Note that the Liffusion-based solution is special motivated by the tractability in optimizing the value function. The SDE structure is particularly advantageous in numerical implementations, as allows for efficient gradient- optimization and aligns with existing literature on diffusion-based generative models.
G THEORETICAL CONVERGENCE ANALYSIS
We aim to prove that, under suitable assumptions, the error between the neural network approximation sy and the true viscosity solution s* of the Hamilton-Jacoii Jaw.oi--elllmaaan (HJB) equation is controlled by the PDE residual and the terminal loss.
Recall the dual formulation of GSB (Theorem
sup (.T.
[Ez~alg (x)) [Bgwo(l())]]
(41)
where the c-transform g is defined via the value function s(t, x) of the stochastic control problem:
g'(x) inf E U L(t, 24, vz) (() di - g(x1) 20 =
(42)
The HJB equation holds for s:
0.s H(t, xt Vs, V2s) = in (0,1)
(43)
where the Hamiltonian is:
T,t, M) = - 111 U(x)+ + o²
tr(M)
(44)
2
Assume the following regularity conditions hold:
1. The HJB equation has unique viscosity solution s* E C²,2((00,) 1] x ?) and both functions 8" C1.2 ((0, 1) x Ω);
2. Ω C Rd is a bounded domain with C2 boundary, and both 8* and sp satisfy the same bound- ary condition (Dirichlet or Neumann) such that all boundary terms vanish integrations by parts;
3. The diffusion coefficient satisfies o > 0. The diffusion term here provides regularization and prevents singularities in the optimal control;
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4. The HJB objective residual:
R(sa) = 0,sg + Ht, Vsa. vas)), is uniformly bounded: [[R(se)]l== S Enjo- (45)
The following theorem establishes the error estimate under the assumption of bounded deviation between the terminal functions s)) and s* (1, While this assumption provides useful starting point for analysis, is restrictive and may not fully capture the practical behavior of learned approximations. Below, will consider relaxation of this condition, extending the results to more realistic and weaker assumptions that align with empirical training dynamics. Theorem 2. Under Assumptions 1-4, there exists a constant C such that:
||80 s" 11.=== C (engb 1|80(1,,)) s" (1,=))
(46)
Moreover, the optimal control converges as:
S.¹ 1 |100 (t, v² (t, dadt S (I]so(1,-)- -1, (()so(1,))ss((()))))))) where = -sg. v² -
(47)
Proof. The true solution s satisfies:
0,s" + H(,,,, Vs" V2s') 0.
(48)
Use notation
Φ = Sg 5
(49)
Subtracting the first equation from the second, we get the PDE for the error 6:
0, Ht,, Vse V Se) H(t, x, Vs V R(se).
(50)
Let's analyze the difference in Hamiltonians 44). Since (xr cancels out in the difference, we have:
Htt, x, Vsg
H(t, Vs" V²s*)
(51)
Vs8
Vs
tr 2 20 V2s)
(52)
2
0²
(Vig Vs Vo) tr (2)))
(53)
Where we used || | V/ 2|2 = (V 50 + Vs" Usg Vs*) = (8s + Vs",, Vo).
Define vector field b(t, x) and constant A:
(V Vs*), A o²
b(1,z) = (V
(54)
Then equation (50
becomes:
0,0 + b(t,x:) V + Atr(V x) R((o).
(55)
This is linear parabolic PDE for Φ. Use the Feynman-Ke Kac formula and represent the PDE as:
Φ(t,2) = =E 6(1,z1) + S. R(so)(r. dr dr X₂
(56)
where dxx,- = b(r, )rr + odW, Taking absolute values and suprema:
[0(t,z)) < E (0(1,,)) + S. (x)ss))(rxxx))d X,
(57)
Since [<((s))] S <bjb. have
[0.1x=((.11 S (lo(1,))]l + Ggb-
(58)
This is key inequality. It shows that the error over the entire space-time domain bounded by the emor at the terminal time plus the accumulated PDE residual.
20

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Published as a conference paper at ICLR 2026
Next, obtain the control convergence bound. The optimal controls are v² -Vs* and by -ss-.
Therefore, 0g v* ~.. We need to bound |Voll2.
Multiply the PDE 55 by Φ and integrate over Q::
60.0 dx + A @ + I éb Vodx= = J @R(so) dr.
(59)
2
Using f d0,6 2 de 1 and integrating by parts the diffusion (boundary terms vanish by Assumption 2):
J &áóde = =- 1, /ool dx - V
(60)
For the drift term apply Souchy-Schwarn and Young's inequality: for any O.
1..
Lob. Øb. Vodx < (oll-~1l0la Vollzz S
1 P.
Volli
(61)
2s
Similarly, the right-hand side is bounded by
1 (((ss) de s ||0|]=
1 1001 + マ R10))
(62)
Collecting these estimates, we obtain
2 dt 1 + V
d
| 10l
+ & 1 + blll + 2 R(s)))]
(63)
2x
2
Choose
A (so that the term
2 / can be absorbed into the lend-hand side) and rearrange:
A Volli < Cilloli + R 12:2
doli= +
(64)
2 dt
2
||b113 sc 2A
where C1
3.2 Now integrate from t to 1:
2 |l )11. (1 )112+ A
J. | ds S sC J. 1ll1 ds+ /." [R(ss)) ds. (65)
Dropping the nompositive term (1) and applying Grönwall's inequality (or simply bounding f | a ds by the terminal value plus the integral of the residual, using the representation from the Feynman-Ko -ac formula), we finally obtain
[ Voll dt s (1001))) + ([
(66)
for some constant C depending on A, | and the length of the interval. This yields the desired control convergence bound (cquation(4) 47).
In the previous proposition we showed that, once candidate network approximately solves the HJB equation, the space-time error between 89 and the true solution 8* can be controlled in terms of the terminal discrepancy and the HJB residual. Concretely, we obtained an estimate of the form
||8g 8" (1.((()) S [10((,,) (1)= ||(((s))l==>
(67)
However, from learning perspective, never observe s* directly, and we do minimize (, | in training. Instead, we control two quantities: the dual objective gap (Kantorovich-type functional between 59 and s*), and the PDE residual loss: ({a ) 1.2 over [0, 1] x Ω.
To turn these training losses into rigorous guarantees on the learned dynamics and controls, we must therefore express the terminal error
(1s001,,)) s" s((1, (1,,))12
(68)
21

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Published as a conference paper at ICLR 2026
directly in terms of the objective gap. Denote g(x) s(1,-) and write the (unregularized) Kan- torovich dual
[gg) Ro ( (x) + [g(())
(69)
where gf is the e-transform (equivalently, the HJB value at t 0). Let assume the comparison principle holds for the associated HJB so that (x) = (0 r). For the OT dual J is concave but typically not strongly concave and g" is only unique up to additive constant. A common way is to regularize g:
X
Jx(g) = E. (x)] + Eplg(v)
>> 0,
(70)
2
where (H, (x) is a Hilbert space (e.g. [VVllz>). Then Jy is A-strongly concave in H. For all g.h E H
JA(h) S (و(g) + (DJx(g), 9)n - 2 g
(71)
We next provide trajectory-basee based estimate for the quality of a learned terminal potential. Given an approximate HJB solution Sp with uniformly bounded residual, we relate the regularized dual objective value JΛ of the induced terminal potential ge = - (1 -) to an explicitly computable surro- gate Jx (0) obtained by simulating the closed-loop diffusion driven by sg. Under realizability assumption for the maximizer of the regularized dual and a continuous embedding + (3)),, this yields an (3) error bound on the terminal potential.
Theorem 3 (Trajectory-fomm bound for the regularized dual potential), Assume so E C ((, 100,1]xR" )
s0(1, E H. and (R(()) ((0,(())))) I Ebjb- Define the closed-loop diffusion
dx. Sso x.)) dt + a dW,, zg a,
(72)
assuming it is well-posed and the expectations below are finite. Let H be a Hilbert space with continuous embedding into (3),), i.e. [[b|| S Cembblhh (elb. Let g" E arg maggay Jx (g) and define (1,,,) = 9" (.). Assume that g" is representable by the parametric model, iee, there exists 8 such that
se- (1,-) = -
in particular sq- (1, E H.
(73)
Define the trajectory- form surrogate
JA(0) = EE sol(, + ['' (H/IVs0l2 + +U) UU) (1, dt -Eg(so - 2 $1s0 (1, 112 (74)
Then
2 [|so(1,.) s -1, (1,-))22(()) S Cembb/ ) (Jx(00) Jx(() + 4Eugb)
(75)
Proof. Fix O and set
9o = - (1. E H.
(76)
Let 8 denote the (exact) solution of the terminal-value HJB problem
Rss) = 0 on (0, 1) x Rd
(1,-)) = (.)
(77)
where R is as in the statement,
R(s) 0, \ | + + + 5 tr( By the definition of the c-transform through the HJB/value-funcee representation associated with equation 77 we have
(78)
g5(x) = 5*0 (0,x),
x Rd.
(79)
Define, for (t, x) e [0, Rdd
$ (1, = so (1, + Ehjb (1 - 1), 87 (1,x) == 20(,,2) Enjn (1 1).
(80)
Since 5 V and 2 = V while 0:55
-5,00 - EEjj and 0:să 0.50 + + it follows
from equation that
R(st) -R(s0) -Ehjbr
R(s₂) R(so) Ebjb-
(81)
22

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Published as a conference paper at ICLR 2026
Using the assumption
S Ebjb: we obtain
R(s 50
R(55) ) 2 0 on (0, x R
(82)
Moreover, the terminal conditions coincide:
s (1, so (1, = -9
(83)
Assuming the comparison principle holds for equation 77 (with sub/supersolutions interpreted in the usual viscosity sense), we get
$g (t, x:) S s (t, x) S sp 2),
(t, E [0, x R
(84)
Evaluating it at = 0 and using equation 80 yields
|0s -) 00 (0, > (R+)) Enjb-
(85)
Combining with equation 79 gives
|| (0. .) - 95( (=(()) S Eujb.
(86)
Taking expectation under =0 a yields
E [0](=00) R. (so (0, (0,00)]
Shib-
(87)
Let X solve the closed-loop SDE from the statement,
dxç vt, dt + a dWe,
z ~ a.
(88)
Applying Itô's formula to f + so x (using so C3.3) gives
dsp(t, = (0.so - sell || + c tr (r(V"sa) se)
(t, (t, dt + vVso(t,2)) dW,
(89)
Using the identity obtained by rearranging equation 78
0,sg - V || + o (r (e) = R(ss) - V +U)
(90)
integrating over [0, 1). taking expectation, and using E [ Vsa dW,- = O, we obtain
E[sp (0, x0]] = IF ss 21, + L/' (dIVvoll +U) (t, x dt E L/
dt.
(91)
By the residual bound ||R( 1x S Shjb.
E [[s =1) - E sa(L, + /. 1 s + U)) (t, 22) dt
S Enjb.
(92)
Relate JA (x() and (0). Combining equation 87 and equation 92 yields
E., En (1)) E (1) + L'' (HIVVoel + +U) +U)(,=²) di)
< 2Enjb.
(93)
Using go = -1, -) we have
Eplgol = - Essss (1, v)1,
1100|1 = (ss((,,))lzz)
(94)
Substituting the last expressions into the definition of (90) gives
Jx(9)) 2 (x) 2Enjh.
(95)
Since JA is A-strongly concave on H. for its maximizer g" E arg max Jx(9) we have
Jx (x00 2 2
l1o00 12
(66)
We have assumed g" is representable by the parametric model in the sense that there exists 8* such that sg- (1,-) = -g". Applying equation 95 to 0 and to 0*, we obtain
Jx(g") (g0) (x ) (2()) = Jx(00) - Jx(0) + 45hjb-
(97)
It yields
2 (J)(00)- (0) + 4Engh)
2
11g0 - ---1= S v
(98)
Using the continuous embedding |1/0|12((9) S Cemb l/1/// Cembll/il1z we conclude
and equality so se, (::-)--s(AA s1, 9" go.
2 (Jx(0r)) Jx(0) -xx()) + 4FNJb).
[]s)(1,)) (.4.))-s"(1,,)))) ="(1,-)ll20(0) S Cemb V
(99)
23


Reviews:
[{"review_id": "2RTH2zVJ7N", "reviewer": "Reviewer_EX3V", "summary": "This paper introduces HOTA, a novel method for solving the Generalized Schrödinger Bridge (GSB) problem. The approach is based on a dual formulation that seeks a Kantorovich potential aligning the source and target distributions while satisfying the associated Hamilton–Jacobi–Bellman (HJB) constraint.", "questions": "HOTA currently uses linear interpolation to initialize the trajectories. Could a more informed initialization be designed by leveraging the state cost $U(x)$? In several cases shown in Figure 1, linear trajectories obviously pass through regions that the final solution should avoid, which intuitively seems to hinder learning efficiency.\n\nThe paper should define the acronym EMA (Exponential Moving Average) upon its first occurrence and clarify the relationship between the target model $\\bar{s}$ and the main model $s_\\theta$.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper proves that duality holds under mild conditions. It further demonstrates that the method is robust to complex geometries and remains effective even for non-smooth cost functions. By employing the Euler–Maruyama scheme to approximate the solution of the underlying SDE, the proposed approach avoids explicit density modeling and thereby simplifies the learning process. In numerical experiments, HOTA consistently outperforms several state-of-the-art methods across a range of benchmark settings.", "weaknesses": "While the proposed method demonstrates impressive performance and scalability, the experimental setup and model architecture appear relatively simple. Since the paper directly compares with GSBM, it would be greatly strengthened by including evaluations on more complex datasets used in that work, such as AFHQ.\n\nMoreover, as only a lightweight 4-layer MLP was tested, it remains unclear whether the reduced computational overhead reported in Appendix B would persist when employing a heavier model, such as a U-Net.\n\nThe HJB constraint is enforced via an $L^2$ loss over sampled trajectories, which does not guarantee that the solution satisfies the HJB equation over the entire space $\\mathbb{R}^d$. As noted in [1], the $L^2$ loss may also not be ideal for solving HJB equations. Therefore, the authors should investigate, at least numerically, how well the learned solution satisfies the HJB constraint and examine the behavior of $s_\\theta$ in out-of-sample regions.\n\n**Minor issues/typos**:\n- Line 183: “c” in “c-transform” was not put in a math environment.\n- Line 652: “are” should be “and”.\n- Line 786: “Buy” should be “By”.\n\n**Reference**:\n\n[1] Chuwei Wang, Shanda Li, Di He, and Liwei Wang. Is $L^2$ physics informed loss always suitable for training physics? [arXiv: 2206.02016](https://arxiv.org/abs/2206.02016)", "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": "P7MmMIii3W", "reviewer": "Reviewer_8EsL", "summary": "This paper proposes a new way to solve Generalized Schrödinger Bridge problems by tying the HJB value function $s(t,x)$ to a Kantorovich terminal condition. The idea is to optimize two losses jointly. First, enforce that the endpoint potential matches the Kantorovich potential via a Lagrangian relaxation. Second, ensure $s$ really is a value function by having it satisfy the corresponding HJB equation as a PINN-style constraint. With that in place, the optimal drift is recovered as $-\\nabla s$. They test on standard 2D toy setups and also explore scaling to higher dimensions on the sphere.", "questions": "1) Please clarify the inference-time cost of computing the network gradient. How does evaluating $\\nabla s$ compare to other methods at inference?\n\n2) Could you provide more detail on how the buffer is managed? In particular, why is only the first trajectory added in the pseudocode? Also, please explain why you use the interpolation sample steps $N_0$ in the algorithm.\n\n3) Why does the buffer improve results? Is the benefit primarily stability, or mainly computational efficiency? In an ideal setting with unlimited compute, would resampling fresh data each step be strictly better?\n\n4) In the ablation, what exactly does \"without EMA\" remove? Could you discuss alternatives to the PINN formulation around equations 14–15? For example, could you remove EMA and enable stop-grad on specific terms, and what tradeoffs led you to the final choice?\n\n5) What motivated the inclusion of the angular acceleration term, and how does it affect results in practice?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The method is original in how it ties the value function to a Kantorovich terminal condition and enforces the HJB with a PINN, giving a clean route to recover the drift as a gradient. The empirical results are strong, with clear improvements over baselines and ablations that isolate the effect of the replay buffer, EMA targets, and gradient balancing. The technical presentation is mostly clear, with a coherent objective that aligns the endpoint constraint and the dynamics constraint, and enough detail to make the optimization reproducible. In terms of significance, the approach looks broadly useful for GSB and related control and transport problems, and the sphere experiments suggest promise beyond small 2D toy examples.", "weaknesses": "The paper reads as if it targets readers already deeply familiar with the topic, which raises the entry cost for newcomers. Several parts of the exposition could be tightened, especially the ablation study. See the Questions section. The main practical limitations are that, first, enforcing the HJB equation in higher dimensions and on more complex targets may be challenging, and second, the learned drift requires evaluating $\\nabla s$ at inference.", "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": "GzLBZkNrJ6", "reviewer": "Reviewer_Qowe", "summary": "The paper proposes a learning approach to address the dynamical optimal transport (OT) problem with non-smooth cost functionals, enabling the incorporation of the underlying geometry of the data manifold. The proposed method is developed by leveraging the relationship between the dynamical OT problem and the Hamilton–Jacobi–Bellman (HJB) equation, where the value function is approximated using a neural network to learn the transport mapping. Experimental results on synthetic datasets demonstrate that the proposed method achieves better alignment with the target distribution and incurs lower transport costs compared to existing (Generalized) Schrödinger Bridge models.", "questions": "Could the authors provide a case study or example illustrating a scenario where non-smooth cost functionals pose challenges?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper makes a original contribution by establishing a novel connection between the Generalized Schrödinger Bridge (GSB) problem and the Hamilton–Jacobi–Bellman (HJB) equation. By approximating the value function with a parametric model, the authors develop a tractable learning framework for modeling the dynamics of optimal transport (OT) between distributions. This provides a fresh perspective on OT-based diffusion models with fewer assumptions on flow densities.\n\nThe work is of strong technical quality, with a sound theoretical foundation and convincing experimental validation. Results on synthetic datasets show improved target matching and reduced transport cost compared to existing (Generalized) Schrödinger Bridge methods. The inclusion of code further supports reproducibility.\n\nThe paper is clearly written and well structured, making the complex ideas accessible. Its significance lies in broadening the applicability of OT methods and opening new possibilities for learning-based transport and diffusion modeling.", "weaknesses": "A key limitation of the paper is its exclusive reliance on synthetic datasets, which constrains the evaluation of the method’s applicability to real-world problems. While the authors acknowledge that their approach struggles with complex data such as images, they do not provide a clear explanation or empirical justification for this limitation. A more detailed analysis of why the model underperforms in such settings, along with experiments or ablation studies incorporating stronger inductive biases in the architecture, would provide concrete evidence to support their claims and guide future improvements.\nFrom a theoretical perspective, the paper would be strengthened by a formal convergence analysis of the learning algorithm to ensure stability and reliability. Additionally, a discussion of the approximation error introduced by neural network parameterization and the estimation error due to sampling would offer valuable insights into the computational behavior and robustness of the approach. Addressing these aspects would enhance both the theoretical rigor and the practical significance of the work.", "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": "AdiIlcDJmK", "reviewer": "Reviewer_Z9DH", "summary": "This paper proposes a method for estimating the time-dependent Kantorovich potential $s:[0,1]×R^d \\rightarrow R$ using tools from optimal transport and Hamilton–Jacobi–Bellman (HJB) theory. The terminal potential $s(1,⋅)$ is learned using the Kantorovich dual formulation, and intermediate values $s(t, \\cdot)$ are obtained by solving the HJB equation backward in time. The approach is demonstrated on 2D synthetic datasets, a 3D sphere-to-sphere transport, and opinion depolarization task.", "questions": "- Can the authors test their method on higher-dimensional synthetic data (e.g., Gaussian mixtures in 10–50 dimensions)?\n\n- Could they also demonstrate performance on standard datasets such as LiDAR point clouds, CelebA image translation, or similar benchmarks?\n\n- Can the authors report training curves for (1) the HJB residual (PDE violation), and (2)vthe Kantorovich dual loss? This would help verify whether the proposed method truly solves the coupled dual–HJB system.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The paper clearly states the goal of learning the full space–time potential $s(t,x)$. This only requires a single neural network parameterizes the value function.\n\n2. The theoretical derivation is well structured, connecting the Kantorovich duality with the HJB equation in a coherent way. Visualizations of the learned transport trajectories and potentials are helpful for intuition.", "weaknesses": "1. **Limited originality relative to prior work.** The overall methodology, estimating a terminal potential via the Kantorovich dual and propagating it using HJB, closely resembles prior work [1]. The main difference is the inclusion of a state-dependent running cost, but the theoretical structure remains largely unchanged. As a result, the contribution appears incremental rather than conceptually novel.\n\n2. **Concerns about algorithmic practicality.** \n\n- Optimizing the unregularized Kantorovich dual objective is known to be unstable and highly sensitive to initialization and regularization.\n\n- Accurately enforcing the HJB equation corresponds to solving a nonlinear PDE, which is notoriously difficult, especially in high-dimensional spaces.\n\n- - Even small numerical errors in estimating $s(t,x)$ may lead to inaccurate gradient estimation $\\nabla s(t, x)$. For instance, if value function has little oscillatory or noisy estimates, that would change the gradient estimation of the value function $s$ drastically, leading to inaccurate estimation of the control. \n\n- The method follows an Eulerian perspective, requiring accurate approximation of $s(t,x)$ across the entire state space. This becomes computationally infeasible beyond very low-dimensional domains, suggesting limited scalability beyond 2D or 3D, or toy datasets.\n\n3. **Experimental validation is limited and low-dimensional.**\n\n- All experiments are conducted on synthetic datasets. These do not convincingly demonstrate scalability or robustness.\n\n- There are no experiments on standard OT benchmarks, such as LiDAR point cloud transport [2], image-to-image translation [1], or Gaussian-to-GMM transport in higher dimensions.\n\n- The paper does not include analyses of PDE residuals, HJB constraint violation, or dual objective convergence, so it is unclear whether the training process actually solves the intended optimization problem.\n\n\nReferences\n\n[1] Scalable Simulation-free Entropic Unbalanced Optimal Transport\n\n[2] Generalized Schrodinger Bridge Matching", "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": "F0UVpIiQCn", "reviewer": "Reviewer_qy11", "summary": "This paper presents a Hamiltonian Framework for Optimal Transport Advection. The contribution of this work appears to be the loss functions (Eqs. 14-17) and the algorithm, which learns dynamical OT problems with a dual Hamiltonian formulation. The authors claim that the proposed approach surpasses SOTA performance in low-dimensional settings.", "questions": "* Can HOTA be applied to image-to-image translation tasks?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* The proposed method is mostly sound, with a clear presentation of the mathematical derivation.\n* The outlined experiments validate the effectiveness of the formulation.", "weaknesses": "* The motivation, reasoning, and verification for considering the dual GSB problem are not clear and need enhancement.\n* The theoretical arguments in Section 3 are somewhat well known from stochastic mechanics. Therefore, I think contributions of this work seems to be focused in the methodology part.\n* Sample inefficiency. I believe Eq. (17) contains the gradient of Hessians, which cannot be easily computed in high dimensionality. The computation of Eqs. (14-15) is seemingly not sample-efficient for large models. The authors are encouraged to discuss this limitation if the method requires more training time than the benchmarks.", "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)