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HOTA: Hamiltonian framework for Optimal Transport Advection

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.6Download Report PDF
Optimal transportoptimal controlgeneralized Schrödinger bridgediffusion models
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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper introduces HOTA, a Hamiltonian-Jacobi-Bellman framework for optimal transport that operates via Kantorovich potentials and avoids explicit density modeling. It resides in the 'Dynamical and Hamiltonian Formulations' leaf, which contains only three papers total, including the original work. This leaf sits within the broader 'Computational Methods and Algorithms' branch, indicating a relatively sparse but active research direction focused on dynamical reformulations of optimal transport. The small sibling count suggests this specific Hamiltonian angle is less crowded than adjacent areas like regularization techniques.

The taxonomy reveals neighboring leaves addressing related computational challenges: 'Regularization and Approximation Techniques' (three papers on entropy-based methods) and 'Discrete and Finite-Dimensional Implementations' (one paper on discrete surfaces). The 'Theoretical Foundations' branch explores regularity and stability questions that underpin when dynamical schemes remain well-posed, while 'Non-Euclidean Extensions' test transport on curved geometries. HOTA bridges computational efficiency with geometric generality, diverging from entropy-regularized approaches by tackling non-smooth costs directly through dual formulations rather than smoothing approximations.

Among 21 candidates examined, the contribution-level analysis shows mixed novelty signals. The dual HJB reformulation examined 10 candidates with 1 refutable match, suggesting some prior overlap in Hamiltonian-dual perspectives. The specialized training procedure examined only 1 candidate but found 1 refutable instance, indicating limited search scope in this area. The overall HOTA framework examined 10 candidates with 2 refutable matches, implying that while the specific combination may be novel, individual components have precedents. The small candidate pool (21 total) means these statistics reflect top-K semantic matches, not exhaustive coverage.

Given the limited search scope and sparse taxonomy leaf, HOTA appears to occupy a moderately explored niche within dynamical optimal transport. The Hamiltonian-dual angle and non-smooth cost handling distinguish it from sibling works, though the contribution-level refutations suggest incremental refinement over existing Hamiltonian or control-theoretic methods. A broader literature search might reveal additional overlaps, particularly in control theory or variational analysis communities not fully captured by the 21-candidate sample.

Taxonomy

Core-task Taxonomy Papers
22
3
Claimed Contributions
21
Contribution Candidate Papers Compared
4
Refutable Paper

Research Landscape Overview

Core task: optimal transport with non-trivial geometry and non-smooth potentials. The field has evolved into several interconnected branches that address both computational challenges and foundational questions. Computational Methods and Algorithms focus on numerical schemes and dynamical formulations that can handle irregular cost functions or complex geometries, often leveraging Hamiltonian or Lagrangian perspectives to design tractable solvers. Theoretical Foundations and Regularity investigate the existence, uniqueness, and smoothness properties of transport maps when classical assumptions break down, drawing on tools from metric geometry and non-smooth analysis. Non-Euclidean and Geometric Extensions explore transport on manifolds, discrete surfaces, and even Lorentzian or synthetic spaces, broadening the scope beyond flat Euclidean settings. Connections to Information Geometry and Applications bridge optimal transport to statistical inference, generative modeling, and physics-inspired problems, demonstrating the framework's versatility across disciplines. Within the computational landscape, a particularly active line of work examines dynamical and Hamiltonian formulations that reformulate transport as a time-evolving system, enabling gradient-based or control-theoretic methods. HOTA[0] sits squarely in this branch, emphasizing Hamiltonian structures to tackle non-smooth potentials and non-trivial geometries. It shares methodological kinship with Neural Lagrangian Transport[1], which also adopts a Lagrangian viewpoint but focuses on neural parameterizations, and with Dynamical Control-Affine[5], which explores control-theoretic angles on similar dynamical systems. These works collectively highlight a trade-off between analytical tractability and the flexibility needed for irregular or high-dimensional settings. Meanwhile, neighboring branches such as Theoretical Foundations probe regularity questions that inform when such dynamical schemes are well-posed, and Non-Euclidean Extensions test these methods on curved or discrete domains. The interplay between computational innovation and geometric rigor remains a central open question, with HOTA[0] contributing a Hamiltonian lens that complements existing Lagrangian and control-based approaches.

Claimed Contributions

Dual reformulation of GSB via Kantorovich potentials and HJB value function

The authors propose a dual formulation of the Generalized Schrödinger Bridge problem that connects Kantorovich potentials with a Hamilton-Jacobi-Bellman value function. This reformulation transforms the HJB constraint into a regularizer, yielding a stable objective suitable for gradient-based optimization in non-Euclidean and non-smooth settings.

10 retrieved papers
Can Refute
Specialized training procedure with data replay and target network stabilization

The authors develop a training procedure that incorporates data replay buffers and exponential moving average (EMA) target networks. This design enables stable joint learning of both the Kantorovich potential and the optimal drift policy in high-dimensional spaces.

1 retrieved paper
Can Refute
HOTA framework for optimal transport advection

The authors introduce HOTA, a method that solves the dual dynamical optimal transport problem using Hamilton-Jacobi-Bellman equations and Kantorovich potentials. The framework enables efficient trajectory optimization without requiring explicit density modeling and handles non-smooth cost functionals.

10 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Dual reformulation of GSB via Kantorovich potentials and HJB value function

The authors propose a dual formulation of the Generalized Schrödinger Bridge problem that connects Kantorovich potentials with a Hamilton-Jacobi-Bellman value function. This reformulation transforms the HJB constraint into a regularizer, yielding a stable objective suitable for gradient-based optimization in non-Euclidean and non-smooth settings.

Contribution

Specialized training procedure with data replay and target network stabilization

The authors develop a training procedure that incorporates data replay buffers and exponential moving average (EMA) target networks. This design enables stable joint learning of both the Kantorovich potential and the optimal drift policy in high-dimensional spaces.

Contribution

HOTA framework for optimal transport advection

The authors introduce HOTA, a method that solves the dual dynamical optimal transport problem using Hamilton-Jacobi-Bellman equations and Kantorovich potentials. The framework enables efficient trajectory optimization without requiring explicit density modeling and handles non-smooth cost functionals.

Table of Contents