Hamiltonian Optimal Transport Advection (HOTA)

• HOTA is a Hamiltonian formulation of dynamical optimal transport that uses a dual HJB solver to directly generate optimal trajectories via potential-based advection, bypassing intermediate density estimation.
• The method leverages a neural network to approximate the Kantorovich potential, serving simultaneously as the control and value function in both stochastic and deterministic regimes.
• It robustly handles complex geometries and non-smooth costs by enforcing viscosity solutions and integrating Riemannian metrics, leading to improved feasibility and optimality across various benchmarks.

Hamiltonian Optimal Transport Advection (HOTA) denotes a Hamiltonian formulation of optimal transport in which probability mass is advected by trajectories induced from a conjugate potential or value function. In its most specific current usage, HOTA refers to the dual, Hamilton–Jacobi–Bellman (HJB) based solver introduced in “HOTA: Hamiltonian framework for Optimal Transport Advection,” which learns Kantorovich potentials directly, generates optimal trajectories by Hamiltonian advection, and remains density-free in the sense that it never estimates intermediate densities along the path (Buzun et al., 23 Jul 2025). Related arXiv work uses the same Hamiltonian-advection viewpoint in discrete Wasserstein geometry on finite graphs (Cui et al., 2021) and in Hamilton–Jacobi characteristic-flow formulations that recover transport maps from viscosity solutions (Park et al., 30 Sep 2025). Across these variants, the central idea is that optimal transport can be expressed through a Hamiltonian or HJB structure, with advection determined by a potential whose gradient or discrete gradient yields the optimal drift.

1. Dynamical optimal transport problem and the role of advection

HOTA is formulated for dynamical optimal transport: given source and target distributions $\rho_0$ and $\rho_1$ on a space that may be Euclidean or manifold-valued, one seeks a probability path $\{\rho_t\}_{t\in[0,1]}$ and a velocity field $v(t,x)$ that transport $\rho_0$ to $\rho_1$ with minimal path cost. In the Euclidean quadratic case, the Benamou–Brenier formulation is

$$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$

More generally, HOTA considers Lagrangian costs $L(x,v,t)$ and, in the generalized Schrödinger Bridge setting emphasized by the 2025 HOTA paper, the cost

$\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$

where $U(x)$ encodes geometry, obstacles, or preferences (Buzun et al., 23 Jul 2025).

The motivation is explicitly geometric. The 2025 formulation argues that many OT-based generative methods assume trivial Euclidean geometry and rely on explicit intermediate densities or simple kinetic-energy costs. According to that paper, such choices can produce straight-line paths, ignore manifold constraints, and become brittle when the state cost $\rho_1$0 is non-smooth. HOTA addresses this by solving the dual dynamical OT problem through an HJB equation for a single potential $\rho_1$1, from which trajectories are obtained directly by advection rather than by explicit density evolution (Buzun et al., 23 Jul 2025).

This yields a specific interpretation of “advection.” In HOTA, advection is the transport of particles or mass elements under an optimal drift derived from the potential. In the continuous Euclidean setting, particles follow the drift $\rho_1$2. In manifold settings, the drift is modified by the metric. In graph-based formulations, mass is advected by a discrete continuity equation with flux induced by a discrete gradient potential.

2. Hamiltonian and HJB formulation

The defining structural step is the passage from a Lagrangian to a Hamiltonian through the Legendre transform

$\rho_1$3

For the cost $\rho_1$4, this becomes

$\rho_1$5

while for a Riemannian quadratic cost $\rho_1$6 one obtains

$\rho_1$7

The 2025 HOTA solver identifies a single function $\rho_1$8 as both the dual Kantorovich potential and the HJB value function. With stochastic dynamics

$\rho_1$9

the value function satisfies

$\{\rho_t\}_{t\in[0,1]}$0

Minimization in $\{\rho_t\}_{t\in[0,1]}$1 yields the optimal control

$\{\rho_t\}_{t\in[0,1]}$2

and the PDE

$\{\rho_t\}_{t\in[0,1]}$3

Equivalently, in Hamiltonian form,

$\{\rho_t\}_{t\in[0,1]}$4

with the Laplace–Beltrami operator replacing $\{\rho_t\}_{t\in[0,1]}$5 on manifolds (Buzun et al., 23 Jul 2025).

The associated dual objective is density-free:

$\{\rho_t\}_{t\in[0,1]}$6

subject to the HJB PDE. In this formulation, the terminal value $\{\rho_t\}_{t\in[0,1]}$7 is free to optimize and is equivalent to the negative of a Kantorovich potential. The 2025 paper states that this objective enforces exact target matching through the terminal term difference, while the HJB equation supplies the optimal drift (Buzun et al., 23 Jul 2025).

For deterministic characteristics with $\{\rho_t\}_{t\in[0,1]}$8, the induced Hamiltonian trajectories satisfy

$\{\rho_t\}_{t\in[0,1]}$9

If $v(t,x)$0, then

$v(t,x)$1

In the Euclidean quadratic case this reduces to $v(t,x)$2, and in the Riemannian quadratic case to $v(t,x)$3.

3. The 2025 HOTA solver: density-free trajectory optimization

The computational realization in “HOTA: Hamiltonian framework for Optimal Transport Advection” parameterizes the Kantorovich/HJB potential by a neural network $v(t,x)$4 and optimizes it jointly for terminal matching and PDE consistency (Buzun et al., 23 Jul 2025). The method is described as density-free because it never estimates intermediate densities $v(t,x)$5 along the path. Instead, samples $v(t,x)$6 are pushed forward by

$v(t,x)$7

and terminal samples are matched to $v(t,x)$8 through the dual potential difference.

The terminal matching term is

$v(t,x)$9

where $\rho_0$0 are generated from $\rho_0$1 under the learned drift and $\rho_0$2 are sampled from $\rho_0$3. PDE consistency is imposed through an HJB residual loss built from $\rho_0$4 and an exponential moving average target $\rho_0$5. The residual contains the time derivative, the quadratic gradient term, the pointwise potential $\rho_0$6, the second-order trace term, and an optional angular acceleration penalty

$\rho_0$7

which is used to straighten trajectories. The total gradient update combines the potential loss and the HJB residual with gradient balancing based on an EMA of gradient-norm ratios.

Advection is discretized by Euler–Maruyama:

$\rho_0$8

with $\rho_0$9. When $\rho_1$0, this becomes standard Euler advection. Replay buffers store trajectory points so that HJB residuals can be sampled in high-probability regions, and gradients are propagated through both the integrator and the network by automatic differentiation. The reported per-iteration cost scales roughly as $\rho_1$1 for $\rho_1$2 particles, $\rho_1$3 time steps, and ambient dimension $\rho_1$4, with Hessian terms in the HJB residual contributing additional cost; the paper states that practical batching and limited feature frequencies keep training under minutes on a single GPU (RTX 3090 in experiments) (Buzun et al., 23 Jul 2025).

The implementation details are correspondingly specific. The potential network is a simple MLP with Fourier feature encoding for time. To stabilize explicit time derivatives, frequencies are limited to $\rho_1$5 and sine/cosine features are normalized by $\rho_1$6. Optimization uses Adam with cosine annealing, initial learning rate $\rho_1$7, EMA target decay $\rho_1$8, batch size $\rho_1$9, and around $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$0k iterations. Warm-up residual sampling uses linear interpolation between $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$1 and $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$2, after which replay-buffer sampling is used.

4. Geometry, non-smooth costs, and theoretical status

A central property of HOTA is that geometry enters through either the state potential $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$3 or a Riemannian metric $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$4. For Riemannian quadratic costs,

$$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$5

and the induced Hamiltonian is $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$6. This extends the method beyond trivial Euclidean geometry. The paper explicitly states that plain MLPs were sufficient in experiments, although manifold-aware layers or geometric features are possible (Buzun et al., 23 Jul 2025).

The handling of non-smooth state costs is another defining claim. In the 2025 HOTA formulation, the HJB residual uses $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$7 only pointwise and does not require $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$8. The PDE is enforced in the viscosity sense, which is intended to tolerate non-smoothness. Under coercivity of the Hamiltonian and $$\min_{\rho,v} \int_0^1 \int_{\mathbb{R}^d} \tfrac{1}{2}\|v(t,x)\|^2 \rho(t,x)\, dx\, dt \quad \text{s.t. } \partial_t \rho + \nabla\cdot(\rho v)=0,\; \rho(0,\cdot)=\rho_0,\;\rho(1,\cdot)=\rho_1.$$9, the paper states that uniqueness in the viscosity sense holds; for bounded $L(x,v,t)$0 and $L(x,v,t)$1, the solution is unique. This is one of the main reasons the method is presented as robust on nearly non-differentiable costs (Buzun et al., 23 Jul 2025).

The formal duality claim appears as Theorem 1 in the paper. With dynamics $L(x,v,t)$2 and cost $L(x,v,t)$3, the dynamical OT problem admits the dual

$L(x,v,t)$4

subject to

$L(x,v,t)$5

The optimal control is $L(x,v,t)$6. The proof sketch reported in the paper combines Kantorovich duality with dynamic programming, derives the HJB equation via Itô’s lemma, and identifies the minimizing $L(x,v,t)$7 (Buzun et al., 23 Jul 2025).

Two common misconceptions are explicitly contradicted by this formulation. First, density-free does not mean unconstrained: feasibility is enforced at the terminal boundary by the dual objective, and the continuity equation is satisfied in the particle sense by advection. Second, Hamiltonian optimal transport advection is not restricted to smooth Euclidean costs: the framework is stated to support non-smooth $L(x,v,t)$8, Riemannian metrics, and stochastic dynamics.

5. Related Hamiltonian OT frameworks and alternative meanings of HOTA

Within the 2025 HOTA paper, the method is positioned against several adjacent frameworks. Benamou–Brenier solvers work in the primal variables $L(x,v,t)$9 and require density evolution and PDE solvers, whereas HOTA operates in the dual and in particle space. Schrödinger bridge and entropic OT methods introduce entropic regularization and stochastic processes, and many such methods compute intermediate densities or relax terminal constraints; HOTA is described as maintaining strict terminal matching while remaining density-free. Flow matching, diffusion, and score-based methods are described as relying on intermediate densities or score fields and often assuming Euclidean geometry, whereas HOTA optimizes a single value function and uses the Hamiltonian drift $\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$0 (Buzun et al., 23 Jul 2025).

A distinct but related meaning appears in “What is a stochastic Hamiltonian process on finite graph? An optimal transport answer.” There, Hamiltonian optimal transport advection is the advection of probability mass on a finite graph driven by a Hamiltonian vector field on the Wasserstein manifold of graph-supported densities (Cui et al., 2021). The finite-graph setting begins with

$\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$1

a discrete gradient

$\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$2

and a continuity equation

$\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$3

The corresponding discrete Hamiltonian for OT geodesics is

$\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$4

yielding coupled Hamilton equations for $\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$5. That paper further shows that not every discrete OT formulation induces a valid stochastic Markov process, because the transition-rate matrix $\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$6 must have nonnegative off-diagonal entries. Arithmetic or logarithmic means can fail this requirement, while an upwind choice $\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$7 together with $\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$8 ensures positivity. This is a genuine structural constraint rather than a numerical detail (Cui et al., 2021).

A second neighboring formulation appears in “Neural Hamilton--Jacobi Characteristic Flows for Optimal Transport.” That work does not use the 2025 HOTA architecture, but it instantiates the same Hamiltonian-advection viewpoint through characteristics of the Hamilton–Jacobi equation (Park et al., 30 Sep 2025). For translation-invariant costs $\mathcal{L}(t,x,v)=\tfrac{1}{2}\|v\|^2+U(x),$9 with Legendre transform $U(x)$0, the HJ equation is

$U(x)$1

with Hamiltonian $U(x)$2. Because $U(x)$3 is independent of $U(x)$4, the characteristic momentum is constant:

$U(x)$5

This produces straight-line characteristics and closed-form bidirectional transport maps

$U(x)$6

The method therefore eliminates ODE integration entirely and learns a single network through an implicit HJ loss plus an MMD or energy-distance alignment term. A plausible implication is that “Hamiltonian Optimal Transport Advection” is developing into a broader family of methods unified by Hamiltonian transport geometry rather than by a single solver design.

6. Empirical results, ablations, limitations, and open directions

The empirical evaluation reported for the 2025 HOTA solver covers smooth and nearly non-differentiable potentials in two-dimensional benchmarks, higher-dimensional sphere datasets, and a high-dimensional opinion-depolarization problem (Buzun et al., 23 Jul 2025). Feasibility is measured by $U(x)$7 with quadratic cost, and optimality by the integral trajectory cost

$U(x)$8

The paper states that HOTA consistently outperforms baselines on both feasibility and optimality.

Benchmark	HOTA	Comparison reported
Stunnel	feasibility $U(x)$9; optimality $\rho_1$00	GSBM $\rho_1$01; $\rho_1$02
Vneck	feasibility $\rho_1$03; optimality $\rho_1$04	GSBM $\rho_1$05; $\rho_1$06
GMM	feasibility $\rho_1$07; optimality $\rho_1$08	GSBM $\rho_1$09; $\rho_1$10

For nearly non-differentiable costs, the paper reports that on BabyMaze, Slit, and Box, HOTA achieves feasibility of approximately $\rho_1$11, $\rho_1$12, and $\rho_1$13, respectively, with lower integral costs than baselines. On sphere datasets, it is reported to maintain feasibility and optimality trends consistently as dimension increases, even when the potential barrier is an $\rho_1$14-dimensional unit sphere and is non-smooth. In a $\rho_1$15 opinion-depolarization task, the controlled drift is reported to achieve lower kinetic cost $\rho_1$16 and more uniform directional similarities than DeepGSB and GSBM (Buzun et al., 23 Jul 2025).

The ablation results identify three operational dependencies. Removing the replay buffer severely degrades feasibility, with Vneck $\rho_1$17 reported to jump to $\rho_1$18. Removing gradient balancing harms feasibility and stability. Tuning $\rho_1$19 and $\rho_1$20 reveals trade-offs: higher $\rho_1$21 straightens trajectories and lowers path cost with a slight feasibility bias, while $\rho_1$22 must be tuned to avoid instability (Buzun et al., 23 Jul 2025).

The reported limitations are likewise concrete. Fourier time features can increase $\rho_1$23 magnitudes, so careful normalization is required. A single MLP must serve both as dual potential and as control, suggesting that stronger inductive biases such as manifold-aware layers or physics-informed nets may improve performance. The computation of second-order derivatives in the HJB residual adds overhead, motivating exploration of PDE-consistent losses that avoid Hessians or use stochastic estimators. The future directions listed in the paper include symplectic integrators for deterministic Hamiltonian flows, explicit Riemannian divergences and Laplace–Beltrami enforcement, adaptive replay buffers, theoretical convergence rates, and extensions to underdamped dynamics and interacting particles (Buzun et al., 23 Jul 2025).

Taken together, these results place HOTA at the intersection of dynamical OT, Hamilton–Jacobi theory, and particle-based generative transport. In the narrow sense of the 2025 solver, it is a dual HJB method that learns a single Kantorovich/value potential, advects particles without density estimation, and supports non-smooth costs through viscosity solutions. In the broader sense developed by related graph and characteristic-flow work, it denotes a Hamiltonian view of transport in which optimal advection is generated by a conjugate potential, with the continuity equation appearing either explicitly, discretely, or implicitly through characteristic dynamics.