HOTA: Hamiltonian framework for Optimal Transport Advection: what it means for business leaders

HOTA finds optimal probability paths without density estimation, delivering faster, geometry-aware transport plans that respect obstacles and non-smooth costs—helping data-science teams deploy scalable generative, robotic, or logistics models with confidence.

1. What the method is

HOTA casts dynamic optimal transport as a Hamiltonian control problem. Two neural Kantorovich potentials generate a drift field whose trajectories move stochastic particles from source to target distributions while minimising kinetic energy and arbitrary state costs. The approach is density-free, handles curved manifolds and obstacles, and slots into existing flow-matching or diffusion pipelines whenever path fidelity matters.

2. Why the method was developed

Classical OT ignores path geometry; bridge models assume Gaussian marginals or run slow Sinkhorn loops. Teams modelling robot fleets, supply chains or latent generative flows thus juggle fidelity versus runtime. HOTA resolves that tension with a stable, high-dimensional solver that needs no density estimation and tolerates non-smooth costs, delivering controllable, energy-efficient transports at production speed.

3. Who should care

• Autonomy leads planning obstacle-rich robot or drone paths
• Generative-AI engineers seeking geometry-aware latent flows
• Logistics analysts optimising mass movement through networks
• Quant researchers modelling state-cost finance scenarios

4. How the method works

Paired boundary samples seed Brownian bridges parameterised by a neural drift. A Hamiltonian loss—kinetic energy + state potential − Kantorovich constraint—is minimised via stochastic gradient descent. Variance-reduction schedules keep gradients stable, and automatic differentiation updates both drift and potential networks. No closed-form conditionals or matrix scaling are required, enabling GPU-friendly training in tens of dimensions.

5. How it was evaluated

Benchmarks covered 2-D obstacle navigation, ImageNet-64 latent transport and 128-D opinion dynamics. Baselines were Flow-Matching, DeepGSB and Sinkhorn-based solvers. Key metrics: marginal KL, energy cost, FID for images, convergence steps and GPU minutes. Ablations removed Hamiltonian coupling or variance control to gauge their impact.

6. How it performed

HOTA matched marginals within 0.1 % KL, cut energy up to 35 % versus Flow-Matching, and trained 42 % faster while improving ImageNet FID by 3 points. In 128-D trials where DeepGSB diverged, HOTA remained stable. Both coupling and variance control proved essential—dropping either halved the gains. (Source: arXiv 2507.17513, 2025)