# Generative Stochastic Optimal Transport: Guided Harmonic Path-Integral Diffusion

arXiv: https://arxiv.org/abs/2512.11859 · HTML: https://mlsanity.com/paper/2512.11859/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 5 Dec 2025.

## Builds on (most cited first, 9 of 9)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023)
- [Adaptive importance sampling for control and inference](https://mlsanity.com/paper/1505.01874/) (2015)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [Stochastic Optimal Control as Non-equilibrium Statistical Mechanics: Calculus of Variations over Density and Current](https://mlsanity.com/paper/1306.6572/) (2013)
- [Space-Time Diffusion Bridge](https://mlsanity.com/paper/2402.08847/) (2024)
- [Physics-Guided Actor-Critic Reinforcement Learning for Swimming in Turbulence](https://mlsanity.com/paper/2406.10242/) (2024)

## Built on by (most cited first, 3 of 3)

- [Mean-Field Path-Integral Diffusion: From Samples to Interacting Agents](https://mlsanity.com/paper/2605.00007/) (2026)
- [Analytic Bridge Diffusions for Controlled Path Generation](https://mlsanity.com/paper/2605.02961/) (2026)
- [Temporal Memory for Resource-Constrained Agents: Continual Learning via Stochastic Compress-Add-Smooth](https://mlsanity.com/paper/2604.00067/) (2026)

## Abstract (arXiv)

We introduce Guided Harmonic Path-Integral Diffusion (GH-PID), a linearly-solvable framework for guided Stochastic Optimal Transport (SOT) with a hard terminal distribution and soft, application-driven path costs. A low-dimensional guidance protocol shapes the trajectory ensemble while preserving analytic structure: the forward and backward Kolmogorov equations remain linear, the optimal score admits an explicit Green-function ratio, and Gaussian-Mixture Model (GMM) terminal laws yield closed-form expressions. This enables stable sampling and differentiable protocol learning under exact terminal matching. We develop guidance-centric diagnostics -- path cost, centerline adherence, variance flow, and drift effort -- that make GH-PID an interpretable variational ansatz for empirical SOT. Three navigation scenarios illustrated in 2D: (i) Case A: hand-crafted protocols revealing how geometry and stiffness shape lag, curvature effects, and mode evolution; (ii) Case B: single-task protocol learning, where a PWC centerline is optimized to minimize integrated cost; (iii) Case C: multi-expert fusion, in which a commander reconciles competing expert/teacher trajectories and terminal beliefs through an exact product-of-experts law and learns a consensus protocol. Across all settings, GH-PID generates geometry-aware, trust-aware trajectories that satisfy the prescribed terminal distribution while systematically reducing integrated cost.