# Mean-Field Path-Integral Diffusion: From Samples to Interacting Agents

arXiv: https://arxiv.org/abs/2605.00007 · HTML: https://mlsanity.com/paper/2605.00007/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 19 of its 35 references that are in the map.
- First posted on arXiv 23 Feb 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 19 of 19)

- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Optimal transport over a linear dynamical system](https://mlsanity.com/paper/1502.01265/) (2015)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [The mean field Schr"odinger problem: ergodic behavior, entropy estimates and functional inequalities](https://mlsanity.com/paper/1905.02393/) (2019)
- [Propagation of chaos for mean field Schr"odinger problems](https://mlsanity.com/paper/2304.09340/) (2023)
- [On the Contraction Coefficient of the Schr"odinger Bridge for Stochastic Linear Systems](https://mlsanity.com/paper/2309.06622/) (2023)
- [Generative Stochastic Optimal Transport: Guided Harmonic Path-Integral Diffusion](https://mlsanity.com/paper/2512.11859/) (2025)
- [Mean Field Control for Efficient Mixing of Energy Loads](https://mlsanity.com/paper/1810.00450/) (2018)
- [Dynamic multi-agent assignment via discrete optimal transport](https://mlsanity.com/paper/1910.10748/) (2019)
- [Statistical Mechanics of Thermostatically Controlled Multi-Zone Buildings](https://mlsanity.com/paper/2208.13099/) (2022)
- [Learning to flock in open space by avoiding collisions and staying together](https://mlsanity.com/paper/2506.15587/) (2025)

## Built on by (most cited first, 2 of 2)

- [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)

Independent sample generation is the prevailing paradigm in modern diffusion-based generative models of AI. We ask a different question: can samples coordinate through shared population statistics to transport probability mass more efficiently? We introduce Mean-Field Path-Integral Diffusion (MF-PID), a framework in which samples are promoted to interacting agents whose drift depends self-consistently on the evolving population density. The coupling converts distribution matching into a McKean--Vlasov extension of the stochastic optimal transport problem, unifying generative modeling and multi-agent control under the same Hamilton--Jacobi--Bellman/Kolmogorov--Fokker--Planck duality. We identify two analytically tractable regimes: a Linear--Quadratic--Gaussian (LQG) benchmark in which the infinite-dimensional mean-field system reduces to a finite set of Riccati and linear ODEs, and a Gaussian-mixture regime governed by a piecewise-constant protocol that preserves closed-form solvability. For a quadratic interaction potential with schedule βt and zero base drift we prove that the self-consistent MF guidance is the exact linear interpolant between initial and target global means -- a result that holds for arbitrary initial and target densities and any βt. Applied to demand-response control of energy systems, where agents aggregated into an ensemble are energy consumers (e.g. thermal zones within a building), MF-PID achieves 19--24% reductions in cumulative control energy over independent-agent baselines while matching the prescribed terminal distribution exactly, and reveals how coordination redistributes actuation effort across heterogeneous sub-populations.