# Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems

arXiv: https://arxiv.org/abs/2504.09836 · HTML: https://mlsanity.com/paper/2504.09836/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 14 Apr 2025.

## Builds on (most cited first, 17 of 17)

- [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)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [Metropolis-Hastings Generative Adversarial Networks](https://mlsanity.com/paper/1811.11357/) (2018)
- [Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems](https://mlsanity.com/paper/1909.12511/) (2019)
- [Mean-Field Optimal Control](https://mlsanity.com/paper/1306.5913/) (2013)
- [Optimal Covariance Control for Stochastic Systems Under Chance Constraints](https://mlsanity.com/paper/1804.02829/) (2018)
- [Benamou-Brenier Formulation of Optimal Transport for Nonlinear Control Systems on Rd](https://mlsanity.com/paper/2407.16088/) (2024)
- [Continuification-based control of large multiagent systems in a ring](https://mlsanity.com/paper/2209.15060/) (2022)
- [Score matching for sub-Riemannian bridge sampling](https://mlsanity.com/paper/2404.15258/) (2024)
- [Dynamics-aware Diffusion Models for Planning and Control](https://mlsanity.com/paper/2504.00236/) (2025)
- [Density Stabilization Strategies for Nonholonomic Agents on Compact Manifolds](https://mlsanity.com/paper/2308.15755/) (2023)
- [Denoising Diffusion-Based Control of Nonlinear Systems](https://mlsanity.com/paper/2402.02297/) (2024)
- [A Time-Reversal Control Synthesis for Steering the State of Stochastic Systems](https://mlsanity.com/paper/2504.00238/) (2025)

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

- [Learning Sampled-data Control for Swarms via MeanFlow](https://mlsanity.com/paper/2603.20189/) (2026)
- [Notes on Generative Modeling for Feedback Control and Planning](https://mlsanity.com/paper/2609.33164/) (2026)

## Abstract (arXiv)

In this paper, we propose a deterministic diffusion-based framework for controlling the probability density of nonlinear control-affine systems, with theoretical guarantees for drift-free and linear time-invariant (LTI) dynamics. The central idea is to first excite the system with white noise so that a forward diffusion process explores the reachable regions of state space, and then to design a deterministic feedback law that acts as a denoising mechanism driving the system back toward a desired target distribution supported on the target set. This denoising phase provides a feedback controller that steers the control system to the target set. In this framework, control synthesis reduces to constructing a deterministic reverse process that reproduces the desired evolution of state densities. We derive existence conditions ensuring such deterministic realizations of time-reversals for controllable drift-free and LTI systems, and show that the resulting feedback laws provide a tractable alternative to nonlinear control by viewing density control as a relaxation of controlling a system to target sets. Numerical studies on a unicycle model with obstacles, a five-dimensional driftless system, and a four-dimensional LTI system demonstrate reliable diffusion-inspired density control.