# Denoising Diffusion-Based Control of Nonlinear Systems

arXiv: https://arxiv.org/abs/2402.02297 · HTML: https://mlsanity.com/paper/2402.02297/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 3 Feb 2024.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## 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)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [3D Shape Generation and Completion through Point-Voxel Diffusion](https://mlsanity.com/paper/2104.03670/) (2021)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [Diffusion Model is an Effective Planner and Data Synthesizer for Multi-Task Reinforcement Learning](https://mlsanity.com/paper/2305.18459/) (2023)
- [Metropolis-Hastings Generative Adversarial Networks](https://mlsanity.com/paper/1811.11357/) (2018)
- [Generating Behaviorally Diverse Policies with Latent Diffusion Models](https://mlsanity.com/paper/2305.18738/) (2023)
- [Optimal transport over a linear dynamical system](https://mlsanity.com/paper/1502.01265/) (2015)
- [Technical Report: Convex Optimization of Nonlinear Feedback Controllers via Occupation Measures](https://mlsanity.com/paper/1305.7484/) (2013)
- [Bilinear Controllability of a Class of Advection-Diffusion-Reaction Systems](https://mlsanity.com/paper/1711.01696/) (2017)
- [Controlled density transport using Perron Frobenius generators](https://mlsanity.com/paper/2304.13829/) (2023)
- [Density Stabilization Strategies for Nonholonomic Agents on Compact Manifolds](https://mlsanity.com/paper/2308.15755/) (2023)

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

- [Dynamics-aware Diffusion Models for Planning and Control](https://mlsanity.com/paper/2504.00236/) (2025)
- [Score Matching Diffusion Based Feedback Control and Planning of Nonlinear Systems](https://mlsanity.com/paper/2504.09836/) (2025)

## Abstract (arXiv)

We propose a novel approach based on Denoising Diffusion Probabilistic Models (DDPMs) to control nonlinear dynamical systems. DDPMs are the state-of-art of generative models that have achieved success in a wide variety of sampling tasks. In our framework, we pose the feedback control problem as a generative task of drawing samples from a target set under control system constraints. The forward process of DDPMs constructs trajectories originating from a target set by adding noise. We learn to control a dynamical system in reverse such that the terminal state belongs to the target set. For control-affine systems without drift, we prove that the control system can exactly track the trajectory of the forward process in reverse, whenever the the Lie bracket based condition for controllability holds. We numerically study our approach on various nonlinear systems and verify our theoretical results. We also conduct numerical experiments for cases beyond our theoretical results on a physics-engine.