# Model-Based Diffusion for Trajectory Optimization

arXiv: https://arxiv.org/abs/2407.01573 · HTML: https://mlsanity.com/paper/2407.01573/

## Key facts

- Cited by 32 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 39 of its 68 references that are in the map.
- First posted on arXiv 28 May 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, 30 of 39)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (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)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Human Motion Diffusion Model](https://mlsanity.com/paper/2209.14916/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Is Conditional Generative Modeling all you need for Decision-Making?](https://mlsanity.com/paper/2211.15657/) (2022)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulation](https://mlsanity.com/paper/2106.13281/) (2021)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Scalable Global Optimization via Local Bayesian Optimization](https://mlsanity.com/paper/1910.01739/) (2019)
- [Goal-Conditioned Imitation Learning using Score-based Diffusion Policies](https://mlsanity.com/paper/2304.02532/) (2023)
- [MotionDiffuser: Controllable Multi-Agent Motion Prediction using Diffusion](https://mlsanity.com/paper/2306.03083/) (2023)
- [Motion Planning Diffusion: Learning and Planning of Robot Motions with Diffusion Models](https://mlsanity.com/paper/2308.01557/) (2023)
- [AdaptDiffuser: Diffusion Models as Adaptive Self-evolving Planners](https://mlsanity.com/paper/2302.01877/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Learning Human-to-Humanoid Real-Time Whole-Body Teleoperation](https://mlsanity.com/paper/2403.04436/) (2024)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Information Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving](https://mlsanity.com/paper/1707.02342/) (2017)
- [Learning Search Space Partition for Black-box Optimization using Monte Carlo Tree Search](https://mlsanity.com/paper/2007.00708/) (2020)
- [Parallel and Distributed Thompson Sampling for Large-scale Accelerated Exploration of Chemical Space](https://mlsanity.com/paper/1706.01825/) (2017)
- [SafeDiffuser: Safe Planning with Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.00148/) (2023)

## Built on by (most cited first, 30 of 32)

- [Diffusion Model Predictive Control](https://mlsanity.com/paper/2410.05364/) (2024)
- [Maximum Entropy Reinforcement Learning with Diffusion Policy](https://mlsanity.com/paper/2502.11612/) (2025)
- [SPIDER: Scalable Physics-Informed Dexterous Retargeting](https://mlsanity.com/paper/2511.09484/) (2025)
- [Reverse Flow Matching: A Unified Framework for Online Reinforcement Learning with Diffusion and Flow Policies](https://mlsanity.com/paper/2601.08136/) (2026)
- [Constrained Diffusers for Safe Planning and Control](https://mlsanity.com/paper/2506.12544/) (2025)
- [Guided Flow Policy: Learning from High-Value Actions in Offline Reinforcement Learning](https://mlsanity.com/paper/2512.03973/) (2025)
- [Diffusion-Sharpening: Fine-tuning Diffusion Models with Denoising Trajectory Sharpening](https://mlsanity.com/paper/2502.12146/) (2025)
- [Ego-Vision World Model for Humanoid Contact Planning](https://mlsanity.com/paper/2510.11682/) (2025)
- [Equality Constrained Diffusion for Direct Trajectory Optimization](https://mlsanity.com/paper/2410.01939/) (2024)
- [Generative Predictive Control: Flow Matching Policies for Dynamic and Difficult-to-Demonstrate Tasks](https://mlsanity.com/paper/2502.13406/) (2025)
- [Generative Models From and For Sampling-Based MPC: A Bootstrapped Approach For Adaptive Contact-Rich Manipulation](https://mlsanity.com/paper/2510.14643/) (2025)
- [Conformal Prediction in The Loop: A Feedback-Based Uncertainty Model for Trajectory Optimization](https://mlsanity.com/paper/2510.16376/) (2025)
- [InterReal: A Unified Physics-Based Imitation Framework for Learning Human-Object Interaction Skills](https://mlsanity.com/paper/2603.07516/) (2026)
- [Can a Robot Walk the Robotic Dog: Triple-Zero Collaborative Navigation for Heterogeneous Multi-Agent Systems](https://mlsanity.com/paper/2603.21723/) (2026)
- [FlowRL: A Taxonomy and Modular Framework for Reinforcement Learning with Diffusion Policies](https://mlsanity.com/paper/2603.27450/) (2026)
- [Planning with Language and Generative Models: Toward General Reward-Guided Wireless Network Design](https://mlsanity.com/paper/2602.00357/) (2026)
- [On the Learnability of Offline Model-Based Optimization: A Ranking Perspective](https://mlsanity.com/paper/2603.04000/) (2026)
- [Diffusion-based learning framework for Constrained Nonconvex Optimization with Weighted Bootstrapped Refinement](https://mlsanity.com/paper/2502.10330/) (2025)
- [Diffusing Trajectory Optimization Problems for Recovery During Multi-Finger Manipulation](https://mlsanity.com/paper/2510.07030/) (2025)
- [DualShield: Safe Model Predictive Diffusion via Reachability Analysis for Interactive Autonomous Driving](https://mlsanity.com/paper/2601.15729/) (2026)
- [Motion Planning with Model-Based Diffusion via Constraint Optimization and Adaptive Scheduling](https://mlsanity.com/paper/2607.14455/) (2026)
- [D4orm: Multi-Robot Trajectories with Dynamics-aware Diffusion Denoised Deformations](https://mlsanity.com/paper/2503.12204/) (2025)
- [PegasusFlow: Parallel Rolling-Denoising Score Sampling for Robot Diffusion Planner Flow Matching](https://mlsanity.com/paper/2509.08435/) (2025)
- [PAD-TRO: Projection-Augmented Diffusion for Direct Trajectory Optimization](https://mlsanity.com/paper/2510.04436/) (2025)
- [Adaptive Linear Path Model-Based Diffusion](https://mlsanity.com/paper/2602.02831/) (2026)
- [STRIDE: Structured Lagrangian and Stochastic Residual Dynamics via Flow Matching](https://mlsanity.com/paper/2603.08478/) (2026)
- [Tempered Sequential Monte Carlo for Trajectory and Policy Optimization with Differentiable Dynamics](https://mlsanity.com/paper/2604.21456/) (2026)
- [Muninn: Your Trajectory Diffusion Model But Faster](https://mlsanity.com/paper/2605.09999/) (2026)
- [Global Convergence of Sampling-Based Nonconvex Optimization through Diffusion-Style Smoothing](https://mlsanity.com/paper/2605.16520/) (2026)
- [Dynamic Neural Koopman Distillation for Real-Time Robot Control Using Diffusion Models](https://mlsanity.com/paper/2605.24924/) (2026)

## Abstract (arXiv)

Recent advances in diffusion models have demonstrated their strong capabilities in generating high-fidelity samples from complex distributions through an iterative refinement process. Despite the empirical success of diffusion models in motion planning and control, the model-free nature of these methods does not leverage readily available model information and limits their generalization to new scenarios beyond the training data (e.g., new robots with different dynamics). In this work, we introduce Model-Based Diffusion (MBD), an optimization approach using the diffusion process to solve trajectory optimization (TO) problems without data. The key idea is to explicitly compute the score function by leveraging the model information in TO problems, which is why we refer to our approach as model-based diffusion. Moreover, although MBD does not require external data, it can be naturally integrated with data of diverse qualities to steer the diffusion process. We also reveal that MBD has interesting connections to sampling-based optimization. Empirical evaluations show that MBD outperforms state-of-the-art reinforcement learning and sampling-based TO methods in challenging contact-rich tasks. Additionally, MBD's ability to integrate with data enhances its versatility and practical applicability, even with imperfect and infeasible data (e.g., partial-state demonstrations for high-dimensional humanoids), beyond the scope of standard diffusion models.