# Aligning Diffusion Model with Problem Constraints for Trajectory Optimization

arXiv: https://arxiv.org/abs/2504.00342 · HTML: https://mlsanity.com/paper/2504.00342/

## Key facts

- Cited by 7 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 32 of its 44 references that are in the map.
- First posted on arXiv 1 Apr 2025.

## 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 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Is Conditional Generative Modeling all you need for Decision-Making?](https://mlsanity.com/paper/2211.15657/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [DIFUSCO: Graph-based Diffusion Solvers for Combinatorial Optimization](https://mlsanity.com/paper/2302.08224/) (2023)
- [Design-Bench: Benchmarks for Data-Driven Offline Model-Based Optimization](https://mlsanity.com/paper/2202.08450/) (2022)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Diffusion Models Beat GANs on Topology Optimization](https://mlsanity.com/paper/2208.09591/) (2022)
- [Diffusion Models for Black-Box Optimization](https://mlsanity.com/paper/2306.07180/) (2023)
- [Compositional Diffusion-Based Continuous Constraint Solvers](https://mlsanity.com/paper/2309.00966/) (2023)
- [Learning to Warm-Start Fixed-Point Optimization Algorithms](https://mlsanity.com/paper/2309.07835/) (2023)
- [The Voice of Optimization](https://mlsanity.com/paper/1812.09991/) (2018)
- [DiffDock-PP: Rigid Protein-Protein Docking with Diffusion Models](https://mlsanity.com/paper/2304.03889/) (2023)
- [DiffuSolve: Diffusion-based Solver for Non-convex Trajectory Optimization](https://mlsanity.com/paper/2403.05571/) (2024)
- [Safe and Near-Optimal Policy Learning for Model Predictive Control using Primal-Dual Neural Networks](https://mlsanity.com/paper/1906.08257/) (2019)
- [Trajectory Generation, Control, and Safety with Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.15512/) (2023)
- [Amortized Global Search for Efficient Preliminary Trajectory Design with Deep Generative Models](https://mlsanity.com/paper/2308.03960/) (2023)
- [Predictive Planner for Autonomous Driving with Consistency Models](https://mlsanity.com/paper/2502.08033/) (2025)
- [Predicting Long-Term Human Behaviors in Discrete Representations via Physics-Guided Diffusion](https://mlsanity.com/paper/2405.19528/) (2024)

## Built on by (most cited first, 7 of 7)

- [Unified Generation-Refinement Planning: Bridging Guided Flow Matching and Sampling-Based MPC for Social Navigation](https://mlsanity.com/paper/2508.01192/) (2025)
- [Human Centric General Physical Intelligence for Agile Manufacturing Automation](https://mlsanity.com/paper/2508.11960/) (2025)
- [Self-supervised diffusion model fine-tuning for costate initialization using Markov chain Monte Carlo](https://mlsanity.com/paper/2510.02527/) (2025)
- [GLENS: Global Search via Learning from Solver Iterates with Diffusion Models](https://mlsanity.com/paper/2606.00366/) (2026)
- [Complexity Bounds and Approaches to Learning Projected Gradient Descent Solver Iterates](https://mlsanity.com/paper/2607.22467/) (2026)
- [Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion](https://mlsanity.com/paper/2608.29507/) (2026)
- [Spacecraft Rendezvous Trajectory Generation with Modular Constraints via Diffusion Model Composition](https://mlsanity.com/paper/2610.05642/) (2026)

## Abstract (arXiv)

Diffusion models have recently emerged as effective generative frameworks for trajectory optimization, capable of producing high-quality and diverse solutions. However, training these models in a purely data-driven manner without explicit incorporation of constraint information often leads to violations of critical constraints, such as goal-reaching, collision avoidance, and adherence to system dynamics. To address this limitation, we propose a novel approach that aligns diffusion models explicitly with problem-specific constraints, drawing insights from the Dynamic Data-driven Application Systems (DDDAS) framework. Our approach introduces a hybrid loss function that explicitly measures and penalizes constraint violations during training. Furthermore, by statistically analyzing how constraint violations evolve throughout the diffusion steps, we develop a re-weighting strategy that aligns predicted violations to ground truth statistics at each diffusion step. Evaluated on a tabletop manipulation and a two-car reach-avoid problem, our constraint-aligned diffusion model significantly reduces constraint violations compared to traditional diffusion models, while maintaining the quality of trajectory solutions. This approach is well-suited for integration into the DDDAS framework for efficient online trajectory adaptation as new environmental data becomes available.