# Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation

arXiv: https://arxiv.org/abs/2305.18470 · HTML: https://mlsanity.com/paper/2305.18470/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 56 of its 114 references that are in the map.
- First posted on arXiv 29 May 2023.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 56)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [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)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Label-Efficient Semantic Segmentation with Diffusion Models](https://mlsanity.com/paper/2112.03126/) (2021)
- [Diffusion Autoencoders: Toward a Meaningful and Decodable Representation](https://mlsanity.com/paper/2111.15640/) (2021)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [Ladder Variational Autoencoders](https://mlsanity.com/paper/1602.02282/) (2016)
- [Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Images](https://mlsanity.com/paper/2011.10650/) (2020)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning](https://mlsanity.com/paper/1705.07761/) (2017)
- [On Fast Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.00132/) (2021)
- [Learning to Efficiently Sample from Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.03802/) (2021)

## Built on by (most cited first, 30 of 52)

- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Enhancing Deep Reinforcement Learning: A Tutorial on Generative Diffusion Models in Network Optimization](https://mlsanity.com/paper/2308.05384/) (2023)
- [From Concept to Manufacturing: Evaluating Vision-Language Models for Engineering Design](https://mlsanity.com/paper/2311.12668/) (2023)
- [Diffusion Predictive Control with Constraints](https://mlsanity.com/paper/2412.09342/) (2024)
- [Training Free Guided Flow Matching with Optimal Control](https://mlsanity.com/paper/2410.18070/) (2024)
- [CGD: Constraint-Guided Diffusion Policies for UAV Trajectory Planning](https://mlsanity.com/paper/2405.01758/) (2024)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Training-Free Constrained Generation With Stable Diffusion Models](https://mlsanity.com/paper/2502.05625/) (2025)
- [Adding Conditional Control to Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2406.12120/) (2024)
- [DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories](https://mlsanity.com/paper/2502.15043/) (2025)
- [Constraining Generative Models for Engineering Design with Negative Data](https://mlsanity.com/paper/2306.15166/) (2023)
- [ShipGen: A Diffusion Model for Parametric Ship Hull Generation with Multiple Objectives and Constraints](https://mlsanity.com/paper/2311.06315/) (2023)
- [PRESTO: Fast Motion Planning Using Diffusion Models Based on Key-Configuration Environment Representation](https://mlsanity.com/paper/2409.16012/) (2024)
- [Constraint-Aware Diffusion Guidance for Robotics: Real-Time Obstacle Avoidance for Autonomous Racing](https://mlsanity.com/paper/2505.13131/) (2025)
- [Neuro-Symbolic Generative Diffusion Models for Physically Grounded, Robust, and Safe Generation](https://mlsanity.com/paper/2506.01121/) (2025)
- [Composition and Alignment of Diffusion Models using Constrained Learning](https://mlsanity.com/paper/2508.19104/) (2025)
- [A conditional latent autoregressive recurrent model for generation and forecasting of beam dynamics in particle accelerators](https://mlsanity.com/paper/2403.13858/) (2024)
- [State-observation augmented diffusion model for nonlinear assimilation with unknown dynamics](https://mlsanity.com/paper/2407.21314/) (2024)
- [Fast Direct: Query-Efficient Online Black-box Guidance for Diffusion-model Target Generation](https://mlsanity.com/paper/2502.01692/) (2025)
- [Agentic Large Language Models for Conceptual Systems Engineering and Design](https://mlsanity.com/paper/2507.08619/) (2025)
- [PhysGen: Physically Grounded 3D Shape Generation for Industrial Design](https://mlsanity.com/paper/2512.00422/) (2025)
- [UniConFlow: A Unified Constrained Flow-Matching Framework for Certified Motion Planning](https://mlsanity.com/paper/2506.02955/) (2025)
- [C-ShipGen: A Conditional Guided Diffusion Model for Parametric Ship Hull Design](https://mlsanity.com/paper/2407.03333/) (2024)
- [NITO: Neural Implicit Fields for Resolution-free Topology Optimization](https://mlsanity.com/paper/2402.05073/) (2024)
- [Pick-and-place Manipulation Across Grippers Without Retraining: A Learning-optimization Diffusion Policy Approach](https://mlsanity.com/paper/2502.15613/) (2025)
- [On Efficiency-Effectiveness Trade-off of Diffusion-based Recommenders](https://mlsanity.com/paper/2510.17245/) (2025)
- [Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization](https://mlsanity.com/paper/2510.23667/) (2025)
- [SAD-Flower: Flow Matching for Safe, Admissible, and Dynamically Consistent Planning](https://mlsanity.com/paper/2511.05355/) (2025)
- [GIFT: Bootstrapping Image-to-CAD Program Synthesis via Geometric Feedback](https://mlsanity.com/paper/2603.27448/) (2026)

## Abstract (arXiv)

Generative models have had a profound impact on vision and language, paving the way for a new era of multimodal generative applications. While these successes have inspired researchers to explore using generative models in science and engineering to accelerate the design process and reduce the reliance on iterative optimization, challenges remain. Specifically, engineering optimization methods based on physics still outperform generative models when dealing with constrained environments where data is scarce and precision is paramount. To address these challenges, we introduce Diffusion Optimization Models (DOM) and Trajectory Alignment (TA), a learning framework that demonstrates the efficacy of aligning the sampling trajectory of diffusion models with the optimization trajectory derived from traditional physics-based methods. This alignment ensures that the sampling process remains grounded in the underlying physical principles. Our method allows for generating feasible and high-performance designs in as few as two steps without the need for expensive preprocessing, external surrogate models, or additional labeled data. We apply our framework to structural topology optimization, a fundamental problem in mechanical design, evaluating its performance on in- and out-of-distribution configurations. Our results demonstrate that TA outperforms state-of-the-art deep generative models on in-distribution configurations and halves the inference computational cost. When coupled with a few steps of optimization, it also improves manufacturability for out-of-distribution conditions. By significantly improving performance and inference efficiency, DOM enables us to generate high-quality designs in just a few steps and guide them toward regions of high performance and manufacturability, paving the way for the widespread application of generative models in large-scale data-driven design.