# Diffusing the Optimal Topology: A Generative Optimization Approach

arXiv: https://arxiv.org/abs/2303.09760 · HTML: https://mlsanity.com/paper/2303.09760/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 54 references that are in the map.
- First posted on arXiv 17 Mar 2023.

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

- [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)
- [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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (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)
- [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022)
- [ILVR: Conditioning Method for Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2108.02938/) (2021)
- [Crystal Diffusion Variational Autoencoder for Periodic Material Generation](https://mlsanity.com/paper/2110.06197/) (2021)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [On Fast Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.00132/) (2021)
- [BEGAN: Boundary Equilibrium Generative Adversarial Networks](https://mlsanity.com/paper/1703.10717/) (2017)
- [Deep Generative Models in Engineering Design: A Review](https://mlsanity.com/paper/2110.10863/) (2021)
- [TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain](https://mlsanity.com/paper/2003.04685/) (2020)
- [Deep neural networks for the evaluation and design of photonic devices](https://mlsanity.com/paper/2007.00084/) (2020)
- [Deep learning for determining a near-optimal topological design without any iteration](https://mlsanity.com/paper/1801.05463/) (2018)
- [Few-Shot Diffusion Models](https://mlsanity.com/paper/2205.15463/) (2022)
- [On the use of Artificial Neural Networks in Topology Optimisation](https://mlsanity.com/paper/2208.02563/) (2022)
- [3D Topology Optimization using Convolutional Neural Networks](https://mlsanity.com/paper/1808.07440/) (2018)
- [Generating 3D structures from a 2D slice with GAN-based dimensionality expansion](https://mlsanity.com/paper/2102.07708/) (2021)
- [Multi-modal Machine Learning in Engineering Design: A Review and Future Directions](https://mlsanity.com/paper/2302.10909/) (2023)
- [Real-Time Topology Optimization in 3D via Deep Transfer Learning](https://mlsanity.com/paper/2102.07657/) (2021)
- [De-homogenization using Convolutional Neural Networks](https://mlsanity.com/paper/2105.04232/) (2021)
- [A novel topology design approach using an integrated deep learning network architecture](https://mlsanity.com/paper/1808.02334/) (2018)

## Built on by (most cited first, 10 of 10)

- [Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation](https://mlsanity.com/paper/2305.18470/) (2023)
- [Drag-guided diffusion models for vehicle image generation](https://mlsanity.com/paper/2306.09935/) (2023)
- [Constraining Generative Models for Engineering Design with Negative Data](https://mlsanity.com/paper/2306.15166/) (2023)
- [On the Generalization in Topology Optimization via Sensitivity-Conditioned Bernoulli Flow Matching](https://mlsanity.com/paper/2606.02179/) (2026)
- [NITO: Neural Implicit Fields for Resolution-free Topology Optimization](https://mlsanity.com/paper/2402.05073/) (2024)
- [Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization](https://mlsanity.com/paper/2510.23667/) (2025)
- [Inverse design with conditional cascaded diffusion models](https://mlsanity.com/paper/2408.08526/) (2024)
- [Trajectory-Aware Flow Matching for Topology Optimisation](https://mlsanity.com/paper/2607.14652/) (2026)
- [Flow Battery Manifold Design with Heterogeneous Inputs Through Generative Adversarial Neural Networks](https://mlsanity.com/paper/2508.08863/) (2025)
- [How the Classical Baseline Shapes Machine-Learning Speed Claims in Topology Optimization](https://mlsanity.com/paper/2610.16051/) (2026)

## Abstract (arXiv)

Topology Optimization seeks to find the best design that satisfies a set of constraints while maximizing system performance. Traditional iterative optimization methods like SIMP can be computationally expensive and get stuck in local minima, limiting their applicability to complex or large-scale problems. Learning-based approaches have been developed to accelerate the topology optimization process, but these methods can generate designs with floating material and low performance when challenged with out-of-distribution constraint configurations. Recently, deep generative models, such as Generative Adversarial Networks and Diffusion Models, conditioned on constraints and physics fields have shown promise, but they require extensive pre-processing and surrogate models for improving performance. To address these issues, we propose a Generative Optimization method that integrates classic optimization like SIMP as a refining mechanism for the topology generated by a deep generative model. We also remove the need for conditioning on physical fields using a computationally inexpensive approximation inspired by classic ODE solutions and reduce the number of steps needed to generate a feasible and performant topology. Our method allows us to efficiently generate good topologies and explicitly guide them to regions with high manufacturability and high performance, without the need for external auxiliary models or additional labeled data. We believe that our method can lead to significant advancements in the design and optimization of structures in engineering applications, and can be applied to a broader spectrum of performance-aware engineering design problems.