# HPG-Diff: Hierarchical physics-guided diffusion with differentiable connectivity constraints for topology optimization

arXiv: https://arxiv.org/abs/2607.07233 · HTML: https://mlsanity.com/paper/2607.07233/

## Key facts

- Cited by 1 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 24 of its 70 references that are in the map.
- First posted on arXiv 8 Jul 2026.

## 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, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [T2I-Adapter: Learning Adapters to Dig out More Controllable Ability for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.08453/) (2023)
- [GLIGEN: Open-Set Grounded Text-to-Image Generation](https://mlsanity.com/paper/2301.07093/) (2023)
- [clDice -- A Novel Topology-Preserving Loss Function for Tubular Structure Segmentation](https://mlsanity.com/paper/2003.07311/) (2020)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Sketch-Guided Text-to-Image Diffusion Models](https://mlsanity.com/paper/2211.13752/) (2022)
- [Deep Generative Models in Engineering Design: A Review](https://mlsanity.com/paper/2110.10863/) (2021)
- [DiLightNet: Fine-grained Lighting Control for Diffusion-based Image Generation](https://mlsanity.com/paper/2402.11929/) (2024)
- [Beyond the Pixel-Wise Loss for Topology-Aware Delineation](https://mlsanity.com/paper/1712.02190/) (2017)
- [Diffusion Models Beat GANs on Topology Optimization](https://mlsanity.com/paper/2208.09591/) (2022)
- [TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domain](https://mlsanity.com/paper/2003.04685/) (2020)
- [Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation](https://mlsanity.com/paper/2305.18470/) (2023)
- [Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy](https://mlsanity.com/paper/2012.15744/) (2020)
- [Self-Directed Online Machine Learning for Topology Optimization](https://mlsanity.com/paper/2002.01927/) (2020)
- [Pi-fusion: Physics-informed diffusion model for learning fluid dynamics](https://mlsanity.com/paper/2406.03711/) (2024)
- [DiffFluid: Plain Diffusion Models are Effective Predictors of Flow Dynamics](https://mlsanity.com/paper/2409.13665/) (2024)
- [Optimize Any Topology: A Foundation Model for Shape- and Resolution-Free Structural Topology Optimization](https://mlsanity.com/paper/2510.23667/) (2025)
- [Topology Optimization with linearized buckling criteria in 250 lines of Matlab](https://mlsanity.com/paper/2101.02973/) (2021)
- [Generative method for aerodynamic optimization based on classifier-free guided denoising diffusion probabilistic model](https://mlsanity.com/paper/2503.07056/) (2025)
- [Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws](https://mlsanity.com/paper/2603.12365/) (2026)

## Built on by (most cited first, 1 of 1)

- [TopoMamba: A Load-Support Relation-Guided Multi-Directional State-Space Model for Topology Optimization](https://mlsanity.com/paper/2609.33688/) (2026)

## Abstract (arXiv)

Deep generative models offer a promising paradigm for topology optimization, enabling rapid design exploration. However, these approaches lack intrinsic physics guidance, often leading to poor generalizability across unseen boundary conditions and the formation of floating material artifacts. To address these limitations, we propose Hierarchical Physics-Guided Diffusion (HPG-Diff), a novel diffusion framework that enforces physics consistency through two synergistic mechanisms. First, we introduce a hierarchical physics-guided strategy that aligns different precomputed physics features with the denoising process, guiding material distribution toward optimal load paths to enhance generalizability. Second, we propose a floating material suppression loss as a differentiable connectivity constraint inspired by thermal conduction to improve topological connectivity. By simulating a virtual heat propagation process from load positions, this mechanism explicitly penalizes floating material during training. Quantitative evaluations demonstrate that HPG-Diff achieves average compliance errors of 0.87% (in-distribution) and 5.29% (out-of-distribution), while reducing floating material ratios to 2.90% and 2.44%, respectively. Furthermore, case studies on a 3:1 rectangular domain, including cantilever and bridge benchmarks, provide preliminary evidence that lightweight LoRA fine-tuning with a small dataset can support the adaptation of HPG-Diff to rectangular non-square domains.