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

Traditional topology optimization (TO) methods, such as the Solid Isotropic Material with Penalization (SIMP), are computationally expensive because they require iterative Finite Element Analysis (FEA). While deep generative models offer a faster alternative for design exploration, they often struggle with physics consistency, leading to designs with poor mechanical performance, anomalous stress concentrations, or disconnected "floating" material artifacts. This paper asks: can a diffusion-based generative model be designed to enforce physical principles and connectivity constraints while maintaining the speed and diversity of generative design?

To address these limitations, the authors introduce HPG-Diff, a hierarchical physics-guided diffusion framework. The model employs two primary mechanisms:

1. Hierarchical Physics Guidance: Instead of treating physics data as a global input, the model injects specific physics features—Displacement (U) at shallow layers, Principal Stress Lines (PSL) at mid-layers, and Strain Energy Density (SED) at deep layers—using cross-attention modules. This aligns the model's receptive fields with the appropriate level of mechanical abstraction.
2. Floating Material Suppression (FMS) Loss: Inspired by thermal conduction, this loss simulates a virtual heat propagation process starting from load positions. By penalizing regions where "heat" cannot reach, the model explicitly discourages the formation of disconnected material during training.

Findings and Significance

HPG-Diff demonstrates significant improvements in both mechanical accuracy and manufacturability. Quantitative evaluations show that the model achieves low compliance errors (0.87% in-distribution and 5.29% out-of-distribution) while drastically reducing the ratio of floating material compared to standard generative approaches. Furthermore, the authors demonstrate that the model can be adapted to non-square design domains using lightweight LoRA fine-tuning, making it a versatile tool for practical engineering scenarios where rapid, diverse, and physically sound design generation is required.