On Diffusion Process in SE(3)-invariant Space

Sampling viable 3D structures (e.g., molecules and point clouds) with SE(3)-invariance using diffusion-based models proved promising in a variety of real-world applications, wherein SE(3)-invariant properties can be naturally characterized by the inter-point distance manifold. However, due to the non-trivial geometry, we still lack a comprehensive understanding of the diffusion mechanism within such SE(3)-invariant space. This study addresses this gap by mathematically delineating the diffusion mechanism under SE(3)-invariance, via zooming into the interaction behavior between coordinates and the inter-point distance manifold through the lens of differential geometry. Upon this analysis, we propose accurate and projection-free diffusion SDE and ODE accordingly. Such formulations enable enhancing the performance and the speed of generation pathways; meanwhile offering valuable insights into other systems incorporating SE(3)-invariance.

Carbon Emission Prediction and Clean Industry Transformation Based on Machine Learning: A Case Study of Sichuan Province

This study preprocessed 2000-2019 energy consumption data for 46 key Sichuan industries using matrix normalization. DBSCAN clustering identified 16 feature classes to objectively group industries. Penalized regression models were then applied for their advantages in overfitting control, high-dimensional data processing, and feature selection - well-suited for the complex energy data. Results showed the second cluster around coal had highest emissions due to production needs. Emissions from gasoline-focused and coke-focused clusters were also significant. Based on this, emission reduction suggestions included clean coal technologies, transportation management, coal-electricity replacement in steel, and industry standardization. The research introduced unsupervised learning to objectively select factors and aimed to explore new emission reduction avenues. In summary, the study identified industry groupings, assessed emissions drivers, and proposed scientific reduction strategies to better inform decision-making using algorithms like DBSCAN and penalized regression models.