Uncertainty is inherent in data generation processes, whether arising from stochastic dynamics, limited samples, or complex multi-scale interactions. Understanding how structured patterns emerge from such uncertainties is a central challenge in generative modeling. This report explores this question through the lens of dynamics, geometry, and topology, with a particular focus on early warning prediction. We investigate the mechanisms underlying critical transitions in generative models, including mode collapse and vector field splitting, which manifest as topological changes across scales. Building upon the Onsager–Machlup action functional and Schrödinger bridge theory, we introduce entropy-based indicators defined in the space of probability measures to assess and anticipate such transitions.  In parallel, we examine how geometric properties of latent spaces can be exploited to improve few-shot generation, where data scarcity amplifies uncertainty. By imposing geometric constraints on latent flows, we achieve more stable training and better mode coverage. Together, these perspectives—dynamical, geometrical, and topological—offer a unified framework for understanding how certainties emerge from uncertainties, and suggest new directions for building robust and interpretable generative models.

高婷, 华中科技大学数学与统计学院、数学中心副教授。2015年毕业于伊利诺伊理工大学, 获博士学位。研究方向: 非高斯随机动力系统与深度学习交叉及在脑科学、信息通信与金融中的应用。在SIAM、NSR等期刊发表SCI论文40余篇。主持和作为项目骨干参与多项研究项目。