Chapter 17: Score Function and Energy-Based Models. In the chapter mind map this icon labels Diffusion Preview: Time-Indexed Scores. The discussion below is excerpted and lightly edited from § Preview: Diffusion as Time-Indexed Scores in Mathematics for AI and Machine Learning.

The score function framework developed provides the foundation for understanding modern diffusion models. In the subsequent chapters, we will extend the score function concept to time-dependent settings, where noise levels $\sigma_t$ define a continuous time axis. The score function becomes $\mathbf{s}_\theta(\mathbf{x}, t)$, encoding the geometry of probability distributions at different noise scales.

This time-dependent extension enables a complete mathematical description of diffusion processes. Chapter the referenced section will show how score functions are used in Langevin dynamics for sampling. Chapter the referenced section will formalize the continuous-time limit as stochastic differential equations. Chapter the referenced section will reveal the connection between stochastic and deterministic probability flows. Finally, Chapter the referenced section will describe how probability distributions evolve through the Fokker-Planck equation.

Together, these chapters demonstrate that diffusion models are not ad hoc neural network architectures, but rather principled applications of stochastic calculus and differential geometry to probability distributions. The score function is the mathematical thread that connects all these concepts, revealing the deep geometric structure underlying modern generative modeling.