This talk presents a unifying applied mathematics/theoretical physics framework that bridges core components of modern generative AI -- diffusion models, reinforcement learning, and transformers -- through the lens of contemporary applied mathematics. Central to this framework are the concepts of Decision Flows and Path Integral Diffusions, which offer structured approaches to sequential sampling over discrete, continuous, and hybrid spaces. These approaches are rooted in Green-function-based control, Schrödinger bridges, and non-equilibrium statistical physics.

Building on recent work, we explore analytically tractable and algorithmically efficient regimes -- often requiring minimal use of neural networks -- where sampling from complex distributions becomes both explainable and extrapolative. We highlight connections between score-based diffusion, linearly-solvable Markov Decision Processes, and energy-based models, including emerging insights into phase transitions in generative AI (e.g., memorization and speciation dynamics).

Applications span inference/sampling in Ising models, CIFAR-10 image generation, physics-informed reinforcement learning in turbulent flows, and auto-regressive modeling of statistical hydrodynamics. We also touch on decision-making under uncertainty in energy systems.