Abstract: Modern machine learning increasingly relies on high-dimensional stochastic dynamics, many of which can be naturally interpreted through the lens of partial differential equations and diffusion processes. In this talk, I will present two directions where analytical techniques lead to accelerated algorithms. The first direction concerns constrained sampling, which can be viewed as the study of stochastic processes with boundary conditions. We develop reflected and replica-exchange Langevin dynamics, as well as non-reversible perturbations, and analyze their mixing behavior in bounded domains. These methods can be interpreted as approximations of Fokker–Planck-type equations with reflection, where non-reversibility enhances convergence by breaking detailed balance and improving exploration of the state space. The second direction addresses a fundamental computational challenge in generative modeling: the estimation of traces of large operators arising from high-dimensional transport and diffusion models. We introduce a variance-optimal estimator based on Hutch++ and provide a theoretical framework for variance reduction, connecting randomized numerical linear algebra with operator-theoretic perspectives.