Langevin Dynamics

Langevin dynamics combines motion along a potential’s gradient with noise, allowing it to sample complex distributions and support Bayesian inference, optimization, and generative modeling. Common approaches include overdamped and underdamped dynamics, stochastic-gradient variants, and annealed or score-guided samplers; researchers also use preconditioning, non-reversible forces, and constraints to improve exploration. Much of the work develops discretizations and convergence guarantees, often measured in Wasserstein distance, total variation, or sampling error on high-dimensional and multimodal problems. A central tension is making sampling faster and more scalable without introducing bias or losing reliable coverage, especially with approximate scores, nonsmooth targets, or nonconvex landscapes.