Langevin SDEs have unique transient dynamics

The overdamped Langevin stochastic differential equation (SDE) is a classical physical model used for chemical, genetic, and hydrological dynamics. In this work, we prove that the drift and diffusion terms of a Langevin SDE are jointly identifiable from temporal marginal distributions if and only if the process is observed out of equilibrium. This complete characterization of structural identifiability removes the long-standing assumption that the diffusion must be known to identify the drift. We then complement our theory with experiments in the finite sample setting and study the practical identifiability of the drift and diffusion, in order to propose heuristics for optimal data collection.

Learning Diffusion at Lightspeed

Diffusion regulates a phenomenal number of natural processes and the dynamics of many successful generative models. Existing models to learn the diffusion terms from observational data rely on complex bilevel optimization problems and properly model only the drift of the system. We propose a new simple model, JKOnet*, which bypasses altogether the complexity of existing architectures while presenting significantly enhanced representational capacity: JKOnet* recovers the potential, interaction, and internal energy components of the underlying diffusion process. JKOnet* minimizes a simple quadratic loss, runs at lightspeed, and drastically outperforms other baselines in practice. Additionally, JKOnet* provides a closed-form optimal solution for linearly parametrized functionals. Our methodology is based on the interpretation of diffusion processes as energy-minimizing trajectories in the probability space via the so-called JKO scheme, which we study via its first-order optimality conditions, in light of few-weeks-old advancements in optimization in the probability space.