Simulation-Free Differential Dynamics Through Neural Conservation Laws

We present a novel simulation-free framework for training continuous-time diffusion processes over very general objective functions. Existing methods typically involve either prescribing the optimal diffusion process—which only works for heavily restricted problem formulations—or require expensive simulation to numerically obtain the time-dependent densities and sample from the diffusion process. In contrast, we propose a coupled parameterization which jointly models a time-dependent density function, or probability path, and the dynamics of a diffusion process that generates this probability path. To accomplish this, our approach directly bakes in the Fokker-Planck equation and density function requirements as hard constraints, by extending and greatly simplifying the construction of Neural Conservation Laws. This enables simulation-free training for a large variety of problem formulations, from data-driven objectives as in generative modeling and dynamical optimal transport, to optimality-based objectives as in stochastic optimal control, with straightforward extensions to mean-field objectives due to the ease of accessing exact density functions. We validate our method in a diverse range of application domains from modeling spatio-temporal events, to learning optimal dynamics from population data.