Simulation-Free Differential Dynamics through Neural Conservation Laws

Mengjian Hua, Eric Vanden-Eijnden, Ricky T. Q. Chen

Keywords: Generative Modeling; Simulation-free Methods; Diffusion Models;

Abstract:

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.

Primary Area: generative models

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Submission Number: 3330