Simulation-Free Differential Dynamics through Neural Conservation Laws

• URL: http://arxiv.org/abs/2506.18604v1
• Date: Mon, 23 Jun 2025 13:04:23 GMT
• Title: Simulation-Free Differential Dynamics through Neural Conservation Laws
• Authors: Mengjian Hua, Eric Vanden-Eijnden, Ricky T. Q. Chen,
• Abstract summary: We present a novel simulation-free framework for training continuous-time diffusion processes over very general objective functions.<n>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.
• Score: 22.4113724471297
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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