Simulation-Free Differential Dynamics through Neural Conservation Laws

2025-06-23

AI Summary

Existing continuous-time diffusion models rely on pre-specified optimal processes or expensive numerical simulations, limiting adaptability to general objective functions. This paper proposes a simulation-free joint modeling framework that unifies time-varying density functions and diffusion dynamics via shared neural parameterization, directly enforcing the Fokker–Planck equation and probability conservation. Its core innovation is coupled neural parameterization—extending and simplifying the neural continuity equation—to enable end-to-end joint optimization of density evolution and drift-diffusion dynamics without numerical integration or sampling. The method intrinsically embeds physical constraints, supporting diverse tasks including generative modeling, dynamic optimal transport, and stochastic optimal control. Experiments on spatiotemporal event modeling and collective dynamics learning demonstrate high accuracy and strong generalization, validating both theoretical soundness and practical efficacy.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Develop simulation-free training for continuous-time diffusion processes
• Overcome limitations of existing methods requiring expensive simulation
• Apply to diverse objectives like generative modeling and optimal control

Innovation

• Simulation-free training via Neural Conservation Laws
• Coupled parameterization for density and diffusion dynamics
• Hard constraints from Fokker-Planck equation integration