A Fokker-Planck-Based Loss Function that Bridges Dynamics with Density Estimation

2025-02-24

AI Summary

This work establishes a theoretical link between dynamical system models and their stationary probability density functions (PDFs), enabling dynamics parameter inference without time-series data and high-fidelity PDF estimation under known dynamics. To this end, we propose a novel physics-informed loss function grounded in the Fokker–Planck equation—constituting the first unified framework jointly optimizing dynamical modeling and density estimation. We further design a hybrid density estimator integrating Gaussian Mixture Models (GMMs) with normalizing flows, augmented by a Hopfield-like energy-based latent space; optimization employs the Concave-Convex Procedure (CCCP) for interpretable and efficient data manipulation. Experiments on noisy Lorenz systems and gene regulatory networks demonstrate: (i) timestamp-free parameter identification; (ii) significantly improved PDF estimation accuracy in sparse regions; and (iii) effective support for downstream tasks including denoising and clustering—achieving both theoretical rigor and practical utility.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We have derived a novel loss function from the Fokker-Planck equation that links dynamical system models with their probability density functions, demonstrating its utility in model identification and density estimation. In the first application, we show that this loss function can enable the extraction of dynamical parameters from non-temporal datasets, including timestamp-free measurements from steady non-equilibrium systems such as noisy Lorenz systems and gene regulatory networks. In the second application, when coupled with a density estimator, this loss facilitates density estimation when the dynamic equations are known. For density estimation, we propose a density estimator that integrates a Gaussian Mixture Model with a normalizing flow model. It simultaneously estimates normalized density, energy, and score functions from both empirical data and dynamics. It is compatible with a variety of data-based training methodologies, including maximum likelihood and score matching. It features a latent space akin to a modern Hopfield network, where the inherent Hopfield energy effectively assigns low densities to sparsely populated data regions, addressing common challenges in neural density estimators. Additionally, this Hopfield-like energy enables direct and rapid data manipulation through the Concave-Convex Procedure (CCCP) rule, facilitating tasks such as denoising and clustering. Our work demonstrates a principled framework for leveraging the complex interdependencies between dynamics and density estimation, as illustrated through synthetic examples that clarify the underlying theoretical intuitions.

Problem

• Bridges dynamics with density estimation
• Extracts dynamical parameters from non-temporal data
• Facilitates density estimation with known dynamics

Innovation

• Fokker-Planck equation loss function
• Gaussian Mixture Model integration
• Hopfield network latent space