Learning Population-Level Dynamics through a Latent Fokker--Planck Model and Discrepancy Transport Maps

2026-07-23

AI Summary

This work addresses the challenge of inferring unknown population dynamics solely from snapshots of time-series probability distributions, without access to individual trajectories or prescribed dynamical equations. To this end, it introduces a novel paradigm that decomposes the dynamics into a latent Ornstein–Uhlenbeck stochastic process and a geometric transport map. The former yields an analytically tractable Fokker–Planck equation, while the latter employs monotone neural networks to implement a Knothe–Rosenblatt rearrangement that captures distributional deformations. A deformation energy regularizer grounded in hyperelasticity theory is incorporated to enhance solution uniqueness and physical interpretability. Experiments demonstrate that the method accurately reconstructs complex probabilistic dynamics in nonlinear, multimodal distribution evolution tasks, while maintaining a compact and analytically manageable latent representation.

Abstract

Many scientific and engineering systems are observed as time-indexed probability distributions whose governing dynamics are unknown and whose individual trajectories are unavailable. These settings challenge conventional system-identification approaches that rely on trajectory correspondence or prescribed evolution equations. This work presents a population-level inference framework that recovers latent stochastic dynamics directly from snapshot probability distributions by decomposing the observed evolution into an intrinsic latent stochastic process and a discrepancy transport map that captures geometric deformation between the latent and observed probability spaces. The latent dynamics are modeled using an Ornstein--Uhlenbeck process, providing a closed-form solution to the associated Fokker--Planck equation, while the discrepancy transport map is parameterized through the Knothe--Rosenblatt rearrangement with monotone neural networks. To mitigate the non-uniqueness inherent in the latent--transport decomposition, the transport map is regularized using a deformation energy motivated by hyperelasticity, promoting smooth, physically interpretable deformations while reducing unnecessary complexity. The latent stochastic model and discrepancy transport map are learned jointly through a unified optimization problem defined over probability distributions. Numerical examples involving nonlinear and multimodal distributional dynamics demonstrate that the proposed framework accurately reconstructs complex probability evolution while preserving a compact and analytically tractable latent representation. The proposed formulation provides a general framework for population-level dynamical inference and establishes a foundation for extending latent stochastic models and transport-based learning to more general and higher-dimensional systems.

Problem

• population-level dynamics
• stochastic dynamics
• probability distributions
• system identification
• trajectory-free inference

Innovation

• Latent Fokker--Planck Model
• Discrepancy Transport Maps
• Monotone Neural Networks
• Population-Level Dynamics
• Knothe--Rosenblatt Rearrangement