通过潜在福克 - 普朗克模型和差异传输映射学习总体水平动力学
Learning Population-Level Dynamics through a Latent Fokker--Planck Model and Discrepancy Transport Maps
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中文总结 AI 辅助
针对科学和工程系统概率分布动力学未知问题，提出总体水平推理框架，通过潜在福克 - 普朗克模型和差异传输映射恢复潜在随机动力学，经正则化联合学习，能准确重建复杂概率演化，为动力学推理提供通用框架。
中文摘要 AI 辅助
许多科学和工程系统以时间索引概率分布形式被观测，其控制动力学未知且个体轨迹不可用，这对传统系统识别方法提出挑战。本文提出一个总体水平推理框架，通过将观测演化分解为内在潜在随机过程和捕获潜在与观测概率空间几何变形的差异传输映射，直接从快照概率分布中恢复潜在随机动力学。潜在动力学用奥恩斯坦 - 乌伦贝克过程建模，差异传输映射通过带有单调神经网络的克诺特 - 罗森布拉特重排参数化。为减轻潜在 - 传输分解中的非唯一性，传输映射用超弹性激发的变形能量正则化。通过在概率分布上定义的统一优化问题联合学习潜在随机模型和差异传输映射。数值例子表明该框架能准确重建复杂概率演化并保留紧凑且可解析处理的潜在表示，为总体水平动力学推理提供通用框架。
英文摘要
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.
