基于边际观测推断隐藏动力学的特征敏感性集成方法
Characteristic Sensitivity Ensembles for Inference of Hidden Dynamics from Marginal Observations
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中文总结 AI 辅助
该研究提出一种基于边际观测推断隐藏动力学的随机梯度方法，利用特征表示结合交叉U统计量生成无偏梯度估计器，经四类实验验证其有效性。
中文摘要 AI 辅助
本文开发了一个框架，用于推断由广义常微分方程组描述的动力学。提出了一种随机梯度方法，该方法利用可观测变量与潜变量的联合概率密度函数，从观测到的边际概率密度函数中推断动力学。在低维状态下观测到的扩散及其他不可逆过程，可在足够增广的状态空间中重构为确定性可逆流，其中联合密度满足双曲刘维尔方程；观测到的边际分布是这些双曲动力学在观测坐标上的投影，潜变量分量承载随机性与记忆。这种重构将不可逆或随机动力学的推断转化为从边际观测中恢复确定性常微分方程（ODE）。算法未求解高维联合密度的刘维尔方程，而是利用其特征表示：从初始分布采样的粒子沿特征线输运，通过沿特征线传播敏感性得到关于参数的欧拉敏感性，结合交叉U统计量生成无偏梯度估计器，支持随机梯度下降。通过四个实验验证该方法：从单模边际观测恢复三模线性系统；含隐藏模的非线性冈珀茨增长模型；隐藏模使单模边际变为双模的双稳系统；细胞流中粒子的斯托克斯-奥森阻力定律恢复，并分析了不同设置下的收敛行为。
英文摘要
A framework is developed for the inference of dynamics described by a generalized system of ordinary differential equations. A stochastic gradient method is coined that infers dynamics from observed marginal probability density functions using the joint probability density function of the observable and latent variables. Diffusion and other irreversible processes observed in a low-dimensional state can be recast as deterministic, reversible flows in a sufficiently augmented state space, where the joint density satisfies the hyperbolic Liouville equation. The marginal distribution observed is the projection of these hyperbolic dynamics onto the observed coordinates, with the latent components carrying the randomness and memory. This reframing allows inference for irreversible or stochastic dynamics into the recovery of a deterministic Ordinary Differential Equation (ODE) from marginal observations. Instead of solving the high-dimensional Liouville equation for the joint density, the algorithm exploits its characteristic representation. Particles sampled from the initial distribution are transported along characteristic lines. The Eulerian sensitivity with respect to parameters is obtained by sensitivity propagation along the characteristic lines, with a crossed U-statistic producing an unbiased gradient estimator, which enables stochastic gradient descent. Four experiments validate the method: recovery of a three-mode linear system observed through the marginal of a single mode; a nonlinear Gompertz growth model with a hidden mode; a bistable system whose hidden mode turns a unimodal marginal bimodal; and Stokes--Oseen drag law recovery for particles in a cellular flow. Convergence behavior is analyzed across these settings.
