结合流映射优化的高斯过程学习用于动力系统的参数估计
Gaussian process learning with flow map refinement for parameter estimation in dynamical systems
- Suzhou University of Science and Technology（苏州科技大学）
- Xi’an Jiaotong-Liverpool University（西交利物浦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出结合流映射优化的高斯过程学习（GPL-FMR）两阶段框架，用于动力系统参数估计，可在观测稀缺有噪时提升估计精度，经多个数值示例验证有效。
AI中文摘要：
参数估计是动力系统数据驱动学习中的核心任务，旨在从观测到的时间序列数据中恢复底层物理参数，从而为支配该系统的物理机制提供可解释的见解。基于高斯过程的梯度/导数匹配方法为参数估计提供了一种高效途径，这类方法避免了重复的数值积分，并强制实现局部导数一致性。然而，这种局部匹配可能导致与支配流映射的全局不一致，尤其是在观测数据稀缺且存在噪声的情况下。为解决这一局限，我们提出了基于流映射优化的高斯过程学习（GPL-FMR）框架，这是一个两阶段参数估计框架。第一阶段基于高斯过程学习算法，将得到的后验作为信息先验传递给基于流映射优化的第二阶段；第二阶段通过基于全局动力学约束的优化进一步改进参数估计。我们在多个数值示例上验证并分析了其性能，包括范德波尔振子、洛特卡-沃尔泰拉模型和洛伦兹-63系统。结果表明，所提框架能持续提升参数估计的准确性，尤其在观测数据稀缺且含噪声的情况下表现显著。
英文摘要：
Parameter estimation is a central task in data-driven learning of dynamical systems. It aims to recover the underlying physical parameters from observed time-series data, thereby providing interpretable insights into the physical mechanisms governing the system. Gradient/derivative matching methods based on Gaussian process provide an efficient way to perform parameter estimation. Those methods avoid repeated numerical integration and enforce local derivative consistency. However, such local matching may result in global inconsistency with the governing flow map, particularly under scarce and noisy observations. To address this limitation, we propose a framework based on Gaussian process learning with flow map refinement (GPL-FMR), a two-stage parameter estimation framework. The first stage is based on Gaussian process learning algorithm and the posterior obtained from which is transferred as an informative prior to the second stage based on flow-map refinement. The second stage further improves the parameter estimation via optimisation based on global dynamical constraints. We demonstrate and analyse its performance on multiple numerical examples, including the Van der Pol oscillator, the Lotka-Volterra model, and the Lorenz-63 system. The results show that the proposed framework consistently improves parameter estimation accuracy, particularly under scarce and noisy observations.
