用分层贝叶斯模型从稀疏数据中高效学习动力系统参数
Learning Dynamical Systems from Multiple Sparse Datasets: A Hierarchical Bayesian Modeling Approach
- 将多组数据的参数视为共享分布的抽样，建模共性与差异
- 嵌入数值求解器的梯度MCMC实现高效后验推断
- 在稀疏数据下显著优于独立拟合方法，适合小样本场景
从稀疏、噪声大且不规则采样的数据中估计动力系统的参数通常严重病态。当存在多个相关数据集时，若能恰当建模共享结构与变异性，可提供额外信息。本文提出一种分层贝叶斯框架，用于动力系统中的概率元学习，将每个数据集的特定参数视为来自共享总体分布的抽样。在基于梯度的MCMC中嵌入数值常微分方程求解器，实现对共享总体和各数据集特异性参数分布的高效后验推断。实验表明，该方法在预测性能上优于未共享参数的方法，凸显了在稀疏数据条件下实现数据高效系统辨识的潜力。
原文摘要 · Abstract (English)
Estimating parameters of dynamical systems from sparse, noisy, and irregularly sampled data is often severely ill-conditioned. When multiple related datasets are available, they provide additional information if the shared structure and variability are properly modeled. We propose a hierarchical Bayesian framework for probabilistic meta-learning in dynamical systems, modeling dataset-specific parameters as draws from a shared population distribution. A numerical ODE solver is embedded within gradient-based MCMC to enable efficient posterior inference of the shared population and dataset-specific parameter distribution. Experiments show improved predictive performance over unpooled methods, highlighting the potential for data-efficient system identification in settings with sparse data.
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