Learning dynamical models from data is not only fundamental but also holds great promise for advancing principle discovery, time-series prediction, and controller design. Among various approaches, Gaussian Process State-Space Models (GPSSMs) have recently gained significant attention due to their combination of flexibility and interpretability. However, for online learning, the field lacks an efficient method suitable for scenarios where prior information regarding data distribution and model function is limited. To address this issue, this paper proposes a recursive GPSSM method with adaptive capabilities for both operating domains and Gaussian process (GP) hyperparameters. Specifically, we first utilize first-order linearization to derive a Bayesian update equation for the joint distribution between the system state and the GP model, enabling closed-form and domain-independent learning. Second, an online selection algorithm for inducing points is developed based on informative criteria to achieve lightweight learning. Third, to support online hyperparameter optimization, we recover historical measurement information from the current filtering distribution. Comprehensive evaluations on both synthetic and real-world datasets demonstrate the superior accuracy, computational efficiency, and adaptability of our method compared to state-of-the-art online GPSSM techniques.

翻译：从数据中学习动态模型不仅具有基础性意义，而且在推动原理发现、时间序列预测和控制器设计方面展现出巨大潜力。在各种方法中，高斯过程状态空间模型（GPSSMs）因其灵活性与可解释性的结合而近期受到广泛关注。然而，对于在线学习领域，目前尚缺乏一种适用于数据分布和模型函数先验信息有限场景的高效方法。为解决这一问题，本文提出一种具有运行域和高斯过程（GP）超参数自适应能力的递归GPSSM方法。具体而言，我们首先利用一阶线性化推导系统状态与GP模型联合分布的贝叶斯更新方程，实现闭式且与域无关的学习。其次，基于信息准则开发了诱导点的在线选择算法，以实现轻量化学习。第三，为支持在线超参数优化，我们从当前滤波分布中恢复历史测量信息。在合成数据集和真实数据集上的综合评估表明，相较于最先进的在线GPSSM技术，本方法在精度、计算效率和适应性方面均表现出优越性。