Accurate learning of system dynamics is becoming increasingly crucial for advanced control and decision-making in engineering. However, real-world systems often exhibit multiple channels and highly nonlinear transition dynamics, challenging traditional modeling methods. To enable online learning for these systems, this paper formulates the system as Gaussian process state-space models (GPSSMs) and develops a recursive learning method. The main contributions are threefold. First, a heterogeneous multi-output kernel is designed, allowing each output dimension to adopt distinct kernel types, hyperparameters, and input variables, improving expressiveness in multi-dimensional dynamics learning. Second, an inducing-point management algorithm enhances computational efficiency through independent selection and pruning for each output dimension. Third, a unified recursive inference framework for GPSSMs is derived, supporting general moment matching approaches, including the extended Kalman filter (EKF), unscented Kalman filter (UKF), and assumed density filtering (ADF), enabling accurate learning under strong nonlinearity and significant noise. Experiments on synthetic and real-world datasets show that the proposed method matches the accuracy of SOTA offline GPSSMs with only 1/100 of the runtime, and surpasses SOTA online GPSSMs by around 70% in accuracy under heavy noise while using only 1/20 of the runtime.

翻译：系统动力学的精确学习对于工程中的先进控制与决策正变得日益关键。然而，现实世界系统通常表现出多通道和高度非线性的转移动力学，这对传统建模方法构成了挑战。为使这些系统能够进行在线学习，本文将系统建模为高斯过程状态空间模型（GPSSMs），并开发了一种递归学习方法。主要贡献有三方面。首先，设计了一种异构多输出核，允许每个输出维度采用不同的核类型、超参数和输入变量，从而提高了多维动力学学习的表达能力。其次，一种诱导点管理算法通过为每个输出维度进行独立选择和剪枝，提升了计算效率。第三，推导了一个统一的GPSSM递归推理框架，支持通用的矩匹配方法，包括扩展卡尔曼滤波器（EKF）、无迹卡尔曼滤波器（UKF）和假设密度滤波（ADF），从而能够在强非线性和显著噪声下实现精确学习。在合成和真实数据集上的实验表明，所提方法仅用1/100的运行时间即可达到最先进（SOTA）离线GPSSM的精度，并且在重度噪声下，仅用1/20的运行时间，其精度比最先进的在线GPSSM高出约70%。