Gaussian processes provide a flexible, non-parametric framework for the approximation of functions in high-dimensional spaces. The covariance kernel is the main engine of Gaussian processes, incorporating correlations that underpin the predictive distribution. For applications with spatiotemporal datasets, suitable kernels should model joint spatial and temporal dependence. Separable space-time covariance kernels offer simplicity and computational efficiency. However, non-separable kernels include space-time interactions that better capture observed correlations. Most non-separable kernels that admit explicit expressions are based on mathematical considerations (admissibility conditions) rather than first-principles derivations. We present a hybrid spectral approach for generating covariance kernels which is based on physical arguments. We use this approach to derive a new class of physically motivated, non-separable covariance kernels which have their roots in the stochastic, linear, damped, harmonic oscillator (LDHO). The new kernels incorporate functions with both monotonic and oscillatory decay of space-time correlations. The LDHO covariance kernels involve space-time interactions which are introduced by dispersion relations that modulate the oscillator coefficients. We derive explicit relations for the spatiotemporal covariance kernels in the three oscillator regimes (underdamping, critical damping, overdamping) and investigate their properties.

翻译：高斯过程为高维空间中的函数逼近提供了灵活的非参数框架。协方差核作为高斯过程的核心引擎，整合了支撑预测分布的相关性。针对时空数据集的应用，合适的核函数应能建模联合时空依赖性。可分离的时空协方差核具有简单性和计算效率优势，而非可分离核则包含更准确反映观测相关性的时空相互作用。现有大多数具有显式表达的非可分离核基于数学考量（可容许性条件）而非第一性原理推导。我们提出了一种基于物理论证的混合谱方法用于生成协方差核，并以此推导出一类新的、具有物理动机的非可分离协方差核，其根源在于随机线性阻尼谐振子（LDHO）。新核函数兼具时空相关性的单调衰减与振荡衰减特性。LDHO协方差核通过调节谐振子系数的色散关系引入时空相互作用，我们推导了三种振荡状态（欠阻尼、临界阻尼、过阻尼）下时空协方差核的显式关系，并研究了其性质。