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结构。对于时空数据集的应用场景，合适的核函数应能联合建模空间与时间依赖性。可分离的时空协方差核具有简洁性和计算效率优势，但非可分离核包含的时空交互作用能更准确地捕捉观测相关性。目前多数具有显式表达式的非可分离核均基于数学考量（可容许性条件）而非第一性原理推导得出。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，并据此推导出以随机线性阻尼谐振子（LDHO）为理论根源的物理驱动型非可分离协方差核新类别。该新型核函数同时包含时空相关性的单调衰减与振荡衰减特性。LDHO协方差核通过色散关系调制谐振子系数实现时空交互作用，我们推导了三种谐振子状态（欠阻尼、临界阻尼、过阻尼）下的时空协方差核显式表达式，并系统研究了各状态下的核函数性质。