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. We further illustrate the hybrid spectral method by deriving covariance kernels that are based on the Ornstein-Uhlenbeck model.

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化框架。协方差核是高斯过程的核心引擎，它整合了支撑预测分布的相关性。对于时空数据集的应用场景，合适的核应当能对空间与时间的联合依赖性进行建模。可分离的时空协方差核具有简洁性和计算高效性，但非可分离核包含能更好捕捉观测相关性的时空相互作用。目前大多数允许显式表达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。我们提出了一种基于物理论证的混合谱方法以生成协方差核。利用该方法，我们推导出一类新的具有物理动机的非可分离协方差核，其根源可追溯至随机线性阻尼谐振子（LDHO）模型。该新核整合了兼具单调与振荡衰减特征的时空相关性函数。LDHO协方差核通过调制谐振子系数的色散关系引入时空相互作用，我们推导了三种谐振子状态（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其性质，进一步通过基于奥恩斯坦-乌伦贝克模型推导协方差核，展示了混合谱方法的应用。