We introduce a novel framework for constructing scalable and flexible covariance kernels for Gaussian processes (GPs) by directly learning the covariance structure under a regression-type parameterization induced by Vecchia approximations, using deep neural architectures. Specifically, we model kriging coefficients and conditional standard deviations, deterministic quantities that uniquely characterize the covariance, providing stable and informative learning targets. Exploiting the permutation-equivariant structure of conditioning sets in the Vecchia factorization, we derive a universal representation for permutation-preserving functions and design neural architectures that respect this symmetry, leading to improved training stability and data efficiency. The proposed approach enables expressive, non-stationary kernel learning while maintaining computational scalability, thereby bridging classical GP methodology with modern deep learning.

翻译：我们提出了一种构建高斯过程（GP）可扩展且灵活协方差核的新框架，该框架通过深度神经架构在Vecchia近似诱导的回归型参数化下直接学习协方差结构。具体而言，我们对克里金系数和条件标准差（唯一刻画协方差的确定性量）进行建模，从而提供稳定且信息丰富的学习目标。利用Vecchia分解中条件集的置换等变结构，我们推导出保序函数的通用表示，并设计了尊重该对称性的神经架构，从而提升了训练稳定性与数据效率。所提方法能够实现具有表达力的非平稳核学习，同时保持计算可扩展性，由此弥合了经典GP方法与现代深度学习之间的鸿沟。