This paper develops the Dynamic Gaussian Process (DGP), a framework for estimating functions governed by integro-difference equations (IDEs). IDEs model continuous functions that evolve with discrete-time dynamics and arise naturally from time-discretization of linear partial differential equations (PDEs). The DGP extends Gaussian process regression to time-varying functions and extends Kalman filtering to infinite-dimensional states. The DGP posterior remains a Gaussian process with closed-form mean and covariance updates, and separable kernel structure reduces the problem to a finite-dimensional Kalman filter on basis function coefficients. This paper extends the DGP to vector-valued states, enabling the treatment of higher-order PDEs, and provides a stability and approximation error analysis for the basis function approximation. The functional L2 estimation error decomposes exactly into in-subspace and out-of-subspace contributions, and all approximation errors vanish as the number of basis functions grows. The framework is demonstrated on the heat equation and on the wave equation, the latter with a vector-valued state. Code is available at https://github.com/JvHulst/Dynamic_Gaussian_Processes.

翻译：本文提出了动态高斯过程（DGP），一种用于估计由积分差分方程（IDEs）支配的函数的框架。IDEs建模具有离散时间演化规律的连续函数，自然产生于线性偏微分方程（PDEs）的时间离散化。DGP将高斯过程回归扩展到时变函数，并将卡尔曼滤波扩展到无限维状态。DGP的后验仍为高斯过程，具有闭式均值与协方差更新公式，且可分离核结构将问题简化为基函数系数上的有限维卡尔曼滤波。本文将DGP扩展至向量值状态，从而能够处理高阶PDEs，并对基函数近似进行了稳定性与逼近误差分析。函数L2估计误差精确分解为子空间内部与子空间外部贡献，且所有逼近误差随基函数数量增加而消失。该框架在热传导方程和波动方程上得到验证，其中波动方程涉及向量值状态。代码见https://github.com/JvHulst/Dynamic_Gaussian_Processes。