Gaussian processes with prior-model-informed kernel for dynamical system modeling

Gaussian processes (GPs) are widely used for modeling dynamical systems due to their ability to provide principled uncertainty estimation and to incorporate prior knowledge via the mean or kernel function. However, GPs typically exhibit poor extrapolation performance outside the training region. To address this limitation while preserving the favorable uncertainty behavior of standard GPs, we propose a novel kernel design that integrates prior models-either analytical physical models or imperfect simulators-into the kernel function of the GP. Our approach is motivated by the assumption that similarity in the prior model implies similarity in the true system dynamics. Specifically, we transform the input space using the prior model and apply a base kernel (e.g., Radial Basis Function or Matérn) to construct a prior-model-informed kernel that reflects this assumption. To further enhance modeling flexibility, we add a standard residual kernel to correct discrepancies between the prior model and the true system. This yields our final model: a GP with prior-model-informed kernel (GP-PI-K). Through extensive experiments on benchmark dynamical systems, we demonstrate that GP-PI-K consistently outperforms existing baselines, including standard GPs, GPs with physics-informed or neural network mean functions, and deep kernel learning models, in terms of one-step and multi-step prediction accuracy, uncertainty estimation, and uncertainty calibration. Moreover, GP-PI-K achieves superior performance in downstream applications such as active learning and model-based reinforcement learning.