Physical systems can often be described via a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. This can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. Higher-order numerical integrators provide the necessary tools to address this problem by discretizing the dynamics function with arbitrary accuracy. Many higher-order integrators require dynamics evaluations at intermediate time steps making exact GP inference intractable. In previous work, this problem is often tackled by approximating the GP posterior with variational inference. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to make direct inference tractable, we propose to leverage multistep and Taylor integrators. We demonstrate how to derive flexible inference schemes for these types of integrators. Further, we derive tailored sampling schemes that allow to draw consistent dynamics functions from the learned posterior. This is crucial to sample consistent predictions from the dynamics model. We demonstrate empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

翻译：物理系统通常可以通过连续时间动力系统来描述。在实践中，真实系统往往未知，需要从测量数据中学习。由于数据通常以离散时间形式采集（例如通过传感器），大多数高斯过程动力学模型学习方法都基于一步前向预测进行训练。这在多种场景中可能产生问题，例如当测量值以不规则采样时间步提供，或需要守恒物理系统属性时。因此，我们旨在建立真实连续时间动力学的高斯过程模型。高阶数值积分器通过以任意精度对动力学函数进行离散化，提供了解决该问题的必要工具。许多高阶积分器需要在中间时间步评估动力学，这使得精确高斯过程推断变得不可行。在先前工作中，该问题常通过变分推断近似高斯过程后验来解决。然而，精确高斯过程推断在多数场景中更受青睐，例如因其具有数学保证性。为使直接推断具有可解性，我们提出利用多步积分器和泰勒积分器。我们展示了如何针对这些类型的积分器推导灵活的推断方案。此外，我们推导了定制化采样方案，使得能够从学得后验中抽取一致的动力学函数。这对于从动力学模型中生成一致的预测至关重要。我们通过经验和理论论证表明，我们的方法能够准确地表示连续时间系统。