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学系统来描述。实际中，真实系统往往未知且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单步预测进行训练。这在多种场景下可能产生问题，例如当测量值以非均匀采样的时间步长提供时，或需要保持物理系统特性时。因此，我们旨在构建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离散化动力学函数，为解决该问题提供了必要工具。许多高阶积分器需要在中间时间步进行动力学评估，这使得精确高斯过程推断变得不可解。先前工作中，该问题常通过变分推断近似高斯过程后验来处理。然而，精确高斯过程推断因其数学保证而在许多场景中更受青睐。为使直接推断可行，我们提出利用多步积分器和泰勒积分器。我们展示了如何为这类积分器推导灵活的推断方案，并进一步推导了定制化采样方案，从而能够从学习到的后验中生成一致的动力学函数——这对于从动力学模型采样一致预测至关重要。我们通过实验和理论证明，该方法能够准确表征连续时间系统。