A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a kernel. In particular, this strategy is highly efficient (both in terms of accuracy and complexity) when the kernel is data-adapted using Kernel Flows (KF)\cite{Owhadi19} (which uses gradient-based optimization to learn a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used for interpolation). Despite its previous successes, this strategy (based on interpolating the vector field driving the dynamical system) breaks down when the observed time series is not regularly sampled in time. In this work, we propose to address this problem by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the (KF) data-adapted kernels. We compare our approach with the classical one over different benchmark dynamical systems and show that it significantly improves the forecasting accuracy while remaining simple, fast, and robust.

翻译：从数据中学习动力系统的一种简单且可解释的方法是使用核函数对其向量场进行插值。特别是，当使用核流（KF）\cite{Owhadi19}（该方法基于梯度优化学习核函数，其核心前提是：如果一个核函数性能良好，那么即使仅使用一半数据进行插值，精度也不会显著下降）对核函数进行数据自适应时，该策略在精度和复杂度方面均表现出高效性。尽管此前取得了成功，但当观测到的时间序列在时间上非均匀采样时，这种基于对驱动动力系统的向量场进行插值的策略便会失效。在本工作中，我们提出通过将观测间的时间差纳入（KF）数据自适应核函数中，直接逼近动力系统的向量场，以解决此问题。我们在不同的基准动力系统上比较了所提方法与经典方法，结果表明，新方法在保持简单、快速和鲁棒性的同时，显著提高了预测精度。