In this paper, we consider modeling missing dynamics with a nonparametric non-Markovian model, constructed using the theory of kernel embedding of conditional distributions on appropriate reproducing kernel Hilbert spaces (RKHS), equipped with orthonormal basis functions. Depending on the choice of the basis functions, the resulting closure model from this nonparametric modeling formulation is in the form of parametric model. This suggests that the success of various parametric modeling approaches that were proposed in various domains of applications can be understood through the RKHS representations. When the missing dynamical terms evolve faster than the relevant observable of interest, the proposed approach is consistent with the effective dynamics derived from the classical averaging theory. In the linear Gaussian case without the time-scale gap, we will show that the proposed non-Markovian model with a very long memory yields an accurate estimation of the nontrivial autocovariance function for the relevant variable of the full dynamics. The supporting numerical results on instructive nonlinear dynamics show that the proposed approach is able to replicate high-dimensional missing dynamical terms on problems with and without the separation of temporal scales.

中文翻译：

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缺少动力系统的建模：使用非参数框架推导参数模型

在本文中，我们考虑使用非参数非马尔可夫模型对缺失的动力学进行建模，该模型是使用条件分布的核嵌入理论在适当的具有正交正态基函数的适当再现核希尔伯特空间（RKHS）上进行的。取决于基础函数的选择，从该非参数建模公式生成的闭合模型为参数模型的形式。这表明可以通过RKHS表示来理解在各种应用领域中提出的各种参数建模方法的成功。当缺少的动力学项的发展速度快于相关可观察的相关动态项时，所提出的方法与从经典平均理论得出的有效动力学相一致。在没有时间尺度间隙的线性高斯情况下，我们将表明，所提出的具有非常长的记忆力的非马尔可夫模型对完整动力学的相关变量产生了非平凡的自协方差函数的准确估计。关于指导性非线性动力学的支持数值结果表明，该方法能够在时间尺度分离和不分离的情况下，对问题中的高维缺失动力学项进行复制。