Modeling of missing dynamical systems: deriving parametric models using a nonparametric framework
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RESEARCH
Modeling of missing dynamical systems: deriving parametric models using a nonparametric framework Shixiao W. Jiang1 and John Harlim1,2,3* * Correspondence:
[email protected] Department of Mathematics, The Pennsylvania State University, 109 McAllister Building, University Park, PA 16802-6400, USA Full list of author information is available at the end of the article 1
Abstract 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. Keywords: Missing dynamical systems, Closure model, Nonparametric non-Markovian model, Kernel embedding Mathematics Subject Classification: 37M10, 37M25, 41A10, 41A45, 60J22
1 Introduction One of the long-standing issues in modeling dynamical systems is model errors arising from incomplete understanding of the physics. The progress in tackling this problem goes under different names depending on the scientific fields. In applied mathematics and engineering sciences, some of these approaches are known as the reduced-order modeling, which the ultimate goal is to derive an effective model with low computational complexity from the first principle, assuming that the full dynamics is known. They include the Mori–Zwanzig formalism [44,53,54] and its approximations [4,5,12,17,25]; the averaging/homogenization when there is an apparent scale separation between the relevant and irrelevant variables [10,42,43,50]. In domain sciences, various methods for
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subgrid-scale parameterization were proposed to handle the same problem that arises in applications such as material science, molecular dynamics, and climate dynamics, just to name a few. They include the Markov chain type modeling [7,23]; stochastic parameterization [1,7,18,29,32,34,37,51]; superparameteriz
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