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In this paper, we propose a novel kernel method for the online identification of stochastic nonlinear spatiotemporal dynamical systems using the robust control approach. By the difference method, the stochastic spatiotemporal (SST) systems driven by multiplicative noise are first transformed into a class of multi-input-multi-output-partially linear kernel models (PLKMs) with heterogeneous random terms. With the help of techniques established for reproducing kernel Hilbert space, the online learning problem is reasonably considered as an output feedback control problem for a group of time varying linear dynamical systems. We develop an effective algorithm to address the learning problem of PLKM and SST systems by employing the model predictive control theory. Compared with the existing learning methods, the new one can achieve adaptive, robust, and fast convergent online modeling performance for the spatiotemporal dynamics with multiplicative noise, which greatly facilitates the characterization of physical characteristics of the system. Moreover, this investigation for the first time addresses the learning problems for SST systems with novel robust control techniques, which can provide some novel insights into the design of kernel machine learning methods from the perspective of optimal control theory. Numerical studies for benchmark systems are presented to illustrate the effectiveness and efficiency of our new method. [ABSTRACT FROM AUTHOR]

The identification of nonlinear spatiotemporal dynamical systems given by partial differential equations has attracted a lot of attention in the past decades. Several methods, such as searching principle-based algorithms, partially linear kernel methods, and coupled lattice methods, have been developed to address the identification problems. However, most existing methods have some restrictions on sampling processes in that the sampling intervals should usually be very small and uniformly distributed in spatiotemporal domains. These are actually not applicable for some practical applications. In this paper, to tackle this issue, a novel kernel-based learning algorithm named integral least square regularization regression (ILSRR) is proposed, which can be used to effectively achieve accurate derivative estimation for nonlinear functions in the time domain. With this technique, a discretization method named inverse meshless collocation is then developed to realize the dimensional reduction of the system to be identified. Thereafter, with this novel inverse meshless collocation model, the ILSRR, and a multiple-kernel-based learning algorithm, a multistep identification method is systematically proposed to address the identification problem of spatiotemporal systems with pointwise nonuniform observations. Numerical studies for benchmark systems with necessary discussions are presented to illustrate the effectiveness and the advantages of the proposed method. [ABSTRACT FROM AUTHOR]