Identification of Nonlinear Spatiotemporal Dynamical Systems with Nonuniform Observations Using Reproducing-Kernel-Based Integral Least Square Regulation

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.