Kernel functions embed into the autoencoder to identify the sparse models of nonlinear dynamics

•Identify the governing equations of nonlinear dynamics directly from measurements.•The autoencoder can automatically find the appropriate sparse subspace to implement coordinate transformation.•The kernel functions and the sparse identification of nonlinear dynamics (SINDy) model are embedded into the hidden space of the autoencoder to kernelize and to promote the sparsity of the salient features of nonlinear dynamics.•The error back-forward propagation and the feature forward propagation are leveraged to optimize the sparse models of nonlinear dynamics. Numerous researches have shown that there are three main challenges in data-driven model identification methods: high-dimensional measurements, system complexity and unknown underlying dynamical properties. For most nonlinear dynamics, the feature space defined by the coefficients of their control equations is sparse. Therefore, sparse regression methods are used to learn the sparse coefficients of the control equations of nonlinear dynamics. However, this method strongly depends on the appropriate selection of the sparse basis vectors. In this essay, the autoencoder is combined with the sparse regression method to simultaneously identify the sparse coordinate and a parsimonious, interpretable and generalizable model of the specified system. It also integrates kernel functions to map the intractable measurements in the hidden space of the autoencoder into a linearly distinguishable kernel space, which kernelizes the candidate function library of the sparse identification of nonlinear dynamics (SINDy) model as the sparse dictionaries. Therefore, the flexible representation of neural networks, the simplicity of sparse regression methods and the implicit non-linear representation of kernel functions are consolidated in this article. To inspect the reliability of the proposed model in this paper, a set of nonlinear dynamics formulated by ordinary differential equations (ODEs), second-order trigonometric functions and partial differential equations (PDEs) are utilized as test cases. And the comparisons between the proposed model and other model identification methods illustrate that the performance of the former is the best.