Research On The Sparse Identification Of Nonlinear Dynamics Model

Since a large amounts of engineering dynamics have abundant observations and the complex intrinsic structure,the underlying governing equations cannot be available only from input and output data of nonlinear dynamics.Recently,most researchers have developed data-driven model identification methods derived from the model-driven approaches.Of the note is that there exist a few active elements in the governing equations of dynamical systems.Therefore,sparse regression techniques are introduced to learn parsimonious models from high-dimensional measurements alone.The sparse identification of nonlinear dynamics(SINDy)model was proposed by Brunton’s team at the University of Washington.It makes an assumption that the feature space of nonlinear dynamics is sparse,such that the problem for governing equations will be transformed into a minimisation issue for systematically sparse coefficients.Thus,this thesis makes uses of the machine learning models,statistics learning methods and the distinct nonlinear systems to modify the raw structure of the SINDy model in terms of the sparse model identification.The main researches are shown as follows:1)The nonlinear function candidate library of the original SINDy model will be augmented in virtue of the different levels of complexity for nonlinear dynamics.The dominant improvements are in three areas: i)first,the group sparsity method has been introduced to identify the partial differential equations(PDEs),whereby the regulariser will be enforced on the sparse coefficients through the connection between each group of measurements;ii)second,the kernel regression methods will be incorporated into the original framework to handle intractable high-dimensional observations via kernel mapping;iii)the connection between the SINDy model and the Koopman linear operator will be constructed to simulate a linear model and a forcing activation for nonlinear systems.2)The architecture of the SINDy model will be expanded with the introduction of statistical learning models or machine learning models.In terms of statistical learning,the SINDy model is integrated with the information criterion to optimise the selection of sparse models through ranking the information supports in combinatorially candidate model library.Subsequently,the autoencoder and the kernel functions embed into the SINDy model to identify the sparse models and simultaneously to transform coordinate.3)Virtually,the noise is an inevitable factor in engineering fields.Thus,the noise should be considered during the overall training process for the SINDy model.The measurements will be departed into the past state observations and the future state observations on the basis of the adaptive time-stepper technique and the Kalman filter method.In such case,both the clean observed estimations and the noise probability distribution will be learned.In summary,a wide range of canonical nonlinear systems are utilized to verify the performance of the SINDy model and its derivatives,whose governing equations are enunciated in multiple forms,such as: the ordinary differential equations(ODEs),PDEs,and trigonometric second order equations.It is found that all these models can trustworthily identify the sparse models of the nonlinear dynamics and have strong robustness.