Using Machine Learning for the Precise Experimental Modeling of Catastrophe Phenomena: Taking the Establishment of an Experimental Mathematical Model of a Cusp-Type Catastrophe for the Zeeman Catastrophe Machine as an Example

When catastrophe theory is applied to the experimental modeling of catastrophe phenomena, it is impossible to know in advance the corresponding relationship and mapping form between the parameters of the actual catastrophe mathematical model and the parameters of the canonical catastrophe mathematical model. This gives rise to the problem in which the process of experimental modeling cannot be completed in many instances. To solve this problem, an experimental modeling method of catastrophe theory is proposed. It establishes the quantitative relationship between the actual catastrophe mathematical model and the canonical catastrophe mathematical model by assuming that the actual potential function is equal to the canonical potential function, and it uses a machine learning model to represent the diffeomorphism that can realize the error-free transformation of the two models. The method is applied to establish the experimental mathematical model of a cusp-type catastrophe for the Zeeman catastrophe machine. Through programming calculation, it is found that the prediction errors of the potential function, manifold, and bifurcation set of the established model are 0.0455%, 0.0465%, and 0.1252%, respectively. This indicates that the established model can quantitatively predict the catastrophe phenomenon.