Data-driven reconstruction of limit cycle position provides side information for improved model identification with SINDy

Many important systems in nature are characterized by oscillations. To understand and interpret such behavior, researchers use the language of mathematical models, often in the form of differential equations. Nowadays, these equations can be derived using data-driven machine learning approaches, such as the white-box method 'Sparse Identification of Nonlinear Dynamics' (SINDy). In this paper, we show that to ensure the identification of sparse and meaningful models, it is crucial to identify the correct position of the system limit cycle in phase space. Therefore, we propose how the limit cycle position and the system's nullclines can be identified by applying SINDy to the data set with varying offsets, using three model evaluation criteria (complexity, coefficient of determination, generalization error). We successfully test the method on an oscillatory FitzHugh-Nagumo model and a more complex model consisting of two coupled cubic differential equations. Finally, we demonstrate that using this additional side information on the limit cycle in phase space can improve the success of model identification efforts in oscillatory systems.