Reconstruction of governing equation for nonlinear dynamical system based on Universal Differential Equation

In the sense of black-box approaches, data-driven models can be used for the mathematical description of complex dependencies of (multi-)physical processes. A specific or prior physical knowledge inside the model is not required and is compensated by cost-intensive data amounts. Due to the restrictive accessibility of data in some engineering fields, black-box models are limited regarding their applicability. The incorporation of physical knowledge into Machine Learning methods counteracts data limitations and leads to data efficient modelling approaches. The Universal Differential Equation (UDE) approach seeks for data reduction by combining physical based models and Machine Learning. This enables semi-automated and in some special cases fully-automated modelling. The resulting models and their evaluation are valuable for gaining knowledge during development, production and operating phase. In this paper, UDE is applied to reconstruct the governing equation of a nonlinear dynamical system represented by a forced duffing oscillator and compared with black-box approaches afterwards. In contrast to the approximation of the entire governing equation by an Universal Approximator, UDE aims to approximate only unknown terms inside the differential equation using Universal Approxiamtors such as Neural Networks. Based on sparse regression methods these unknown terms are reconstructable in a targeted manner. The methodology is applied and validated on a nonlinear dynamical system considering robustness and sensitivity aspects against uncertainty-prone training data (e.g. measurement data) and different data-driven modelling approaches are compared regarding their forecast capabilities. Afterwards, potentials for further fields of application in automotive development are shown.