Data Driven prediction of forced nonlinear vibrations using stabilised Autoregressive Neural Networks

In this work, we propose a novel approach to the data-driven prediction of vibration responses of nonlinear systems. The main idea is based on Autoregressive Neural Networks (ARNN) to model the nonlinear transfer behaviour between an external excitation and the system response. We propose an autoregressive network architecture with embedded symmetry using bias-free tanh activation and guarantee Input-to-State-Stability (ISS) by enforcing a special penalty term to the weights. The resulting training procedure is analysed for the example of a DUFFING oscillator with white noise excitation. In a BAYESian optimisation, it is found that beyond enforcing input-to-state-stability, the stabilising penalty term also decreases sensitivity with respect to other training parameters compared to other classical techniques. Furthermore, we show that the stabilised ARNN is able to give excellent approximations of the nonlinear response of the DUFFING oscillator for a wide range of excitation intensities. In contrast, linear models, such as autoregressive models with exogenous input (ARX) in time domain or linear transfer functions in frequency domain, will only find some linear approximation. In particular, by construction, they will not be able to capture nonlinear effects for arbitrary amplitudes and excitation levels.