Augmenting mechanistic ordinary differential equation (ODE) models with machine-learnable
structures is a novel approach to create highly accurate, low-dimensional models of
engineering systems incorporating both expert knowledge and reality through measurement
data. Our exploratory study focuses on training universal differential equation (UDE)
models for physical nonlinear dynamical systems with limit cycles: an aerofoil undergoing
flutter oscillations and an electrodynamic nonlinear oscillator. We consider examples
where training data is generated by numerical simulations, whereas we also employ
the proposed modelling concept to physical experiments allowing us to investigate
problems with a wide range of complexity. To collect the training data, the method
of control-based continuation is used as it captures not just the stable but also
the unstable limit cycles of the observed system. This feature makes it possible to
extract more information about the observed system than the open-loop approach (surveying
the steady state response by parameter sweeps without using control) would allow.
We use both neural networks and Gaussian processes as universal approximators alongside
the mechanistic models to give a critical assessment of the accuracy and robustness
of the UDE modelling approach. We also highlight the potential issues one may run
into during the training procedure indicating the limits of the current modelling
framework.
