Using scientific machine learning for experimental bifurcation analysis of dynamic systems

TL;DR

This paper explores the use of universal differential equations combining mechanistic models with machine learning to analyze nonlinear dynamical systems, utilizing control-based continuation for comprehensive data collection.

Contribution

It introduces a novel approach integrating control-based continuation with UDE models for physical systems, assessing neural networks and Gaussian processes for improved accuracy.

Findings

Control-based continuation captures both stable and unstable limit cycles.

UDE models demonstrate promising accuracy in nonlinear system modeling.

Identifies potential training issues and limitations of current frameworks.

Abstract

Augmenting mechanistic ordinary differential equation (ODE) models with machine-learnable structures is an 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.