Chapter 3: Identifying Nonlinear Dynamics
Jason J. Bramburger · Society for Industrial and Applied Mathematics eBooks · 2024
In the previous chapter we saw that DMD and its variants produce linear models to forecast and interpret nonlinear dynamics. Although DMD is a powerful method in data analysis and dynamical systems, the models it produces often lack the ability to generalize beyond the training set. To see this for yourself, use the DMD matrix from Example 2.1 to forecast a different initial state of the NLS (2.8), and compare it to the result from integrating the system with the same initial condition. This will almost certainly result in a DMD prediction that quickly diverges from the true dynamics of the system. This should be expected since the NLS is a nonlinear equation, while DMD produces a linear system based only on the training data given to it. To produce models that can forecast and interpret outside of the training set, we require nonlinearity.