Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding

This letter presents a Koopman lifting linearization method that is applicable to nonlinear dynamical systems having both stable and unstable regions. It is known that Dynamic Mode Decomposition (DMD) and its extended methods are often unable to model unstable systems accurately and reliably. Here we solve the problem through merging three methodologies: decomposition of a lifted linear system into stable and unstable modes, deep learning of a dictionary of observable functions in the separated subspaces, and a new formula for obtaining the Koopman operator, called Direct Encoding. Two sets of effective observable functions are obtained through neural net training where the training data are separated into stable and unstable trajectories. The resultant learned observables are used for lifting the state space, and a linear state transition matrix is constructed using Direct Encoding where inner products of the learned observables are computed. The proposed method shows a dramatic improvement over existing DMD and data-driven methods. Furthermore, a method is developed for determining the boundaries between stable and unstable regions.