Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired
  Embeddings for Nonlinear Canonical Hamiltonian Dynamics
Pawan Goyal Süleyman Yıldız Peter Benner
Discovering a suitable coordinate transformation for nonlinear systems
enables the construction of simpler models, facilitating prediction, control,
and optimization for complex nonlinear systems. To that end, Koopman operator
theory offers a framework for global linearization for nonlinear systems,
thereby allowing the usage of linear tools for design studies. In this work, we
focus on the identification of global linearized embeddings for canonical
nonlinear Hamiltonian systems through a symplectic transformation. While this
task is often challenging, we leverage the power of deep learning to discover
the desired embeddings. Furthermore, to overcome the shortcomings of Koopman
operators for systems with continuous spectra, we apply the lifting principle
and learn global cubicized embeddings. Additionally, a key emphasis is paid to
enforce the bounded stability for the dynamics of the discovered embeddings. We
demonstrate the capabilities of deep learning in acquiring compact symplectic
coordinate transformation and the corresponding simple dynamical models,
fostering data-driven learning of nonlinear canonical Hamiltonian systems, even
those with continuous spectra.