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Discovering a suitable coordinate transformation for nonlinear systemsenables the construction of simpler models, facilitating prediction, control,and optimization for complex nonlinear systems. To that end, Koopman operatortheory offers a framework for global linearization for nonlinear systems,thereby allowing the usage of linear tools for design studies. In this work, wefocus on the identification of global linearized embeddings for canonicalnonlinear Hamiltonian systems through a symplectic transformation. While thistask is often challenging, we leverage the power of deep learning to discoverthe desired embeddings. Furthermore, to overcome the shortcomings of Koopmanoperators for systems with continuous spectra, we apply the lifting principleand learn global cubicized embeddings. Additionally, a key emphasis is paid toenforce the bounded stability for the dynamics of the discovered embeddings. Wedemonstrate the capabilities of deep learning in acquiring compact symplecticcoordinate transformation and the corresponding simple dynamical models,fostering data-driven learning of nonlinear canonical Hamiltonian systems, eventhose with continuous spectra.

Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired Embeddings for Nonlinear Canonical Hamiltonian Dynamics