The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of existing methods for data-driven discovery, e.g. extended dynamic mode decomposition. This requires a combinatorially large basis set to adequately describe many nonlinear systems of interest. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a deep learning framework for learning Koopman operators of nonlinear dynamical systems, known as deep dynamic mode decomposition (deep DMD). We show that this novel method automatically selects efficient deep dictionaries, outperforming state-of-the-art methods. Furthermore, we show that a class of basis functions discovered using deep DMD exhibit properties of invariance, which provide a mathematical explanation for dictionary closure under properties of differentiation, multiplication, and addition. This finding illustrates that specific bases exhibit a closure property that enables state-inclusion and ultimately state recovery for direct analysis of the underlying physical system. We illustrate the method with several simulation systems and the applicability of deepDMD for discovering operational envelopes for synthetic biological circuits.
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