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Data-driven approximations of the Koopman operator are promising forpredicting the time evolution of systems characterized by complex dynamics.Among these methods, the approach known as extended dynamic mode decompositionwith dictionary learning (EDMD-DL) has garnered significant attention. Here wepresent a modification of EDMD-DL that concurrently determines both thedictionary of observables and the corresponding approximation of the Koopmanoperator. This innovation leverages automatic differentiation to facilitategradient descent computations through the pseudoinverse. We also address theperformance of several alternative methodologies. We assess a 'pure' Koopmanapproach, which involves the direct time-integration of a linear,high-dimensional system governing the dynamics within the space of observables.Additionally, we explore a modified approach where the system alternatesbetween spaces of states and observables at each time step -- this approach nolonger satisfies the linearity of the true Koopman operator representation. Forfurther comparisons, we also apply a state space approach (neural ODEs). Weconsider systems encompassing two and three-dimensional ordinary differentialequation systems featuring steady, oscillatory, and chaotic attractors, as wellas partial differential equations exhibiting increasingly complex and intricatebehaviors. Our framework significantly outperforms EDMD-DL. Furthermore, thestate space approach offers superior performance compared to the 'pure' Koopmanapproach where the entire time evolution occurs in the space of observables.When the temporal evolution of the Koopman approach alternates between statesand observables at each time step, however, its predictions become comparableto those of the state space approach.

Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches