Heterogeneous mixtures of dictionary functions to approximate subspace invariance in Koopman operators
This work provides a theoretical explanation for the success of deep neural networks in learning Koopman operators, which is incremental but offers practical improvements in efficiency for dynamical systems modeling.
The paper tackled the problem of approximating Koopman operators for nonlinear dynamics by analyzing learned dictionaries from deep dynamic mode decomposition (deepDMD) and discovering a novel class of heterogeneous dictionary functions. The result showed that mixing these functions achieves the same accuracy and dimensional scaling as deepDMD with an order of magnitude reduction in parameters while maintaining interpretability.
Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a \textit{dictionary}. A widely used algorithm, is \textit{Extended Dynamic Mode Decomposition}, where the dictionary functions are drawn from a fixed, homogeneous class of functions. Recently, deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We discover a novel class of dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. We discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as deepDMD. This mixed dictionary does so with an order of magnitude reduction in parameters, while maintaining geometric interpretability. Our results provide a hypothesis to explain the success of deep neural networks in learning numerical approximations to Koopman operators.