  Complex nonlinear dynamics are ubiquitous in many fields. Moreover, we rarely
have access to all of the relevant state variables governing the dynamics.
Delay embedding allows us, in principle, to account for unobserved state
variables. Here we provide an algebraic approach to delay embedding that
permits explicit approximation of error. We also provide the asymptotic
dependence of the first order approximation error on the system size. More
importantly, this formulation of delay embedding can be directly implemented
using a Recurrent Neural Network (RNN). This observation expands the
interpretability of both delay embedding and RNN and facilitates principled
incorporation of structure and other constraints into these approaches.
