  Reduced models for the (defocusing) nonlinear Schr\"odinger equation are
developed. In particular, we develop reduced models that only involve the
low-frequency modes given noisy observations of these modes. The ansatz of the
reduced parametric models are obtained by employing a rational approximation
and a colored noise approximation, respectively, on the memory terms and the
random noise of a generalized Langevin equation that is derived from the
standard Mori-Zwanzig formalism. The parameters in the resulting reduced models
are inferred from noisy observations with a recently developed ensemble Kalman
filter-based parameterization method. The forecasting skill across different
temperature regimes are verified by comparing the moments up to order four, a
two-time correlation function statistics, and marginal densities of the
coarse-grained variables.
