  We present a novel framework for learning cost-efficient latent
representations in problems with high-dimensional state spaces through
nonlinear dimension reduction. By enriching linear state approximations with
low-order polynomial terms we account for key nonlinear interactions existing
in the data thereby reducing the problem's intrinsic dimensionality. Two
methods are introduced for learning the representation of such low-dimensional,
polynomial manifolds for embedding the data. The manifold parametrization
coefficients can be obtained by regression via either a proper orthogonal
decomposition or an alternating minimization based approach. Our numerical
results focus on the one-dimensional Korteweg-de Vries equation where
accounting for nonlinear correlations in the data was found to lower the
representation error by up to two orders of magnitude compared to linear
dimension reduction techniques.
