Learning latent representations in high-dimensional state spaces using polynomial manifold constructions
For researchers working on dimension reduction in high-dimensional dynamical systems, this method offers a simple way to incorporate nonlinearity, though it is incremental over existing polynomial-based approaches.
The paper introduces a framework for learning low-dimensional latent representations in high-dimensional state spaces by enriching linear approximations with polynomial terms to capture nonlinear interactions. Applied to the Korteweg-de Vries equation, the method reduces representation error by up to two orders of magnitude compared to linear techniques.
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.