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.

翻译：我们提出了一种新颖框架，通过非线性降维技术解决高维状态空间问题中的低成本隐式表征学习。通过在线性状态近似中引入低阶多项式项，我们捕获了数据中存在的关键非线性相互作用，从而降低了问题的本征维度。本文介绍了两种方法用于学习此类低维多项式流形的表征以嵌入数据。流形参数化系数可通过基于本征正交分解或交替最小化方法的回归求解。数值实验聚焦于一维Korteweg-de Vries方程，结果表明，与线性降维技术相比，考虑数据中的非线性相关性可将表征误差降低两个数量级。