We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

翻译：本文提出一种利用非线性流形学习动力系统降阶模型的新方法。首先，通过通用表示学习问题识别数据中的非线性结构来学习流形。该方法采用低阶多项式形式的嵌入进行驱动。将数据投影至非线性流形可揭示控制所研究问题的降阶空间系统的代数结构，进而通过算子推断从数据中推演出降阶模型的矩阵算子。针对多个非线性问题的数值实验表明，该方法具有良好泛化能力，且相较于采用线性子空间近似的降阶建模方法可获得更高的精度。