Recently developed reduced-order modeling techniques aim to approximate nonlinear dynamical systems on low-dimensional manifolds learned from data. This is an effective approach for modeling dynamics in a post-transient regime where the effects of initial conditions and other disturbances have decayed. However, modeling transient dynamics near an underlying manifold, as needed for real-time control and forecasting applications, is complicated by the effects of fast dynamics and nonnormal sensitivity mechanisms. To begin to address these issues, we introduce a parametric class of nonlinear projections described by constrained autoencoder neural networks in which both the manifold and the projection fibers are learned from data. Our architecture uses invertible activation functions and biorthogonal weight matrices to ensure that the encoder is a left inverse of the decoder. We also introduce new dynamics-aware cost functions that promote learning of oblique projection fibers that account for fast dynamics and nonnormality. To demonstrate these methods and the specific challenges they address, we provide a detailed case study of a three-state model of vortex shedding in the wake of a bluff body immersed in a fluid, which has a two-dimensional slow manifold that can be computed analytically. In anticipation of future applications to high-dimensional systems, we also propose several techniques for constructing computationally efficient reduced-order models using our proposed nonlinear projection framework. This includes a novel sparsity-promoting penalty for the encoder that avoids detrimental weight matrix shrinkage via computation on the Grassmann manifold.

翻译：近期发展的降阶建模技术旨在基于从数据中学习的低维流形来逼近非线性动力系统。该方法对后瞬态阶段（初始条件及其他扰动效应已衰减）的动力学建模十分有效。然而，在实时控制与预测应用所需的底层流形附近对瞬态动力学进行建模时，快速动力学效应及非正规敏感性机制会带来复杂挑战。为解决这些问题，我们引入了一类由约束自编码器神经网络描述的参数化非线性投影，其中流形和投影纤维均从数据中学习。本架构采用可逆激活函数与双正交权重矩阵，确保编码器是解码器的左逆。同时，我们提出了新的动力学感知代价函数，以促进学习能够表征快速动力学与非正规性的斜投影纤维。为展示这些方法及其针对的特定挑战，我们以流体中浸没钝体尾流涡脱落的三个状态模型为详细案例，该模型具有可解析计算的二维慢流形。考虑到未来在高维系统中的应用，我们提出了若干利用所提非线性投影框架构建计算高效降阶模型的技术，其中包括一种针对编码器的创新稀疏促进惩罚项，该方法通过格拉斯曼流形上的计算避免有害的权重矩阵收缩。