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：近年来发展的降阶建模技术旨在利用从数据中学习的低维流形近似非线性动力系统。对于初始条件及其他扰动效应已衰减的瞬态后阶段动力学建模，这是一种有效方法。然而，在实时控制与预测应用中需对底层流形附近的瞬态动力学进行建模，但快速动力学与非正态敏感性机制使得该过程复杂化。为解决这些问题，我们引入了一类由约束自编码神经网络描述的非线性投影参数化方法，其中流形与投影纤维均从数据中学习。该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。我们还引入了考虑快速动力学与非正态性的新型动力学感知代价函数，以促进倾斜投影纤维的学习。通过浸没流体中钝体尾迹涡脱落三状态模型（该模型具有可解析计算的二维慢流形）的详细案例研究，验证了所提方法及其针对的特定挑战。为应对未来高维系统的应用需求，我们进一步提出了利用所提非线性投影框架构建高效降阶模型的多种技术，包括一种基于格拉斯曼流形计算的编码器稀疏性惩罚机制，该机制可避免有害的权重矩阵收缩。