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.

最近开发的降阶建模技术旨在从数据中学习的低维流形上近似非线性动力系统。我们介绍了一种由约束的自动编码器神经网络描述的参数化非线性投影类，其中流形和投影纤维都是从数据中学习得到的。此外，我们还提出了一些新的动力学感知成本函数，以促进学习考虑快速动力学和非正常性的斜投影纤维。为了展示这些方法及其解决的特定挑战，我们提供了一个关于涡街现象的三状态模型的详细案例研究。同时，我们还提出了几种基于我们提出的非线性投影框架构建计算高效的降阶模型的技术。这包括一种用于避免计算 Grassmann 流形上有害的权重矩阵收缩的新型稀疏促进惩罚项的编码器。