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Recently developed reduced-order modeling techniques aim to approximatenonlinear dynamical systems on low-dimensional manifolds learned from data.This is an effective approach for modeling dynamics in a post-transient regimewhere the effects of initial conditions and other disturbances have decayed.However, modeling transient dynamics near an underlying manifold, as needed forreal-time control and forecasting applications, is complicated by the effectsof fast dynamics and nonnormal sensitivity mechanisms. To begin to addressthese issues, we introduce a parametric class of nonlinear projectionsdescribed by constrained autoencoder neural networks in which both the manifoldand the projection fibers are learned from data. Our architecture usesinvertible activation functions and biorthogonal weight matrices to ensure thatthe encoder is a left inverse of the decoder. We also introduce newdynamics-aware cost functions that promote learning of oblique projectionfibers that account for fast dynamics and nonnormality. To demonstrate thesemethods and the specific challenges they address, we provide a detailed casestudy of a three-state model of vortex shedding in the wake of a bluff bodyimmersed in a fluid, which has a two-dimensional slow manifold that can becomputed analytically. In anticipation of future applications tohigh-dimensional systems, we also propose several techniques for constructingcomputationally efficient reduced-order models using our proposed nonlinearprojection framework. This includes a novel sparsity-promoting penalty for theencoder that avoids detrimental weight matrix shrinkage via computation on theGrassmann manifold.

Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders