Nonlinear Oblique Projections for Reduced-Order Modeling using Constrained Autoencoders

Reduced-order modeling techniques based on linear projections are known to provide inefficient dimensionality reduction for systems that evolve near highly curved nonlinear manifolds. We introduce nonlinear projections parameterized by autoencoders, constrained so that composing of the encoder with the decoder is the identity. In existing techniques, the full-order model's time derivative is orthogonally projected onto the tangent space of the nonlinear manifold and is defined only by decoder. In this work, we argue the direction of projection is also important in capturing the dynamics. This is especially true in non-normal systems such as shear-dominated fluid flows. To address this problem, we additionally use the encoder and its derivative to determine the direction of projection which in general may be oblique. Furthermore, we optimize the network using cost functions that balance the reconstruction accuracy of states via the autoencoder against the sensitivity of future outputs of the system to state perturbations. We demonstrate this method on several examples, including a three-dimensional model of flow past a circular cylinder and the complex Ginzburg-Landau equation.