  In the construction of reduced-order models for dynamical systems, linear
projection methods, such as proper orthogonal decompositions, are commonly
employed. However, for many dynamical systems, the lower dimensional
representation of the state space can most accurately be described by a
\textit{nonlinear} manifold. Previous research has shown that deep learning can
provide an efficient method for performing nonlinear dimension reduction,
though they are dependent on the availability of training data and are often
problem-specific \citep[see][]{carlberg_ca}. Here, we utilize randomized
training data to create and train convolutional autoencoders that perform
nonlinear dimension reduction for the wave and Kuramoto-Shivasinsky equations.
Moreover, we present training methods that are independent of full-order model
samples and use the manifold least-squares Petrov-Galerkin projection method to
define a reduced-order model for the heat, wave, and Kuramoto-Shivasinsky
equations using the same autoencoder.
