  This work addresses model order reduction for complex moving fronts, which
are transported by advection or through a reaction-diffusion process. Such
systems are especially challenging for model order reduction since the
transport cannot be captured by linear reduction methods. Moreover, topological
changes, such as splitting or merging of fronts pose difficulties for many
nonlinear reduction methods and the small non-vanishing support of the
underlying partial differential equations dynamics makes most nonlinear
hyper-reduction methods infeasible. We propose a new decomposition method
together with a hyper-reduction scheme that addresses these shortcomings. The
decomposition uses a level-set function to parameterize the transport and a
nonlinear activation function that captures the structure of the front. This
approach is similar to autoencoder artificial neural networks, but additionally
provides insights into the system, which can be used for efficient reduced
order models. We make use of this property and are thus able to solve the
advection equation with the same complexity as the POD-Galerkin approach while
obtaining errors of less than one percent for representative examples.
Furthermore, we outline a special hyper-reduction method for more complicated
advection-reaction-diffusion systems. The capability of the approach is
illustrated by various numerical examples in one and two spatial dimensions,
including real-life applications to a two-dimensional Bunsen flame.
