  We design a physics-aware auto-encoder to specifically reduce the
dimensionality of solutions arising from convection-dominated nonlinear
physical systems. Although existing nonlinear manifold learning methods seem to
be compelling tools to reduce the dimensionality of data characterized by a
large Kolmogorov n-width, they typically lack a straightforward mapping from
the latent space to the high-dimensional physical space. Moreover, the realized
latent variables are often hard to interpret. Therefore, many of these methods
are often dismissed in the reduced order modeling of dynamical systems governed
by the partial differential equations (PDEs). Accordingly, we propose an
auto-encoder type nonlinear dimensionality reduction algorithm. The
unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that
registers the output sequence of the PDEs on a non-uniform
parameter/time-varying grid, such that the Kolmogorov n-width of the mapped
data on the learned grid is minimized. We demonstrate the efficacy and
interpretability of our approach to separate convection/advection from
diffusion/scaling on various manufactured and physical systems.
