  Advection-dominated problems are commonly noticed in nature, engineering
systems, and a wide range of industrial processes. For these problems, linear
approximation methods (proper orthogonal decomposition and reduced basis
method) are not suitable, as the Kolmogorov $n$-width decay is slow, leading to
inefficient and inaccurate reduced order models. There are few non-linear
approaches to accelerate the Kolmogorov $n$-width decay. In this work, we use a
neural-network shift augmented transformation technique, that employs
automatic-shit detection and detects the optimal non-linear transformation of
the full-order model solution manifold $\mathcal{M}$. We exploit a
deep-learning framework to derive parameter-dependent bijective mapping between
the manifold $\mathcal{M}$ and the transformed manifold $\tilde{\mathcal{M}}$.
It consists of two neural networks, 1) ShiftNet, to employ automatic-shift
detection by learning the shift-operator, which finds the optimal shifts for
numerous snapshots of the full-order solution manifold, to accelerate the
Kolmogorov $n$-width decay, and 2) InterpNet, which learns the reference
configuration and can reconstruct the field values of the same, for each
shifted grid distribution. We construct non-intrusive reduced order models on
the resulting transformed linear subspaces and employ automatic-shift detection
for predictions. We test our methodology on advection-dominated problems, such
as 1D travelling waves, 2D isentropic convective vortex and 2D two-phase flow
test cases. This work leads to the development of the complete NNsPOD-ROM
algorithm for model reduction of advection-dominated problems, comprising both
offline-online stages.
