  In this paper, we introduce an uplifted reduced order modeling (UROM)
approach through the integration of standard projection based methods with long
short-term memory (LSTM) embedding. Our approach has three modeling layers or
components. In the first layer, we utilize an intrusive projection approach to
model dynamics represented by the largest modes. The second layer consists of
an LSTM model to account for residuals beyond this truncation. This closure
layer refers to the process of including the residual effect of the discarded
modes into the dynamics of the largest scales. However, the feasibility of
generating a low rank approximation tails off for higher Kolmogorov $n$-width
systems due to the underlying nonlinear processes. The third uplifting layer,
called super-resolution, addresses this limited representation issue by
expanding the span into a larger number of modes utilizing the versatility of
LSTM. Therefore, our model integrates a physics-based projection model with a
memory embedded LSTM closure and an LSTM based super-resolution model. In
several applications, we exploit the use of Grassmann manifold to construct
UROM for unseen conditions. We performed numerical experiments by using the
Burgers and Navier-Stokes equations with quadratic nonlinearity. Our results
show robustness of the proposed approach in building reduced order models for
parameterized systems and confirm the improved trade-off between accuracy and
efficiency.
