  A data-driven parametric model order reduction (MOR) method using a deep
artificial neural network is proposed. The present network, which is the
least-squares hierarchical variational autoencoder (LSH-VAE), is capable of
performing nonlinear MOR for the parametric interpolation of a nonlinear
dynamic system with a significant number of degrees of freedom. LSH-VAE
exploits two major changes to the existing networks: a hierarchical deep
structure and a hybrid weighted, probabilistic loss function. The enhancements
result in a significantly improved accuracy and stability compared against the
conventional nonlinear MOR methods, autoencoder, and variational autoencoder.
Upon LSH-VAE, a parametric MOR framework is presented based on the spherically
linear interpolation of the latent manifold. The present framework is validated
and evaluated on three nonlinear and multiphysics dynamic systems. First, the
present framework is evaluated on the fluid-structure interaction benchmark
problem to assess its efficiency and accuracy. Then, a highly nonlinear
aeroelastic phenomenon, limit cycle oscillation, is analyzed. Finally, the
present framework is applied to a three-dimensional fluid flow to demonstrate
its capability of efficiently analyzing a significantly large number of degrees
of freedom. The performance of LSH-VAE is emphasized by comparing its results
against that of the widely used nonlinear MOR methods, convolutional
autoencoder, and $\beta$-VAE. The present framework exhibits a significantly
enhanced accuracy to the conventional methods while still exhibiting a large
speed-up factor.
