Sangmin Lee

Sangmin Lee, Byeongsu Sim, Jong Chul Ye

To understand learning the dynamics of deep ReLU networks, we investigate the dynamic system of gradient flow $w(t)$ by decomposing it to magnitude $w(t)$ and angle $φ(t):= π- θ(t) $ components. In particular, for multi-layer single ReLU neurons with spherically symmetric data distribution and the square loss function, we provide upper and lower bounds for magnitude and angle components to describe the dynamics of gradient flow. Using the obtained bounds, we conclude that small scale initialization induces slow convergence speed for deep single ReLU neurons. Finally, by exploiting the relation of gradient flow and gradient descent, we extend our results to the gradient descent approach. All theoretical results are verified by experiments.

SiHun Lee, Sangmin Lee, Kijoo Jang et al.

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 $β$-VAE. The present framework exhibits a significantly enhanced accuracy to the conventional methods while still exhibiting a large speed-up factor.