  The engineering design process often relies on mathematical modeling that can
describe the underlying dynamic behavior. In this work, we present a
data-driven methodology for modeling the dynamics of nonlinear systems. To
simplify this task, we aim to identify a coordinate transformation that allows
us to represent the dynamics of nonlinear systems using a common, simple model
structure. The advantage of a common simple model is that customized design
tools developed for it can be applied to study a large variety of nonlinear
systems. The simplest common model -- one can think of -- is linear, but linear
systems often fall short in accurately capturing the complex dynamics of
nonlinear systems. In this work, we propose using quadratic systems as the
common structure, inspired by the lifting principle. According to this
principle, smooth nonlinear systems can be expressed as quadratic systems in
suitable coordinates without approximation errors. However, finding these
coordinates solely from data is challenging. Here, we leverage deep learning to
identify such lifted coordinates using only data, enabling a quadratic
dynamical system to describe the system's dynamics. Additionally, we discuss
the asymptotic stability of these quadratic dynamical systems. We illustrate
the approach using data collected from various numerical examples,
demonstrating its superior performance with the existing well-known techniques.
