  Discovering governing equations from data, in particular high dimensional
data, is challenging in various fields of science and engineering, and it has
potential to revolutionise the science and technology in this big data era.
This paper combines sparse identification and deep learning with non-linear
fluid dynamics, in particular the turbulent flows, to discover governing
equations of nonlinear fluid dynamics in the lower nonlinear manifold space.
The autoencoder deep neural network is used to project the high dimensional
space into a lower dimensional nonlinear manifold space. The Proper Orthogonal
Decomposition (POD) is then used to stabilise the nonlinear manifold space in
order to guarantee a stable manifold space for pattern or equations discovery
for the highly nonlinear problems such as turbulent flows. Sparse regression is
then used to discover the lower dimensional governing equations of fluid
dynamics in the lower dimensional nonlinear manifold space. What distinguishes
this approach is its ability to discover a lower dimensional governing
equations of fluid dynamics in the nonlinear manifold space. We demonstrate
this method on a number of high-dimensional fluid dynamic systems such as lock
exchange, flow past one and two cylinders. The results demonstrate that the
resulting method is capable of discovering lower dimensional governing
equations that took researchers in this community many decades years to
resolve. In addition, this model discovers dynamics in a lower dimensional
manifold space, thus leading to great computational efficiency, model
complexity and avoiding overfitting. It also provides a new insight for our
understanding of sciences such as turbulent flows.
