Data discovery of low dimensional fluid dynamics of turbulent flows

Discovering governing equations from data, in particular high dimensionaldata, is challenging in various fields of science and engineering, and it haspotential to revolutionise the science and technology in this big data era.This paper combines sparse identification and deep learning with non-linearfluid dynamics, in particular the turbulent flows, to discover governingequations of nonlinear fluid dynamics in the lower nonlinear manifold space.The autoencoder deep neural network is used to project the high dimensionalspace into a lower dimensional nonlinear manifold space. The Proper OrthogonalDecomposition (POD) is then used to stabilise the nonlinear manifold space inorder to guarantee a stable manifold space for pattern or equations discoveryfor the highly nonlinear problems such as turbulent flows. Sparse regression isthen used to discover the lower dimensional governing equations of fluiddynamics in the lower dimensional nonlinear manifold space. What distinguishesthis approach is its ability to discover a lower dimensional governingequations of fluid dynamics in the nonlinear manifold space. We demonstratethis method on a number of high-dimensional fluid dynamic systems such as lockexchange, flow past one and two cylinders. The results demonstrate that theresulting method is capable of discovering lower dimensional governingequations that took researchers in this community many decades years toresolve. In addition, this model discovers dynamics in a lower dimensionalmanifold space, thus leading to great computational efficiency, modelcomplexity and avoiding overfitting. It also provides a new insight for ourunderstanding of sciences such as turbulent flows.