Data Discovery of Lower Dimensional Equations of Turbulent Flows

Discovering equations from data, particularly high-dimensional data, is challenging in various fields of science and engineering and has the potential to revolutionize science and technology. This paper presents a new non-intrusive reduced-order modelling (NIROM) method to discover a lower-dimensional version of the equations of fluids from the data. Unlike Navier–Stokes, these equations have a lower dimensional size and are easy to solve. This method provides a different perspective for understanding fluid dynamics, particularly turbulent flows. In this method, the autoencoder deep neural network is used to project the high-dimensional space into a lower-dimensional nonlinear manifold space to find the latent dynamics. 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 equation discovery for highly nonlinear problems such as turbulent flows. Sparse regression is then used to discover the low-dimensional equations of fluid dynamics in the latent nonlinear manifold space. What distinguishes this approach is its ability to discover low-dimensional equations of fluid dynamics in the nonlinear manifold space. We demonstrate this method in several high-dimensional complex fluid dynamic systems, such as lock exchange and two cylinders. The results demonstrate that the resulting method is capable of discovering lower-dimensional equations that researchers in this community took many decades 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 new insight into our understanding of sciences such as turbulent flows.