Data Discovery of Lower Dimensional Equations of Turbulent Flows

ABSTRACT 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.