Explore missing flow dynamics by physics-informed deep learning: The parameterized governing systems

Gaining and understanding flow dynamics have much importance in a wide range of disciplines, e.g., astrophysics, geophysics, biology, mechanical engineering, and biomedical engineering. For turbulent flows, local flow information such as velocity and its statistics, can be measured experimentally. Due to the poor fidelity or experimental limitations, some information may not be resolved in a region of interest. On the other hand, detailed flow features are described by the governing equations, e.g., the Navier–Stokes equations for viscous fluid, and can be resolved numerically, which is heavily dependent on the capability of either computing resources or modeling. Alternatively, we address this problem by employing the physics-informed deep learning and treat the governing equations as a parameterized constraint to recover the missing flow dynamics. We demonstrate that with limited data, no matter from experiment or others, the flow dynamics in the region where the required data are missing or not measured, can be reconstructed with the parameterized governing equations. Meanwhile, a richer dataset, with spatial distribution of the control parameter (e.g., eddy viscosity of turbulence modelings), can be obtained. The method provided in this paper may shed light on the data-driven scale-adaptive turbulent structure recovering and understanding of complex fluid physics and can be extended to other parameterized governing systems beyond fluid mechanics.