Data-driven discovery of governing equations for fluid dynamics based on molecular simulation

Abstract

The discovery of governing equations from data is revolutionizing the development of some research fields, where the scientific data are abundant but the well-characterized quantitative descriptions are probably scarce. In this work, we propose to combine the direct simulation Monte Carlo (DSMC) method, which is a popular molecular simulation tool for gas flows, and machine learning to discover the governing equations for fluid dynamics. The DSMC method does not assume any macroscopic governing equations a priori but just relies on the model of molecular interactions at the microscopic level. The data generated by DSMC are utilized to derive the underlying governing equations using a sparse regression method proposed recently. We demonstrate that this strategy is capable of deriving a variety of equations in fluid dynamics, such as the momentum equation, diffusion equation, Fokker–Planck equation and vorticity transport equation. The data-driven discovery not only provides the right forms of the governing equations, but also determines accurate values of the transport coefficients such as viscosity and diffusivity. This work proves that data-driven discovery combined with molecular simulations is a promising and alternative method to derive governing equations in fluid dynamics, and it is expected to pave a new way to establish the governing equations of non-equilibrium flows and complex fluids.