Uniformly accurate machine learning-based hydrodynamic models for kinetic equations

Significance This paper addresses 2 very important issues of current interest: multiscale modeling in the absence of scale separation and building interpretable and truly reliable physical models using machine learning. We demonstrate that machine learning can indeed help us to build reliable multiscale models for problems with which classical multiscale methods have had trouble. To this end, one has to develop the appropriate models or algorithms for each of the 3 major components in the machine-learning procedure: labeling the data, learning from the data, and exploring the state space. We use the kinetic equation as an example and demonstrate that uniformly accurate moment systems can be constructed this way. A framework is introduced for constructing interpretable and truly reliable reduced models for multiscale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active-learning procedure is introduced to help ensure that the dataset used is representative enough. The reduced system takes the form of a conventional moment system and works regardless of the numerical discretization used. Numerical results are presented for the BGK (Bhatnagar–Gross–Krook) model and binary collision of Maxwell molecules. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow.