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• A neural network enhanced Boltzmann model is proposed. • The mechanical and neural models are unified into a differentiable architecture and the neural- ODE-type training strategy is constructed. • A general numerical scheme is designed to solve the universal Boltzmann equation. • Numerical experiments of homogeneous and inhomogeneous systems are provided to validate the current method. One of the biggest challenges for simulating the Boltzmann equation is the evaluation of fivefold collision integral. Given the recent successes of deep learning and the availability of efficient tools, it is an obvious idea to try to substitute the calculation of the collision operator by the evaluation of a neural network. However, it is unlcear whether this preserves key properties of the Boltzmann equation, such as conservation, invariances, the H-theorem, and fluid-dynamic limits. In this paper, we present an approach that guarantees the conservation properties and the correct fluid dynamic limit at leading order. The concept originates from a recently developed scientific machine learning strategy which has been named "universal differential equations". It proposes a hybridization that fuses the deep physical insights from classical Boltzmann modeling and the desirable computational efficiency from neural network surrogates. The construction of the method and the training strategy are demonstrated in detail. We conduct an asymptotic analysis and illustrate the multi-scale applicability of the method. The numerical algorithm for solving the neural network-enhanced Boltzmann equation is presented as well. Several numerical test cases are investigated. The results of numerical experiments show that the time-series modeling strategy enjoys the training efficiency on this supervised learning task. [ABSTRACT FROM AUTHOR]

The present paper highlights the importance of management of the discontinuous reconstruction in the kinetic schemes for gasdynamic equation systems. Firstly, it is revealed by the analysis of the gas kinetic-BGK scheme [JCP 171 (2001) 289] that a continuous reconstruction created from a discontinuous one is a key to the successful kinetic schemes. When it is applied to a well-resolved region, the numerical flux that takes account of the collision effect correctly becomes Lax–Wendroff-like. When it is applied to an unresolved region, such as a shock layer, an appreciable numerical dissipation, which contributes to the suppression of spurious oscillations, is produced. Secondly, new kinetic schemes for the compressible Navier–Stokes (Euler) equations are developed by using the key. The numerical flux of one of the schemes is computed by using the splitting algorithm, where the effect of the molecular collision is directly taken into account and the undesirable error of the splitting algorithm in the case where the time step is much larger than the mean free time is avoided by a simple modification of the initial data. Although a discontinuous reconstruction is employed in the approximation of the initial data, the continuity is automatically taken into account in the dominant part of the numerical flux. The other schemes are the extensions of the Lax–Wendroff-type scheme to the case of the key reconstruction. Thirdly, the performance of these schemes is tested. It is demonstrated that they work as shock capturing schemes and yield fine boundary-layer profiles with a reasonable number of cells, such as 10 cells in the layer. [Copyright &y& Elsevier]