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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 evaluation 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 its multi-scale applicability. 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., 20 pages, 11 figures

Compressible flows exhibit a diverse set of behaviors, where individual particle transports and their collective dynamics play different roles at different scales. At the same time, the atmosphere is composed of different components that require additional degrees of freedom for representation in computational fluid dynamics. It is challenging to construct an accurate and efficient numerical algorithm to faithfully represent multiscale flow physics across different regimes. In this paper, a unified gas-kinetic scheme (UGKS) is developed to study non-equilibrium multicomponent gaseous flows. Based on the Boltzmann kinetic equation, an analytical space-time evolving solution is used to construct the discretized equations of gas dynamics directly according to cell size and scales of time steps, i.e., the so-called direct modeling method. With the variation in the ratio of the numerical time step to the local particle collision time (or the cell size to the local particle mean free path), the UGKS automatically recovers all scale-dependent flows over the given domain and provides a continuous spectrum of the gas dynamics. The performance of the proposed unified scheme is fully validated through numerical experiments. The UGKS can be a valuable tool to study multiscale and multicomponent flow physics.

Plasmas present a diverse set of behaviors in different regimes. Given the intrinsic multi-scale nature of plasma dynamics, classical theoretical and numerical methods are often employed at separate scales with corresponding assumptions and approximations. Clearly, the coarse-grained modeling may introduce considerable uncertainties between the field solutions of flow and electromagnetic variables, and the real plasma physics. To study the emergence, propagation and evolution of randomness from gyrations of charged particles to magnetohydrodynamics poses great opportunities and challenges to develop both sound theories and reliable numerical algorithms. In this paper, a physics-oriented stochastic kinetic scheme will be developed that includes random inputs from both flow and electromagnetic fields via a hybridization of stochastic Galerkin and collocation methods. Based on the BGK-type relaxation model of the multi-component Boltzmann equation, a scale-dependent kinetic central-upwind flux function is designed in both physical and particle velocity space, and the governing equations in the discrete temporal-spatial-random domain are constructed. By solving Maxwell's equations with the wave-propagation method, the evolutions of ions, electrons and electromagnetic field are coupled throughout the simulation. We prove that the scheme is formally asymptotic-preserving in the Vlasov, magnetohydrodynamical, and neutral Euler regimes with the inclusion of random variables. Therefore, it can be used for the study of multi-scale and multi-physics plasma system under the effects of uncertainties, and provide scale-adaptive physical solutions under different ratios among numerical cell size, particle mean free path and gyroradius (or time step, local particle collision time and plasma period). Numerical experiments including one-dimensional Landau Damping, the two-stream instability and the Brio-Wu shock tube problem with one- to three-dimensional velocity settings, and each under stochastic initial conditions with one-dimensional uncertainty, will be presented to validate the scheme.

The gas dynamics under external force field is essentially associated with multiple scale nature due to the large variations of density and local Knudsen number. Single scale fluid dynamic equations, such as the Boltzmann and Navier-Stokes equations, are valid in their respective modeling scales, and it is challenging for the modeling and computation of a multiple scale problem across different regimes and capture the corresponding non-equilibrium flow physics. Based on the direct modeling of conservation laws in the discretized space, a well-balanced unified gas-kinetic scheme (UGKS) for multiscale flow transport under external force field has been developed and is used in the current study of non-equilibrium gaseous flow under external force field. With the variation of modeling scale, i.e., the cell size and time step, the UGKS is able to recover cross-scale flow physics from particle transport to hydrodynamic wave propagation. Theoretical analysis based on the kinetic model equation is presented to conceptually illustrate the effects of external force on the non-equilibrium heat transport., 36 pages, 17 figures, 2 tables, preprint

The gas dynamics under gravitational field is usually associated with the multiple scale nature due to large density variation and a wide range of local Knudsen number. It is chal- lenging to construct a reliable numerical algorithm to accurately capture the non-equilibrium physical effect in different regimes. In this paper, a well-balanced unified gas-kinetic scheme (UGKS) for all flow regimes under gravitational field will be developed, which can be used for the study of non-equilibrium gravitational gas system. The well-balanced scheme here is defined as a method to evolve an isolated gravitational system under any initial condition to an isothermal hydrostatic equilibrium state and to keep such a solution. To preserve such a property is important for a numerical scheme, which can be used for the study of slowly evolving gravitational system, such as the formation of star and galaxy. Based on the Boltzmann model with external forcing term, an analytic time evolving (or scale-dependent) solution is constructed to provide the corresponding dynamics in the cell size and time step scale, which is subsequently used in the construction of UGKS. As a result, with the varia- tion of the ratio between the numerical time step and local particle collision time, the UGKS is able to recover flow physics in different regimes and provides a continuum spectrum of gas dynamics. For the first time, the flow physics of a gravitational system in the transition regime can be studied using the UGKS, and the non-equilibrium phenomena in such a grav- itational system can be clearly identified. Many numerical examples will be used to validate the scheme. New physical observation, such as the correlation between the gravitational field and the heat flux in the transition regime, will be presented. The current method provides an indispensable tool for the study of non-equilibrium gravitational system., Comment: 21 pages, 9 figures, preprint