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In this study, we introduce a method based on Separable Physics-Informed Neural Networks (SPINNs) for effectively solving the BGK model of the Boltzmann equation. While the mesh-free nature of PINNs offers significant advantages in handling high-dimensional partial differential equations (PDEs), challenges arise when applying quadrature rules for accurate integral evaluation in the BGK operator, which can compromise the mesh-free benefit and increase computational costs. To address this, we leverage the canonical polyadic decomposition structure of SPINNs and the linear nature of moment calculation, achieving a substantial reduction in computational expense for quadrature rule application. The multi-scale nature of the particle density function poses difficulties in precisely approximating macroscopic moments using neural networks. To improve SPINN training, we introduce the integration of Gaussian functions into SPINNs, coupled with a relative loss approach. This modification enables SPINNs to decay as rapidly as Maxwellian distributions, thereby enhancing the accuracy of macroscopic moment approximations. The relative loss design further ensures that both large and small-scale features are effectively captured by the SPINNs. The efficacy of our approach is demonstrated through a series of five numerical experiments, including the solution to a challenging 3D Riemann problem. These results highlight the potential of our novel method in efficiently and accurately addressing complex challenges in computational physics.

In this work, we suggest a system of differential equations that quantitatively models the formulation and evolution of a trend cycle through the consideration of underlying dynamics between the trend participants. Our model captures the five stages of a trend cycle, namely, the onset, rise, peak, decline, and obsolescence. It also provides a unified mathematical criterion/condition to characterize the fad, fashion and classic. We prove that the solution of our model can capture various trend cycles. Numerical simulations are provided to show the expressive power of our model.

Most numerical schemes proposed for solving BGK models for rarefied gas dynamics are based on the discrete velocity approximation. Since such approach uses fixed velocity grids, one must secure a sufficiently large domain with fine velocity grids to resolve the structure of distribution functions. When one treats high Mach number problems, the computational cost becomes prohibitively expensive. In this paper, we propose a velocity adaptation technique in the semi-Lagrangian framework for BGK model. The velocity grid will be set locally in time and space, according to mean velocity and temperature. We apply a weighted minimization approach to impose conservation. We presented several numerical tests that illustrate the effectiveness of our proposed scheme.

Consistent BGK models for inert mixtures are compared, first in their kinetic behavior and then versus the hydrodynamic limits that can be derived in different collision-dominated regimes. The comparison is carried out both analytically and numerically, for the latter using an asymptotic preserving semi-Lagrangian scheme for the BGK models. Application to the plane shock wave in a binary mixture of noble gases is also presented.

In this work, we propose a class of high order semi-Lagrangian scheme for a general consistent BGK model for inert gas mixtures. The proposed scheme not only fulfills indifferentiability principle, but also asymptotic preserving property, which allows us to capture the behaviors of hydrodynamic limit models. We consider two hydrodynamic closure which can be derived from the BGK model at leading order: classical Euler equations for number densities, global velocity and temperature, and a multi-velocities and temperatures Euler system. Numerical simulations are performed to demonstrate indifferentiability principle and asymptotic preserving property of the proposed conservative semi-Lagrangian scheme to the Euler limits.

In this paper, we propose and analyse a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper., Comment: 38 pages

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [S. Y. Cho, et al., Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, 2020]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to rigid body rotation, Vlasov Poisson system and the BGK model of rarefied gas dynamics. In the first two cases operator splitting is adopted, to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model., Comment: 37 pages

In this paper, we propose a new semi-Lagrangian scheme for the polyatomic ellipsoidal BGK model. In order to avoid time step restrictions coming from convection term and small Knudsen number, we combine a semi-Lagrangian approach for the convection term with an implicit treatment for the relaxation term. We show how to explicitly solve the implicit step, thus obtaining an efficient and stable scheme for any Knudsen number. We also derive an explicit error estimate on the convergence of the proposed scheme for every fixed value of the Knudsen number., Comment: 40 pages

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests., Comment: pages 29

As a continuation of the study of the herding model proposed in (Bae et al. in arXiv:1712.01085, 2017), we consider in this paper the derivation of the kinetic version of the herding model, the existence of the measure-valued solution and the corresponding herding behavior at the kinetic level. We first consider the mean-field limit of the particle herding model and derive the existence of the measure-valued solutions for the kinetic herding model. We then study the herding phenomena of the solutions in two different ways by introducing two different types of herding energy functionals. First, we derive a herding phenomena of the measure-valued solutions under virtually no restrictions on the parameter sets using the Barbalat's lemma. We, however, don't get any herding rate in this case. On the other hand, we also establish a Gr\"onwall type estimate for another herding functional, leading to the exponential herding rate, under comparatively strict conditions. These results are then extended to smooth solutions., Comment: 24 pages

In this paper, we study the herding phenomena in financial markets arising from the combined effect of (1) non-coordinated collective interactions between the market players and (2) concurrent reactions of market players to dynamic market signals. By interpreting the expected rate of return of an asset and the favorability on that asset as position and velocity in phase space, we construct an agent-based particle model for herding behavior in finance. We then define two types of herding functionals using this model, and show that they satisfy a Gronwall type estimate and a LaSalle type invariance property respectively, leading to the herding behavior of the market players. Various numerical tests are presented to numerically verify these results., Comment: 31 pages