Seung Yeon Cho

Seung Yeon Cho, Sungsu Park, Seok-Bae Yun

The Vlasov-Poisson-BGK (VPBGK) model is a kinetic model for describing the dynamics of collisional plasmas. Although various numerical schemes have been developed for it, a corresponding convergence theory has been absent. This paper fills this gap by presenting the first convergence analysis for a non-splitting, finite-difference Eulerian scheme discretized on the full phase-space grid. A major theoretical obstacle is the mixing of velocity indices induced by the electric field, which hinders the derivation of a uniform lower bound for the discrete solution. To overcome this stability challenge, we propose a modified lower bound estimate suitable for ionized systems that incorporates the step-wise degradation. Under a truncated velocity domain with a Neumann boundary condition, we establish error estimates for the distribution function in a weighted $L^{\infty}$ norm and for the electric field in a $L^{\infty}$ norm, respectively.

Gyounghun Ko, Sung-Jun Son, Seung Yeon Cho et al.

While Physics-Informed Neural Networks offer a promising framework for solving partial differential equations, the standard $L^2$ loss formulation is fundamentally insufficient when applied to the Bhatnagar-Gross-Krook (BGK) model. Specifically, simply minimizing the standard loss does not guarantee accurate predictions of the macroscopic moments, causing the approximate solutions to fail in capturing the true physical solution. To overcome this limitation, we introduce a velocity-weighted $L^2$ loss function designed to effectively penalize errors in the high-velocity regions. By establishing a stability estimate for the proposed approach, we shows that minimizing the proposed weighted loss guarantees the convergence of the approximate solution. Also, numerical experiments demonstrate that employing this weighted PINN loss leads to superior accuracy and robustness across various benchmarks compared to the standard approach.

Jaemin Oh, Seung Yeon Cho, Seok-Bae Yun et al.

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.