We introduce a deep neural network-based numerical method for solving kinetic Fokker Planck equations, including both linear and nonlinear cases. Building upon the conservative dissipative structure of Vlasov-type equations, we formulate a class of generalized minimizing movement schemes as iterative constrained minimization problems: the conservative part determines the constraint set, while the dissipative part defines the objective functional. This leads to an analog of the classical Jordan-Kinderlehrer-Otto (JKO) scheme for Wasserstein gradient flows, and we refer to it as the kinetic JKO scheme. To compute each step of the kinetic JKO iteration, we introduce a particle-based approximation in which the velocity field is parameterized by deep neural networks. The resulting algorithm can be interpreted as a kinetic-oriented neural differential equation that enables the representation of high-dimensional kinetic dynamics while preserving the essential variational and structural properties of the underlying PDE. We validate the method with extensive numerical experiments and demonstrate that the proposed kinetic JKO-neural ODE framework is effective for high-dimensional numerical simulations.
Discrete hypocoercive estimates for discontinuous Galerkin methods: application to the Vlasov-Poisson-Fokker-Planck system
Xiamen University（厦门大学）；Centre Inria de l’Université de Lille（里尔大学Inria中心）；University of Science and Technology of China（中国科学技术大学）；Institut de Mathématiques de Toulouse, Université de Toulouse（图卢兹数学研究所，图卢兹大学）
