Machine learning surrogate model for solving the Vlasov equation

The Vlasov equation provides a kinetic description of plasma dynamics in phase space. Solving the Vlasov equation is computationally expensive, often leading to the use of fluid models as alternatives. Fluid models describe macroscopic quantities like density, velocity, and pressure, while the Vlasov equation offers a microscopic perspective, capturing the detailed kinetic behavior of plasmas under the influence of electromagnetic fields. This allows for a more precise understanding of phenomena such as wave-particle interactions, instabilities, and nonlinear effects that are critical in plasma dynamics. The computational expense of solving the Vlasov equation and the challenges of traditional numerical methods in practical applications have motivated the exploration of alternative approaches that can balance accuracy and computational efficiency. Machine learning methods have the potential to overcome these challenges, and here we introduce the application of the Deep Operator Network (DeepONet) for solving the Vlasov equation. By training DeepONet on data generated from a Vlasov solver, it is able to output the precise solutions of the Vlasov equation, capturing the evolution of the plasma quantities such as density, pressure as well as electric field energy. This advancement holds significant potential for efficiently and accurately modeling plasma dynamics and advancing both theoretical and applied plasma physics research.