Hamiltonian Normalizing Flows as kinetic PDE solvers: application to the 1D Vlasov-Poisson Equations

Abstract

Many conservative physical systems can be described using the Hamiltonian formalism. A notable example is the Vlasov-Poisson equations, a set of partial differential equations that govern the time evolution of a phase-space density function representing collisionless particles under a self-consistent potential. These equations play a central role in both plasma physics and cosmology. Due to the complexity of the potential involved, analytical solutions are rarely available, necessitating the use of numerical methods such as Particle-In-Cell. In this work, we introduce a novel approach based on Hamiltonian-informed Normalizing Flows, specifically a variant of Fixed-Kinetic Neural Hamiltonian Flows. Our method transforms an initial Gaussian distribution in phase space into the final distribution using a sequence of invertible, volume-preserving transformations derived from Hamiltonian dynamics. The model is trained on a dataset comprising initial and final states at a fixed time T, generated via numerical simulations. After training, the model enables fast sampling of the final distribution from any given initial state. Moreover, by automatically learning an interpretable physical potential, it can generalize to intermediate states not seen during training, offering insights into the system's evolution across time.