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

TL;DR

The paper addresses efficient simulation of Hamiltonian kinetic PDEs, notably the Vlasov–Poisson system, by learning a probabilistic map from initial phase-space distributions to final distributions over a fixed time horizon. It introduces PDE-NHF, a physics-informed Hamiltonian normalizing flow with a fixed kinetic term and a self-consistent potential, integrated via a Leapfrog scheme to preserve the underlying symplectic structure. Empirically, PDE-NHF yields accurate final distributions and interpolates intermediate states, outperforming a baseline flow model in 1D VP while enabling acceleration through fewer Leapfrog steps; the learned Hamiltonian parameter converges toward the true unit value, evidencing correct physics recovery. The approach provides a fast, interpretable surrogate for Hamiltonian dynamics with broad potential applications to higher-dimensional plasmas, cosmology, and gravitational systems, where efficient trajectory-free sampling is valuable.

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.

Figures (2)

• Figure 1: Schematic representation of PDE-NHF.
• Figure 2: Normalized cumulative histograms from the true and the sampled trajectory in phase-space for one example from the test dataset. We used $L=25$ and $\delta t=0.04$ for both training and sampling.