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

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.

Fixed-kinetic Neural Hamiltonian Flows for enhanced interpretability and reduced complexity

Normalizing Flows (NF) are Generative models which are particularly robust and allow for exact sampling of the learned distribution. They however require the design of an invertible mapping, whose Jacobian determinant has to be computable. Recently introduced, Neural Hamiltonian Flows (NHF) are based on Hamiltonian dynamics-based Flows, which are continuous, volume-preserving and invertible and thus make for natural candidates for robust NF architectures. In particular, their similarity to classical Mechanics could lead to easier interpretability of the learned mapping. However, despite being Physics-inspired architectures, the originally introduced NHF architecture still poses a challenge to interpretability. For this reason, in this work, we introduce a fixed kinetic energy version of the NHF model. Inspired by physics, our approach improves interpretability and requires less parameters than previously proposed architectures. We then study the robustness of the NHF architectures to the choice of hyperparameters. We analyze the impact of the number of leapfrog steps, the integration time and the number of neurons per hidden layer, as well as the choice of prior distribution, on sampling a multimodal 2D mixture. The NHF architecture is robust to these choices, especially the fixed-kinetic energy model. Finally, we adapt NHF to the context of Bayesian inference and illustrate our method on sampling the posterior distribution of two cosmological parameters knowing type Ia supernovae observations.