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

arXiv: https://arxiv.org/abs/2505.04471 · HTML: https://mlsanity.com/paper/2505.04471/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 41 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 18 of 18)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [CaloGAN: Simulating 3D High Energy Particle Showers in Multi-Layer Electromagnetic Calorimeters with Generative Adversarial Networks](https://mlsanity.com/paper/1712.10321/) (2017)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)
- [Fast cosmic web simulations with generative adversarial networks](https://mlsanity.com/paper/1801.09070/) (2018)
- [Self Normalizing Flows](https://mlsanity.com/paper/2011.07248/) (2020)
- [A Deep Learning-Based Particle-in-Cell Method for Plasma Simulations](https://mlsanity.com/paper/2107.02232/) (2021)
- [Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation](https://mlsanity.com/paper/2112.01438/) (2021)
- [Fixed-kinetic Neural Hamiltonian Flows for enhanced interpretability and reduced complexity](https://mlsanity.com/paper/2302.01955/) (2023)

## Built on by (most cited first, 1 of 1)

- [Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency](https://mlsanity.com/paper/2605.08454/) (2026)

## Abstract (arXiv)

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.