# Eulerian-Lagrangian Fluid Simulation on Particle Flow Maps

arXiv: https://arxiv.org/abs/2405.09672 · HTML: https://mlsanity.com/paper/2405.09672/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2024.

## Built on by (most cited first, 4 of 4)

- [Generative Modeling with Orbit-Space Particle Flow Matching](https://mlsanity.com/paper/2605.02222/) (2026)
- [SmokeSVD: Smoke Reconstruction from A Single View via Progressive Novel View Synthesis and Refinement with Diffusion Models](https://mlsanity.com/paper/2507.12156/) (2025)
- [A Few-Step Generative Model on Cumulative Flow Maps](https://mlsanity.com/paper/2605.03623/) (2026)
- [Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs](https://mlsanity.com/paper/2605.24774/) (2026)

## Abstract (arXiv)

We propose a novel Particle Flow Map (PFM) method to enable accurate long-range advection for incompressible fluid simulation. The foundation of our method is the observation that a particle trajectory generated in a forward simulation naturally embodies a perfect flow map. Centered on this concept, we have developed an Eulerian-Lagrangian framework comprising four essential components: Lagrangian particles for a natural and precise representation of bidirectional flow maps; a dual-scale map representation to accommodate the mapping of various flow quantities; a particle-to-grid interpolation scheme for accurate quantity transfer from particles to grid nodes; and a hybrid impulse-based solver to enforce incompressibility on the grid. The efficacy of PFM has been demonstrated through various simulation scenarios, highlighting the evolution of complex vortical structures and the details of turbulent flows. Notably, compared to NFM, PFM reduces computing time by up to 49 times and memory consumption by up to 41%, while enhancing vorticity preservation as evidenced in various tests like leapfrog, vortex tube, and turbulent flow.