# Velocity Field Generation for Density Control of Swarms using Heat Equation and Smoothing Kernels

arXiv: https://arxiv.org/abs/1710.00299 · HTML: https://mlsanity.com/paper/1710.00299/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2017.

## Built on by (most cited first, 5 of 5)

- [A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling](https://mlsanity.com/paper/2202.12927/) (2022)
- [Scalable Task-Driven Robotic Swarm Control via Collision Avoidance and Learning Mean-Field Control](https://mlsanity.com/paper/2209.07420/) (2022)
- [A Scalable Reinforcement Learning Approach for Attack Allocation in Swarm to Swarm Engagement Problems](https://mlsanity.com/paper/2210.08319/) (2022)
- [Scalable Planning and Learning Framework Development for Swarm-to-Swarm Engagement Problems](https://mlsanity.com/paper/2212.02909/) (2022)
- [Maximum Principle of Optimal Probability Density Control](https://mlsanity.com/paper/2505.18362/) (2025)

## Abstract (arXiv)

This paper presents a method to control the probability density distribution of a swarm of vehicles via velocity fields. The proposed approach synthesizes smooth velocity fields, which specify a desired velocity as a function of time and position in a decentralized manner i.e., each agent calculates the desired velocity locally by utilizing the number of agents within a prescribed communication distance. Swarm converges to the desired/commanded density distribution by following the velocity field. The local information consists, only, of agents' positions and it is utilized to estimate the density around each agent. Local density estimation is performed by using kernel density estimation. Then the local density estimates and the desired density are utilized to generate the velocity field, which propagates the swarm probability density distribution via the well-known heat equation. The key advantage of using smooth velocity fields to control swarm density with respect to our earlier Markov chain based probabilistic control methods is that the agents move more smoothly in an organized manner and their relative velocities go to zero as they get closer to each other, i.e., they facilitate conflict/collision avoidance. The desired density distribution, which is commanded to the local controllers, can be computed by using a Markov chain that propagates the desired density distribution within prescribed requirements. Along with convergence and stability analysis, the effectiveness of the approach is illustrated via numerical examples.