Particle Swarm Optimisation is a classic algorithm from 1995. You scatter a swarm of points across the search space; each point updates its position using a fixed formula that mixes its own best-found location, the swarm’s collective best, and some inertia. The formula has been hand-tweaked for thirty years — variants for momentum, adaptive coefficients, special cases. What none of those variants do is let the form of the formula itself evolve.

The paper asks the obvious next question: what if we replace the hand-written velocity update with a velocity expression evolved by genetic programming? Start with random arithmetic expressions over the natural inputs (current velocity, distance to personal best, distance to swarm best). Score each candidate by how well a swarm using that expression converges on standard benchmark functions. Mutate winners, repeat. Out comes a velocity rule discovered, not designed.

Published at GECCO 2025 (DOI) — F.J.J.B. Santos, A. De Lorenzo, L. Manzoni, G. Pietropolli. The repository bundles the experimental framework, the evolved velocity expressions, and the analysis notebook. Kept here as the canonical reference for readers of the paper.