  Interacting individuals in complex systems often give rise to coherent motion
exhibiting coordinated global structures. Such phenomena are ubiquitously
observed in nature, from cell migration, bacterial swarms, animal and insect
groups, and even human societies. Primary mechanisms responsible for the
emergence of collective behavior have been extensively identified, including
local alignments based on average or relative velocity, non-local pairwise
repulsive-attractive interactions such as distance-based potentials, interplay
between local and non-local interactions, and cognitive-based inhomogeneous
interactions. However, discovering how to adapt these mechanisms to modulate
emergent behaviours remains elusive. Here, we demonstrate that it is possible
to generate coordinated structures in collective behavior at desired moments
with intended global patterns by fine-tuning an inter-agent interaction rule.
Our strategy employs deep neural networks, obeying the laws of dynamics, to
find interaction rules that command desired collective structures. The
decomposition of interaction rules into distancing and aligning forces,
expressed by polynomial series, facilitates the training of neural networks to
propose desired interaction models. Presented examples include altering the
mean radius and size of clusters in vortical swarms, timing of transitions from
random to ordered states, and continuously shifting between typical modes of
collective motions. This strategy can even be leveraged to superimpose
collective modes, resulting in hitherto unexplored but highly practical hybrid
collective patterns, such as protective security formations. Our findings
reveal innovative strategies for creating and controlling collective motion,
paving the way for new applications in robotic swarm operations, active matter
organisation, and for the uncovering of obscure interaction rules in biological
systems.
