Fish in schooling formations navigate complex flow fields replete with mechanical energy in the vortex wakes of their companions. Their schooling behavior has been associated with evolutionary advantages including energy savings, yet the underlying physical mechanisms remain unknown. We show that fish can improve their sustained propulsive efficiency by placing themselves in appropriate locations in the wake of other swimmers and intercepting judiciously their shed vortices. This swimming strategy leads to collective energy savings and is revealed through a combination of high-fidelity flow simulations with a deep reinforcement learning (RL) algorithm. The RL algorithm relies on a policy defined by deep, recurrent neural nets, with long-short-term memory cells, that are essential for capturing the unsteadiness of the two-way interactions between the fish and the vortical flow field. Surprisingly, we find that swimming in-line with a leader is not associated with energetic benefits for the follower. Instead, "smart swimmer(s)" place themselves at off-center positions, with respect to the axis of the leader(s) and deform their body to synchronize with the momentum of the oncoming vortices, thus enhancing their swimming efficiency at no cost to the leader(s). The results confirm that fish may harvest energy deposited in vortices and support the conjecture that swimming in formation is energetically advantageous. Moreover, this study demonstrates that deep RL can produce navigation algorithms for complex unsteady and vortical flow fields, with promising implications for energy savings in autonomous robotic swarms.

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We study the fluid dynamics of two fish-like bodies with synchronised swimming patterns. Our studies are based on two-dimensional simulations of viscous incompressible flows. We distinguish between motion patterns that are externally imposed on the swimmers and self-propelled swimmers that learn manoeuvres to achieve certain goals. Simulations of two rigid bodies executing pre-specified motion indicate that flow-mediated interactions can lead to substantial drag reduction and may even generate thrust intermittently. In turn we examine two self-propelled swimmers arranged in a leader-follower configuration, with a-priori specified body-deformations. We find that the swimming of the leader remains largely unaffected, while the follower experiences either an increase or decrease in swimming speed, depending on the initial conditions. Finally, we consider a follower that synchronises its motion so as to minimize its lateral deviations from the leader's path. The leader employs a steady gait while the follower uses a reinforcement learning algorithm to adapt its swimming-kinematics. We find that swimming in a synchronised tandem can yield up to about 30% reduction in energy expenditure for the follower, in addition to a 20% increase in its swimming-efficiency. The present results indicate that synchronised swimming of two fish can be energetically beneficial.

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