This paper addresses the learning control problem for a group of robot manipulator systems in the presence of homogeneous nonlinear uncertain dynamics, where all the robots have an identical system structure but the reference signals to be tracked are different. Our control law has two objectives: one is tracking control, and the other is learning/identification. To this end, deterministic learning theory is incorporated with consensus theory to find a common neural network (NN) approximation of the nonlinear uncertain dynamics for the multi-robot system. Specifically, we first present a control scheme called cooperative deterministic learning using adaptive NNs to enable the robotic agents to track their respective reference trajectories on one hand, and to exchange their estimated NN weights online through networked communication on the other. As a result, a consensus about one common NN approximation for the nonlinear uncertain dynamics is achieved for all the agents. Thus, the trained distributed NNs have better generalization capability than those obtained by existing techniques. Numerical simulations on a team of two-degrees-of-freedom robot manipulators have been conducted to demonstrate the effectiveness of the proposed results.