In this paper, we investigate the approximation of completely resonant nonlinear wave systems via deter- ministic learning. The plants are distributed parameter systems （DPS） describing homogeneous and isotropic elastic vibrat- ing strings with fixed endpoints. The purpose of the paper is to approximate the infinite-dimensional dynamics, rather than the parameters of the wave systems. To solve the problem, the wave systems are first transformed into finite-dimensional dynamical systems described by ordinary differential equation （ODE）. The properties of the finite-dimensional systems, including the convergence of the solution, as well as the dominance of partial system dynamics according to point-wise measurements, are analyzed. Based on the properties, second, by using the deterministic learning algorithm, an approxi- mately accurate neural network （NN） approximation of the the finite-dimensional system dynamics is achieved in a local region along the recurrent trajectories. Simulation studies are included to demonstrate the effectiveness of the proposed approach.
