Partitioned neural network approximation based on overlapping or nonoverlapping subdomain partitions of the problem domain has been developed to enhance the training efficiency and solution accuracy of neural network approximation to PDEs (Partial Differential Equations). In the authors’ previous study [1], a gradient descent method was proposed from a classical FETI formulation [2], where partitioned individual neural networks are trained to approximate the PDE solution in each local subdomain for the provided interface value and the interface value is updated by the gradient value of an energy minimization functional related to the FETI formulation. The gradient value is computed by using the normal flux of the trained individual neural networks across the interface. To obtain the iteration convergence, a sufficiently small step size assumption is required for the gradient descent algorithm. To improve its convergence, the resulting updated formula in the gradient descent algorithm can be reformulated into a Robin iterative algorithm by taking one of the normal flux values implicitly. In this work, we propose a Robin iterative algorithm for the partitioned neural network approximation, where for the provided Robin interface condition each individual neural network is trained to approximate the PDE solution and these trained solutions are used to update the Robin interface condition for the next iteration. The advantages of neural network approximation over the classical approximation are; the optimal parameter for the Robin interface condition can be trained just like the parameters for the neural network solutions and extension of the idea into more general nonlinear problems can be done straightforwardly. In our numerical experiment, we will compare the performance of the gradient descent and the Robin iterative methods. In addition, to improve the parallel scalability of the Robin iterative method, a two-level version of the Robin iterative method will be proposed and tested by devising a suitable coarse problem.