The famous filtering problem of estimating the state of a stochastic dynamical system from noisy observations is of central importance in engineering, and high-dimensional nonlinear filtering is still a challenging problem. This problem is reduced to solving the Duncan-Mortensen-Zakai (DMZ) equation which is satisfied by the unnormalized conditional density of the state given the observation history. For general nonlinear filtering problems, we leverage on the representation ability of recurrent neural network and provide a computationally efficient and optimal framework for nonlinear filter design based on Yau-Yau algorithm and recurrent neural network. Theoretically, it can be proved that the size of the neural network required in this algorithm only increases in polynomial (rather than exponentially) with respect to the dimension, which implies that the Yau-Yau algorithm based on recurrent neural network has the capability to overcome the curse of dimensionality. This solves a century old nonlinear filtering problem.