This paper proposes a neural-network-based state estimation (NNSE) method that aims to achieve higher time efficiency, improved robustness against noise, and extended observability when compared with the conventional weighted least squares (WLS) state estimation method. NNSE consists of two parts, the linear state estimation neural network (LSE-net) and the unobservable state estimation neural network (USE-net). The LSE-net functions as an adaptive approximator of linear state estimation (LSE) equations to estimate the nominally observable states. The inputs of LSE-net are the vectors of synchrophasors while the outputs are the estimated states. The USE-net operates as the complementary estimator on the nominally unobservable states. The inputs are the estimated observable states from LSE-net while the outputs are the estimation of nominally unobservable states. USE-net is trained off-line to approximate the veiled relationship between observable states and unobservable states. Two test cases are conducted to validate the performance of the proposed approach. The first case, which is based on the IEEE 118-bus system, shows the comprehensive performance of convergence, accuracy, and robustness of the proposed approach. The second case study adopts real-world synchrophasor measurements, and is based on the Jiangsu power grid, which is one of the largest provincial power systems in China.