State observation of nonlinear systems helps to understand the complex behaviour of the system and plays an important role in fault diagnosis and model predictive control. Traditional methods for state observation of nonlinear systems mainly include methods based on system mechanism models and data-driven methods. Still, these methods are still insufficient in terms of nonlinear feature extraction capability or computational complexity. Therefore, in this paper, a nonlinear system cascade structure state observer based on subspace state reconstruction mapping is proposed to integrate the advantages of model-driven and data-driven methods. The state observer is implemented by a two-layer neural network cascade, where the first and second layer networks are used to fit the subspace state reconstruction mapping and the underlying state transfer mapping of the system, respectively. Further, this paper innovatively proposes the concept of state prediction closure error between the two layers of the network, and minimizes the prediction error and the interlayer closure error as the training objectives of the state observer. Finally, the effectiveness of the method is illustrated by a numerical example.