In order to observe all the unmeasured states of the three-dimensional Autonomous Underwater Vehicles (AUVs) with high performance, this paper proposes a new state observer which has the ability to estimate the rotational angles and velocities of the AUVs by only utilising the positional information. Furthermore, the notable feature of the proposed observer is operating under the effects of the system’s uncertain model and inaccurate output measurement. All the uncertain parameters and structures are combined into a unique vector and compensated by an online training Radial Basis Function (RBF) neural network. Meanwhile, the sensorless issue is addressed by a novel online updating rule to estimate output error. Integrating a neural network to deal with the observation problem with sensor uncertainties is typical in the control sector. The observation scheme constructed by the proposed observer and the RBF neural network is called a Neural-based State Observer. Mathematical solutions are proven theoretically sufficiently to ensure convergence and their effectiveness is verified by simulation in the digital platform.