In this paper, we develop a cell-average based neural network (CANN) method for solving the Hunter-Saxton equation with its zero-viscosity and zero-dispersion limits. Motivated from the finite volume schemes, the cell-average based neural network method is constructed based on the finite volume integrals of the original PDEs. Supervised training is designed to learn the solution average difference between two neighboring time steps. The training data set is generated by the cell average based on a single initial value of the given PDE. The training process employs multiple time levels of cell averages to maintain stability and control temporal accumulation errors. After being well trained based on appropriate meshes, this method can be utilized like a regular explicit finite volume method to evolve the solution under large time steps. Furthermore, it can be applied to solve different type of initial value problems without retraining the neural network. In order to validate the capability and robustness of the CANN method, we also utilize it to deal with the corrupted learning data which is generated from the Gaussian white noise. Several numerical examples of different types of Hunter-Saxton equations are presented to demonstrate the effectiveness, accuracy, capability, and robustness of the proposed method.