In this paper, the artificial neural network method is applied to solve the time-fractional diffusion and diffusion-wave equations. This method combines Taylor series and neural network method, and uses the terms of different power terms of Taylor series as neurons. An error function is given to update the weights of the proposed neural network. In addition, in order to balance the contributions of different error terms in the error function, we propose an adaptive weight adjustment method. In the end, four numerical examples are given to demonstrate the effectiveness of proposed method in solving the time-fractional diffusion and diffusion-wave equations.