In this paper, a family of neural network-based survival models is presented. The models are specified based on piecewise definitions of the hazard function and the density function on a partitioning of the time; both constant and linear piecewise definitions are presented, resulting in a family of four models. The models can be seen as an extension of the commonly used discrete-time and piecewise exponential models and thereby add flexibility to this set of standard models. Using a simulated dataset the models are shown to perform well compared to the highly expressive, state-of-the-art energy-based model, while only requiring a fraction of the computation time.