Data-driven techniques are being increasingly applied to complement physics-based models in fire science. However, the lack of sufficiently large datasets continues to hinder the application of certain machine learning techniques. In this paper, we use simulated data to investigate the ability of neural networks to parameterize dynamics in fire science. In particular, we investigate neural networks that map five key parameters in fire spread to the first arrival time, and the corresponding inverse problem. By using simulated data, we are able to characterize the error, the required dataset size, and the convergence properties of these neural networks. For the inverse problem, we quantify the network's sensitivity in estimating each of the key parameters. The findings demonstrate the potential of machine learning in fire science, highlight the challenges associated with limited dataset sizes, and quantify the sensitivity of neural networks to estimate key parameters governing fire spread dynamics.