In recent years, there has been growing interest in using machine-learning algorithms to assist classical numerical methods for scientific computations, as this data-driven approach could reduce computational cost. While faster execution is attractive, accuracy should be preserved. Perhaps more importantly, our ability to identify when a given machine-learning surrogate is not reliable should make their application more robust. We aim to quantify the uncertainty of predictions through the application of Bayesian and ensemble methods. We apply these methods to approximate a paraboloid and then the solution to the wave equation with both standard neural networks and physics-informed neural networks. We demonstrate that the embedding of physics information in neural networks reduces the model uncertainty while improving the accuracy. Between the two uncertainty quantification methods, our results show that the Bayesian neural networks render overconfident results while model outputs from a well-constructed ensemble are appropriately conservative.