This paper proposes a tractable framework to determine key characteristics of non-linear dynamic systems by converting physics-informed neural networks to a mixed integer linear program. Our focus is on power system applications. Traditional methods in power systems require the use of a large number of simulations and other heuristics to determine parameters such as the critical clearing time, i.e. the maximum allowable time within which a disturbance must be cleared before the system moves to instability. The work proposed in this paper uses physics-informed neural networks to capture the power system dynamic behavior and, through an exact transformation, converts them to a tractable optimization problem which can be used to determine critical system indices. By converting neural networks to mixed integer linear programs, our framework also allows to adjust the conservativeness of the neural network output with respect to the existing stability boundaries. We demonstrate the performance of our methods on the non-linear dynamics of converter-based generation in response to voltage disturbances.