It is well known that building analytical performance models in practice is difficult because it requires a considerable degree of proficiency in the underlying mathematics. In this paper, we propose a machine-learning approach to derive performance models from data. We focus on queuing networks, and crucially exploit a deterministic approximation of their average dynamics in terms of a compact system of ordinary differential equations. We encode these equations into a recurrent neural network whose weights can be directly related to model parameters. This allows for an interpretable structure of the neural network, which can be trained from system measurements to yield a white-box parameterized model that can be used for prediction purposes such as what-if analyses and capacity planning. Using synthetic models as well as a real case study of a load-balancing system, we show the effectiveness of our technique in yielding models with high predictive power.