The recursive Newton-Euler Algorithm (RNEA) is a popular technique for computing the dynamics of robots. RNEA can be framed as a differentiable computational graph, enabling the dynamics parameters of the robot to be learned from data via modern auto-differentiation toolboxes. However, the dynamics parameters learned in this manner can be physically implausible. In this work, we incorporate physical constraints in the learning by adding structure to the learned parameters. This results in a framework that can learn physically plausible dynamics via gradient descent, improving the training speed as well as generalization of the learned dynamics models. We evaluate our method on real-time inverse dynamics control tasks on a 7 degree of freedom robot arm, both in simulation and on the real robot. Our experiments study a spectrum of structure added to the parameters of the differentiable RNEA algorithm, and compare their performance and generalization.