Combining deep learning with classical physics facilitates the efficient creation of accurate dynamical models. In a recent class of neural network, Lagrangian mechanics is hard-coded into the architecture, and training the network learns the given system. However, the current architectures do not facilitate the modelling of dynamical systems that are driven by servomechanisms (e.g. servomotors, stepper motors, current sources, volumetric pumps). This article presents ServoLNN, a new architecture to model dynamical systems that are driven by servomechanisms. ServoLNN is compatible for use in real-time applications, where the driving motion is known only just-in-time. A PyTorch implementation of ServoLNN is provided. The derivations and results reveal the occurrence of a possible family of solutions that the training may converge on. The effect of the family of solutions on the predicted physical quantities is explored, as is the resolution to reduce the family of solutions to a single solution. Resultantly, the architecture can simultaneously accurately find the energies, power, rate of work, mass matrix, generalised accelerations, generalised forces, and the generalised forces that drive the servomechanisms.