Differential Equations as a Model Prior for Deep Learning and its Applications in Robotics

For many decades, much of the scientific knowledge of physics and engineering has been expressed via differential equations. These differential equations describe the underlying phenomena and the relations between different interpretable quantities. Therefore, differential equations are a promising approach to incorporate prior knowledge in machine learning models to obtain robust and interpretable models. In this paper, we summarize a straight forward approach to incorporate deep networks in differential equations to solve first-order non-linear differential equations by minimising the residual end-to-end. We describe the deep differential network that computes the functional value and smooth Jacobians in closed form. Afterwards, we demonstrate that the deep network Jacobians approximate the symbolic Jacboian and apply the proposed approach two robotics applications. These applications use differential equations as model prior for deep networks to learn physically plausible models and optimal feedback control.