Neural learning of vector fields for encoding stable dynamical systems

The data-driven approximation of vector fields that encode dynamical systems is a persistently hard task in machine learning. If data is sparse and given in the form of velocities derived from few trajectories only, state-space regions exist, where no information on the vector field and its induced dynamics is available. Generalization towards such regions is meaningful only if strong biases are introduced, for instance assumptions on global stability properties of the to-be-learned dynamics. We address this issue in a novel learning scheme that represents vector fields by means of neural networks, where asymptotic stability of the induced dynamics is explicitly enforced through utilizing knowledge from Lyapunov׳s stability theory, in a predefined workspace. The learning of vector fields is constrained through point-wise conditions, derived from a suitable Lyapunov function candidate, which is first adjusted towards the training data. We point out the significance of optimized Lyapunov function candidates and analyze the approach in a scenario where trajectories are learned and generalized from human handwriting motions. In addition, we demonstrate that learning from robotic data obtained by kinesthetic teaching of the humanoid robot iCub leads to robust motion generation.