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Meta-Learning Runge-Kutta

Initial value problems, i.e. differential equations with specific, initial conditions, represent a classic problem within the field of ordinary differential equations(ODEs). While the simplest types of ODEs may have closed-form solutions, most interesting cases typically rely on iterative schemes for numerical integration such as the family of Runge-Kutta methods. They are, however, sensitive to the strategy the step size is adapted during integration, which has to be chosen by the experimenter. In this paper, we show how the design of a step size controller can be cast as a learning problem, allowing deep networks to learn to exploit structure in the initial value problem at hand in an automatic way. The key ingredients for the resulting Meta-Learning Runge-Kutta (MLRK) are the development of a good performance measure and the identification of suitable input features. Traditional approaches suggest the local error estimates as input to the controller. However, by studying the characteristics of the local error function we show that including the partial derivatives of the initial value problem is favorable. Our experiments demonstrate considerable benefits over traditional approaches. In particular, MLRK is able to mitigate sudden spikes in the local error function by a faster adaptation of the step size. More importantly, the additional information in the form of partial derivatives and function values leads to a substantial improvement in performance. The source code can be found at https://www.dropbox.com/sh/rkctdfhkosywnnx/AABKadysCR8-aHW_0kb6vCtSa?dl=0

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