Optimal control for neural ODE in a long time horizon

Abstract

We study the optimal control, in a long time horizon, of neural ordinary differential equations which are control-affine or whose activation function is homogeneous. When considering the classical regularized empirical risk minimization problem we show that, in long time and under structural assumption on the activation function, the final state of the optimal trajectories has zero training error if the data can be interpolated and if the error can be taken to zero with a cost proportional to the error. These hypotheses are fulfilled in the classification and ensemble controllability problems for some relevant activation and loss functions.