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@article{faucris.329502150,
abstract = {The finite-time turnpike property describes the situation in an optimal
control problem where an optimal trajectory reaches the desired state
before the end of the time interval and remains there. We consider a
machine learning problem with a neural ordinary differential equation
that can be seen as a homogenization of a deep ResNet. We show that with
the appropriate scaling of the quadratic control cost and the
non-smooth tracking term, the optimal control problem has the
finite-time turnpike property; that is, the desired state is reached
within the time interval and the optimal state remains there until the
terminal time T. The time t0 where the optimal trajectories reach the desired state can serve as an
additional design parameter. Since ResNets can be viewed as
discretizations of neural odes, the choice of t0 corresponds to the choice of the number of layers; that is, the depth of the neural network. The choice of t0 allows to achieve a compromise between the depth of the network and
the size of the optimal system parameters, which we hope will be useful
to determine the optimal depths for neural network architectures in the
future.