  This paper proposes a novel algorithm for training recurrent neural network
models of nonlinear dynamical systems from an input/output training dataset.
Arbitrary convex and twice-differentiable loss functions and regularization
terms are handled by sequential least squares and either a line-search (LS) or
a trust-region method of Levenberg-Marquardt (LM) type for ensuring
convergence. In addition, to handle non-smooth regularization terms such as
$\ell_1$, $\ell_0$, and group-Lasso regularizers, as well as to impose possibly
non-convex constraints such as integer and mixed-integer constraints, we
combine sequential least squares with the alternating direction method of
multipliers (ADMM). We call the resulting algorithm NAILS (nonconvex ADMM
iterations and least squares) in the case line search (LS) is used, or NAILM if
a trust-region method (LM) is employed instead. The training method, which is
also applicable to feedforward neural networks as a special case, is tested in
three nonlinear system identification problems.

Machine Learning;Systems and Control;Systems and Control;Optimization and Control