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

Training Recurrent Neural Networks by Sequential Least Squares and the Alternating Direction Method of Multipliers