  This article introduces the Tensor Network B-spline model for the regularized
identification of nonlinear systems using a nonlinear autoregressive exogenous
(NARX) approach. Tensor network theory is used to alleviate the curse of
dimensionality of multivariate B-splines by representing the high-dimensional
weight tensor as a low-rank approximation. An iterative algorithm based on the
alternating linear scheme is developed to directly estimate the low-rank tensor
network approximation, removing the need to ever explicitly construct the
exponentially large weight tensor. This reduces the computational and storage
complexity significantly, allowing the identification of NARX systems with a
large number of inputs and lags. The proposed algorithm is numerically stable,
robust to noise, guaranteed to monotonically converge, and allows the
straightforward incorporation of regularization. The TNBS-NARX model is
validated through the identification of the cascaded watertank benchmark
nonlinear system, on which it achieves state-of-the-art performance while
identifying a 16-dimensional B-spline surface in 4 seconds on a standard
desktop computer. An open-source MATLAB implementation is available on GitHub.
