On the emergence of numerical instabilities in Next Generation Reservoir Computing

Next Generation Reservoir Computing (NGRC) is a low-cost machine learning method for forecasting chaotic time series from data. However, ensuring the dynamical stability of NGRC models during autonomous prediction remains a challenge. In this work, we uncover a key connection between the numerical conditioning of the NGRC feature matrix -- formed by polynomial evaluations on time-delay coordinates -- and the long-term NGRC dynamics. Merging tools from numerical linear algebra and ergodic theory of dynamical systems, we systematically study how the feature matrix conditioning varies across hyperparameters. We demonstrate that the NGRC feature matrix tends to be ill-conditioned for short time lags and high-degree polynomials. Ill-conditioning amplifies sensitivity to training data perturbations, which can produce unstable NGRC dynamics. We evaluate the impact of different numerical algorithms (Cholesky, SVD, and LU) for solving the regularized least-squares problem.