Surrogate modeling with functional nonlinear autoregressive models (F-NARX)

We propose a novel functional approach to surrogate modeling of dynamical systems with exogenous inputs. This approach, named Functional Nonlinear AutoRegressive with eXogenous inputs ($F$-NARX), approximates the system response based on temporal features of the exogenous inputs and the system response. This marks a major step away from the discrete-time-centric approach of classical NARX models, which determines the relationship between selected time steps of the input/output time series. By modeling the system in a time-feature space, $F$-NARX takes advantage of the temporal smoothness of the process being modeled, providing more stable predictions and reducing the dependence of model performance on the discretization of the time axis.

In this work, we introduce an $F$-NARX implementation based on principal component analysis and polynomial regression. To further improve prediction accuracy, we also introduce a modified hybrid least angle regression approach to identify a sparse model structure and minimize the expected forecast error, rather than the one-step-ahead prediction error.

We investigate the behavior and capabilities of our $F$-NARX implementation on two case studies: an eight-story building under wind loading and a three-story steel frame under seismic loading. Our results demonstrate that $F$-NARX has several favorable properties that make it well-suited to surrogate modeling applications.