Hybrid analytical and neural-network approaches to the non-local short pulse equation

The aim of this study is to examine a non-local short pulse equation and understand how its dynamics differ from those of its classical local counterpart. To achieve this, we introduce a Lax pair and develop a k-fold Darboux transformation, which forms the analytical foundation for constructing exact solutions. Using this method, we derive various solutions, including breathers and stable and unstable solitons over zero and non-zero backgrounds. The exact solutions are then used to train a neural network-based framework using Levenberg–Marquardt artificial neural networks (LM-ANN), which aims to approximate the solitonic structures. Model performance is evaluated using the relative error norm based on the Euclidean distance between the predicted and exact solutions. Surface plots, contour maps, and error visualizations are presented to illustrate approximation quality. To investigate model sensitivity and robustness, we explore multiple activation functions and network architectures, and introduce Gaussian noise at 1%, 2%, and 3% levels to both training and testing datasets. Monte Carlo simulations with multiple noise realizations are conducted, and statistical performance is summarized using the mean and standard deviation of the relative error. Results are reported through comparative tables and graphical analyses. By integrating analytical soliton construction with LM-ANN-based regression and systematic robustness testing, this study presents a hybrid framework for solving the non-local short pulse equation, offering accurate and stable solutions even under noise, a valuable contribution to the analysis of nonlinear systems affected by uncertainty.