A hybrid analytical-machine learning framework for data-driven modeling of soliton solutions in the complex Ginzburg-Landau equation

The extended complex Ginzburg-Landau equation serves as a fundamental model for nonlinear wave dynamics, governing diverse physical phenomena ranging from particle motion in plasmas to pulse propagation in optical fibers. In this study, we introduce a hybrid analytical-machine learning framework that synergizes traditional mathematical techniques with modern data-driven approaches to comprehensively explore soliton dynamics within this system. First, we employ two advanced analytical methods, the modified Riccati extended simple equation methodology and the new modified generalized exponential rational function approach to derive a rich spectrum of nonlinear wave solutions. These include bright, dark, kink, and combined solitons, alongside hyperbolic, periodic, and exponential function solutions, thereby providing a comprehensive analytical foundation. On these analytical insights we construct a hybrid symbolic-numeric model, implemented by a multilayer perceptron regressor neural network, to find and synthesize soliton solutions to data. The key soliton types such as dark and bright solitons, combined solitons, and periodic solitons are well represented by the proposed framework with high levels of concurrence with the analytical benchmarks as seen by the low error measures. This combination provides a common platform that allows connecting classical analysis methods with modern machine learning, allowing creating as well as effectively simulating nonlinear wave behavior. This work paves the way to the frontiers of the physics of higher dimensional nonlinear waves by demonstrating the utility of this hybrid approach and highlighting some fundamental nonlinear dynamical properties of the model under consideration, and provides a paradigm that can be extended to understand the phenomena involving complex waves in interdisciplinary contexts.