A Generalized Nonlinear Grey Differential Framework with Power-Law Transformation for Engineering Systems under Limited Data

Background:This research focuses on developing a generalized nonlinear grey differential framework by extending the differential structure of the conventional GM(1,1) model and incorporating a versatile nonlinear transformation from the power-law family. Grey system theory addresses the problem of insufficient data. The suggested framework tries to understand how nonlinear engineering systems behave over time.
Materials and Methods:
The suggested model differs from typical nonlinear grey models in that it does not impose a specific type of nonlinearity. Instead, it lets the best nonlinear transformation be chosen based on error indicators, making it more adaptable to the data being analyzed. We used the Airfoil Self-Noise dataset from the UCI Machine Learning Repository to test how well the suggested framework worked in a real-world engineering setting with nonlinear dynamic interactions.
Results:The results showed that the best base selection for the nonlinear transformation yielded better performance on some error indicators than the traditional GM(1,1) model. This confirms that the main strengths of the proposed framework are its structural generality and adaptive flexibility, rather than its ability to impose a fixed nonlinearity.
Conclusion:The theoretical study of the suggested model's differential equation looks at whether solutions exist and are unique, as well as if equilibrium points are stable under general conditions on the nonlinear transformation function. This analysis provides the proposed framework with a solid mathematical foundation and makes it more reliable for modeling nonlinear systems. This study is a unique methodological contribution to the theory of grey systems, combining mathematical analysis with practical flexibility. It also lays the groundwork for creating more advanced nonlinear grey models that can solve differential equations in engineering systems with minimal data.