On the Joint Symmetry Neural Network for Optimal Bounds in Fractional Inequalities with H-Godunova–levin Convexity

The gradual developments in mathematical analysis have increased the demand for improving the efficiency of constraints and their validation, which have a significant contribution to resolving many real-world problems. There are many techniques that are used to modify the fractional inequalities, but all approaches are analytical. The adoption of Machine learning (ML) models is one practical approach to optimize the bounds of inequalities and their numerical validations. In this paper, we implement ML models to modify the bounds of Hermite–Hadamard-type inequalities, which consider the h-Godunova–Levin as a weight function. The classical development of inequalities by generalized fractional operators was never systematically studied to determine which value of the weight gives the best bound possible. We prove a sharp lower bound for the Hermite–Hadamard gap uniform over all admissible weights, show that it is achieved exactly by the classical convex weight, and characterize the optimal weights. To complement this analytical result, a parameterization of the weight function by a feedforward neural network is introduced, which is assumed to be admissible in the h-Godunova–Levin framework, and the validity of the resulting inequalities is proved. A Lipschitz-type error analysis is used to relate the approximation accuracy of the network to the tightness of the bound, and it is shown in numerical experiments that the network learns the optimal weight without any knowledge of the analytical weight’s form. The framework thus ensures the rigor of fractional convexity theory while offering a data-driven method of determining the optimal weight functions in cases where they are not explicitly known. The inherent symmetry properties of the fractional operators and the symmetric structure of the Hermite-Hadamard inequalities are preserved, while the neural network framework introduces a symmetry-breaking mechanism that enables the discovery of optimal weights. This dual perspective on symmetry—both preserving and breaking—provides a comprehensive understanding of the underlying mathematical structures and aligns perfectly with the scope of the journal Symmetry. This practical approach opens a new horizon for researchers and yields better results in the field of analysis.