ADVANCED ANALYTICAL AND COMPUTATIONAL FRAMEWORK FOR FRACTIONAL INTEGRAL INEQUALITIES USING MACHINE LEARNING–ENHANCED NUMERICAL METHODS FOR NONLINEAR DYNAMICAL SYSTEMS AND OPTIMIZATION PROBLEMS

Abstract

Fractional integral inequalities provide essential tools for error estimation, stability analysis, and optimization in memory-dependent systems, but classical Hermite–Hadamard-type bounds can be conservative and costly to evaluate numerically. This study develops an integrated analytical-computational framework combining fractional inequality theory with machine learning-enhanced numerical methods to improve bound tightness, approximation accuracy, and computational efficiency. The framework incorporates Riemann–Liouville, Caputo, Caputo–Fabrizio, and Atangana–Baleanu operator families and supports convex, s-convex, and h-convex functions. Its architecture combines fractional operator embedding, neural residual calibration, and an inequality-constrained loss function to preserve analytically established bound directions while learning tighter estimates. Synthetic function families were evaluated across fractional orders of 0.1–0.9 and convexity parameters of 0.1–1.0, with classical quadrature, high-precision numerical references, and component-level ablations used for benchmarking.

The proposed framework achieved an average bound-gap reduction of 55.0%, with a peak of 65% and mean tightening of 60.7% in the low-order, low-convexity-parameter region. At s = 0.2 and 0.8, bound gaps decreased by 44.2% and 32.7%, respectively. Empirical convergence order increased from approximately 2.00 to 2.16, while maximum approximation error decreased by 55% at 128 discretization nodes and by more than 60% at 1,024 nodes. Computational speed-up reached 9.0-fold at one million evaluation points. In a representative fractional-order nonlinear oscillator, the learned controller changed largest Lyapunov exponents from positive values of 0.04–0.21 to negative values of −0.10 to −0.05 across orders of 0.5–1.0. The inequality-guided optimizer reached the target tolerance in approximately 102 iterations, whereas Adam, particle swarm optimization, and gradient descent did not reach it within 200 iterations. Ablation analysis identified the inequality-constrained loss and operator embedding as the largest contributors, with their removal reducing average tightening by 16.5 and 13.8 percentage points, respectively. These findings demonstrate promising gains in fractional numerical analysis, chaos suppression, and constrained optimization, subject to further validation beyond synthetic functions and the representative oscillator. Future work should examine broader function families, additional dynamical systems, and application-specific generalization under realistic computational conditions.