Artificial Intelligence and Neural Network-Driven Quantum Calculus Framework for Nonlinear Optimization of Fuzzy Partial Differential Equations in Fluid Dynamics
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Keywords:
Artificial intelligence–driven optimization; Quantum calculus operators; Fuzzy partial differential equations; Nonlinear stability analysis; Neural network approximationAbstract
This paper introduces a mathematically sound and computationally unified approach to the solution of nonlinear fuzzy partial differential equations in fluid mechanics with quantum calculus and neural network approximation combined together. The intended model redefines the fuzzy nonlinear fluid equation to make use of q-time derivatives and a representation of an equivalent integral operator in a Banach space framework. The existence and uniqueness of the solutions are proved through Banach fixed-point theorem with Lipschitz continuity assumptions whereas the exponential stability is proved by Lyapunov functional analysis. To improve the accuracy of the solutions, a nonlinear optimization functional is presented and a scheme of neural network approximation is integrated into the analytical structure to enhance a faster convergence without breaking any theoretical assurances. It is demonstrated that the neural approximation error can decrease with network size in a polylogarithmic manner, as can be expected in approximation theory. The sensitivity analysis shows that the quantum parameter q has a direct effect on stability decay rates and minimization of residual, which can be used as a controllable balance of discrete-continuous dynamics. The presence
of limited uncertainty propagation, consistent optimization paths, and enhanced convergence behavior with respect to different levels of fuzziness and q-parameters are proved through numerical experiments on fuzzy representations on the α-level. These findings confirm that quantum operator theory, nonlinear optimization, and neural approximation have a stable, convergent and uncertainty consistent computational framework of nonlinear fuzzy fluid systems.
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