AI-Assisted quantum computing and neural network approaches for graph-theoretic nonlinear optimization of fuzzy partial differential equation models
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Keywords:
Quantum optimization, Fuzzy partial differential equations, Graph theory, Neural networks, Variational quantum algorithms, Nonlinear systems.Abstract
This work comes up with a synthetic computational framework, which incorporates quantum-assisted optimization, neural network approximation, and graph-theoretic modeling of the nonlinear fuzzy partial differential equation (FPDE) systems. The strategy transforms nonlinear FPDEs into graph-based operator systems, where the spatial time interactions are represented discretely under
the uncertainty of fuzziness. The methods of quantum computing such as variational quantum eigen solvers, quantum approximate optimization algorithms are applied to find solutions to high-dimensional nonlinear optimization subproblems, which are the results of discretized fuzzy operators. Neural networks are instantiated to provide approximations on nonlinear residual mappings and
fuzzy membership evolution and constitute a hybrid quantum-classical architecture. The suggested framework is tested on reference nonlinear fuzzy diffusion reaction systems and convective transport equations. The convergence stability, computational complexity and the uncertainty propagation robustness are improved as shown by performance metrics. Spectral encoding based on the graph
Laplacian makes it possible to arrange the quantum circuit parameters in a structured way and to enforce consistency of fuzzy boundary constraints with the help of neural residual correction. It is experimentally verified that the hybrid model provides a better optimization accuracy when compared to classical deterministic solvers.
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