Quantum-inspired stochastic geometric hyperstate optimization for explainable deep neural learning under nonlinear manifold dynamics

Irsa Sajjad; Osama Abdulaziz Alamri; Maria Malik; Marwan H. Alhelali; Irsa Sajjad; Osama Abdulaziz Alamri; Maria Malik; Marwan H. Alhelali

doi:10.3934/math.2026986

AIMS Mathematics

2026, Volume 11, Issue 8: 24419-24444. doi: 10.3934/math.2026986

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Quantum-inspired stochastic geometric hyperstate optimization for explainable deep neural learning under nonlinear manifold dynamics

1.
Department of Mathematics, National University of Modern Languages, Islamabad, Pakistan
2.
Department of Statistics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
3.
Department of Statistics, Comsats University, Pakistan

Received: 22 May 2026 Revised: 08 July 2026 Accepted: 20 July 2026 Published: 10 August 2026
MSC : 62P05, 68T07

This study presented a new explainable deep neural learning model by introducing the quantum-inspired stochastic geometric hyperstate optimization (QSGHO) framework under nonlinear manifold dynamics. The proposed framework uses a probabilistic evolution process in which interacting hyperstates evolve on entropy-regularized stochastic manifolds, rather than the conventional deterministic gradient-driven approach. The methodology proposed combines the quantum-inspired hyperstate representation, manifold-guided probabilistic transport, adaptive tunneling diffusion, entropy-controlled stochastic exploration, and explainable probabilistic feature dynamics within a unified mathematical learning architecture. Numerous numerical simulations were performed with nonlinear stochastic benchmark examples in high-dimensional optimization regimes. The proposed QSGHO framework can be compared with SGD, RMSProp, Adam, AdamW, and LION optimization methods, and it performs better in terms of accuracy (98.1%), F1-score (97.6%), and convergence behavior. The entropy–confidence analysis, the behavior of manifold projections, the evolution of free energy, and the dynamics of probabilistic hyperstate further robustly validated and ensured the theoretical consistency of the proposed framework. The statistical analysis also showed that the performance gains were significant under stochastic learning conditions. The general results establish a strong mathematical connection among quantum-inspired optimization, stochastic geometry, information-theoretic learning, and explainable artificial intelligence, offering a scalable and robust framework for nonlinear stochastic deep learning applications.
Citation: Irsa Sajjad, Osama Abdulaziz Alamri, Maria Malik, Marwan H. Alhelali. Quantum-inspired stochastic geometric hyperstate optimization for explainable deep neural learning under nonlinear manifold dynamics[J]. AIMS Mathematics, 2026, 11(8): 24419-24444. doi: 10.3934/math.2026986

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Abstract

This study presented a new explainable deep neural learning model by introducing the quantum-inspired stochastic geometric hyperstate optimization (QSGHO) framework under nonlinear manifold dynamics. The proposed framework uses a probabilistic evolution process in which interacting hyperstates evolve on entropy-regularized stochastic manifolds, rather than the conventional deterministic gradient-driven approach. The methodology proposed combines the quantum-inspired hyperstate representation, manifold-guided probabilistic transport, adaptive tunneling diffusion, entropy-controlled stochastic exploration, and explainable probabilistic feature dynamics within a unified mathematical learning architecture. Numerous numerical simulations were performed with nonlinear stochastic benchmark examples in high-dimensional optimization regimes. The proposed QSGHO framework can be compared with SGD, RMSProp, Adam, AdamW, and LION optimization methods, and it performs better in terms of accuracy (98.1%), F1-score (97.6%), and convergence behavior. The entropy–confidence analysis, the behavior of manifold projections, the evolution of free energy, and the dynamics of probabilistic hyperstate further robustly validated and ensured the theoretical consistency of the proposed framework. The statistical analysis also showed that the performance gains were significant under stochastic learning conditions. The general results establish a strong mathematical connection among quantum-inspired optimization, stochastic geometry, information-theoretic learning, and explainable artificial intelligence, offering a scalable and robust framework for nonlinear stochastic deep learning applications.

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