A Quantum-inspired population optimizer integrating curvature stability and predictive consistency for deep learning

Irsa Sajjad; Osama Abdulaiz Alamri; Maria Malik; Marwan H. Alhelali; Maysoon A. Sultan; Irsa Sajjad; Osama Abdulaiz Alamri; Maria Malik; Marwan H. Alhelali; Maysoon A. Sultan

doi:10.3934/math.20261204

AIMS Mathematics

2026, Volume 11, Issue 9: 30374-30402. doi: 10.3934/math.20261204

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Research article

A Quantum-inspired population optimizer integrating curvature stability and predictive consistency for deep learning

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
4.
Department of Mathematics, Faculty of Science, University of Hafr Al Batin, Saudi Arabia

Received: 29 June 2026 Revised: 15 August 2026 Accepted: 04 September 2026 Published: 17 September 2026
MSC : 68T07, 65K10

Training deep neural networks is often hindered by sharp minima, unstable optimization trajectories, and limited generalization. To tackle these problems, we introduce a novel gradient-free, population-based optimization framework, the Curvature–Interference Quantum Entropic Optimizer (CI-QEO), which integrates the empirical loss, zeroth-order curvature estimation, predictive interference suppression, entropy-driven diversity preservation, and covariance-driven tunnelling into a single energy function. We used a benchmark set of binary and multiclass classification problems, as well as nonlinear regression problems, to test the proposed method with the same network architecture. We found that CI-QEO outperforms strong baseline optimizers in terms of accuracy of binary classification (94.18% vs. 90.58%), accuracy of multiclass classification (91.36% vs. 84.34%), and R² score (0.936 vs. 0.854), with lower test loss (26.46% vs. 37.24%), mean squared error (22.54% vs. 37.24%), predictive interference (64.27% vs. 105.12%), and total optimization energy (73.66% vs. 167.83%). In addition, ablation experiments and statistical significance analysis (p < 0.05) validated each proposed component's contribution to the observed performance gains. The results highlight that CI-QEO is a stable, accurate, and large-scale optimization framework for deep neural network training and a promising alternative to prevalent gradient- and population-based optimization algorithms.
Citation: Irsa Sajjad, Osama Abdulaiz Alamri, Maria Malik, Marwan H. Alhelali, Maysoon A. Sultan. A Quantum-inspired population optimizer integrating curvature stability and predictive consistency for deep learning[J]. AIMS Mathematics, 2026, 11(9): 30374-30402. doi: 10.3934/math.20261204

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Abstract

Training deep neural networks is often hindered by sharp minima, unstable optimization trajectories, and limited generalization. To tackle these problems, we introduce a novel gradient-free, population-based optimization framework, the Curvature–Interference Quantum Entropic Optimizer (CI-QEO), which integrates the empirical loss, zeroth-order curvature estimation, predictive interference suppression, entropy-driven diversity preservation, and covariance-driven tunnelling into a single energy function. We used a benchmark set of binary and multiclass classification problems, as well as nonlinear regression problems, to test the proposed method with the same network architecture. We found that CI-QEO outperforms strong baseline optimizers in terms of accuracy of binary classification (94.18% vs. 90.58%), accuracy of multiclass classification (91.36% vs. 84.34%), and R² score (0.936 vs. 0.854), with lower test loss (26.46% vs. 37.24%), mean squared error (22.54% vs. 37.24%), predictive interference (64.27% vs. 105.12%), and total optimization energy (73.66% vs. 167.83%). In addition, ablation experiments and statistical significance analysis (p < 0.05) validated each proposed component's contribution to the observed performance gains. The results highlight that CI-QEO is a stable, accurate, and large-scale optimization framework for deep neural network training and a promising alternative to prevalent gradient- and population-based optimization algorithms.

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