This report rigorously synthesizes the Geometric Quantum Physics-Informed Neural Network (GQPINN) framework, which leverages group-theoretic twirling to enforce geometric symmetries, with Fractal Algebra and the Pellis-Fractal formalism. The original GQPINN, while superior in efficiency and accuracy for linear partial differential equations (PDEs), struggles fundamentally in regimes dominated by strong dissipation, nonlinearity, or multiscale phenomena due to its reliance on smooth Lie-group symmetries. To overcome this, the novel Fractal Geometric Quantum Physics-Informed Neural Network (FGQPINN) is formulated. This architecture utilizes the Pellis-Fractal Twirling Operator (T̃(α)) to project quantum generators onto a nested hierarchy of scale-weighted commutants, effectively hard-wiring self-similar and non-differentiable symmetries into the variational quantum circuit (VQC) ansatz. Furthermore, the physics-informed loss function is adapted using the Pellis-Fractal Laplacian (ΔPF). This synthesis not only achieves scale-covariant generalization and provides a theoretical mechanism to mitigate barren plateaus by restricting entanglement dynamics Mamun, SM (2026) …………………………….1 Geometric Quantum Physics-Informed Neural Networks: A Synthesis of Ontic Structural Realism and Fractal Algebra (Vol-II) K S C but also operationalizes Ontic Structural Realism, confirming that structural relations are ontologically primary. Also available on Academia.edu: https://www.academia.edu/attachments/132750951/download_file?s=portfolio
This report rigorously synthesizes the Geometric Quantum Physics-Informed Neural Network (GQPINN) framework, which leverages group-theoretic twirling to enforce geometric symmetries, with Fractal Algebra and the Pellis-Fractal formalism. The original GQPINN, while superior in efficiency and accuracy for linear partial differential equations (PDEs), struggles fundamentally in regimes dominated by strong dissipation, nonlinearity, or multiscale phenomena due to its reliance on smooth Lie-group symmetries. To overcome this, the novel Fractal Geometric Quantum Physics-Informed Neural Network (FGQPINN) is formulated. This architecture utilizes the Pellis-Fractal Twirling Operator (T̃(α)) to project quantum generators onto a nested hierarchy of scale-weighted commutants, effectively hard-wiring self-similar and non-differentiable symmetries into the variational quantum circuit (VQC) ansatz. Furthermore, the physics-informed loss function is adapted using the Pellis-Fractal Laplacian (ΔPF). This synthesis not only achieves scale-covariant generalization and provides a theoretical mechanism to mitigate barren plateaus by restricting entanglement dynamics Mamun, SM (2026) …………………………….1 Geometric Quantum Physics-Informed Neural Networks: A Synthesis of Ontic Structural Realism and Fractal Algebra (Vol-II) K S C but also operationalizes Ontic Structural Realism, confirming that structural relations are ontologically primary. Also available on Academia.edu: https://www.academia.edu/attachments/132750951/download_file?s=portfolio
Computational fluid dynamics (CFD) has long been limited by the extreme computational cost and multiscale complexity of the Navier-Stokes equations, particularly in turbulent regimes. We propose that geometric quantum physics-informed neural networks (GQPINNs) — symmetry-embedded variational quantum circuits that enforce finite-group ((K_4)) and compact Lie-group (SO(2), (\mathbb{Z}_2)) invariances via twirling of generator sets — constitute the foundational architecture for next-generation quantum-inspired artificial intelligence in CFD. Drawing on the mechanics of data re-uploading circuits, group-equivariant trainable blocks, and physics-informed loss functions, we show how GQPINNs achieve orders-of-magnitude gains in parameter efficiency and solution accuracy on representative PDE benchmarks while naturally extending to the incompressible and compressible Navier-Stokes system. Ontologically, the framework realises an onto-dynamic structural realism in which fluid structures (vorticity fields, streamlines, conserved quantities) are relationally individuated by symmetry Mamun, SM (2026) …………………………………….1 Navier-Stokes Turbulence as a Computational Laboratory for Next-Generation Quantum-Inspired Artificial Intelligence: Mechanics, Ontology, and Epistemology of Geometric Quantum Physics-Informed Neural Networks (Vol-I) K S C groups rather than by intrinsic point-like properties. Also available on Academia.edu: https://www.academia.edu/attachments/132750817/download_file?s=portfolio
Wai-Hong Tam, Reza Safari and Hiromichi Matsuyama (2026) introduce Geometric Quantum Physics-Informed Neural Networks (GQPINNs) as a symmetry-aware extension of quantum physics-informed neural networks (QPINNs). Their framework embeds finite-group (Klein four-group (K_4)) and compact Lie-group (SO(2) rotational) symmetries directly into the parametrized quantum circuit ansatz using twirling-based construction of equivariant generator sets. This builds on geometric quantum machine learning (GQML) to enforce solution-level invariance/equivariance by construction, rather than through loss penalties or post-hoc constraints. The paper benchmarks GQPINNs on representative linear and nonlinear PDEs (2D Poisson, 2D diffusion, 1D acoustic wave, 1D viscous Burgers), demonstrating superior mean absolute error (MAE) and parameter efficiency compared to baseline QPINNs and symmetry-adapted classical PINNs. Also available on Academia.edu: https://www.academia.edu/attachments/132740816/download_file?s=portfolio
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.
This paper presents a rigorous mathematical and ontological synthesis of solvable quantum systems and emergent fractal geometries. By leveraging the quadratic tensor framework, we demonstrate that disparate families of quantum models—including Clifford circuits, stabilizer codes, Gaussian states, and free-fermion systems—can be unified as quadratic functions over abelian groups or super-commutative Hopf algebras. This algebraic unification permits classical simulation with O(n2) complexity via a generalized Schur complement contraction mechanism, dissolving the boundaries between discrete-variable qudits and continuous-variable bosonic modes. However, physical reality frequently transcends these smooth, integer-dimensional algebraic bases. We integrate a fractal-recursive ontology, identifying that quantum trajectories in path integrals and energy spectra in quantum materials (e.g., Hofstadter's butterfly) exhibit non-integer Hausdorff dimensions and self-similar architectures that challenge traditional linear causality. We further explore the transition from quadratic to i-th order tensors, identifying the Clifford hierarchy's correspondence to higher-order polynomial maps and the resulting computational boundaries of the Gottesman-Knill theorem. Also available on Academia.edu: https://www.academia.edu/attachments/131609415/download_file?s=portfolio
This assessment challenges the prevailing classical, isolationist paradigm of defense quantum engineering, which regards quantum states as inherently fragile and susceptible to thermal decoherence. It introduces an active, topological, and biological framework by synthesizing quantum-biological mechanisms—such as the radical-pair mechanism for coherence stabilization—with advanced mathematical constructs. The methodology integrates Entanglement-Weighted Operator Geometry (EWOG), multi-parameter Topological Data Analysis (TDA) via G-LoG bi-filtration, Information Geometry on Fisher-Rao manifolds, and the spectral realization of Riemann zeta zeros. This synthesis constructs a blueprint for resilient, emergent informational manifolds capable of active decoherence suppression. The resulting hybrid architectures accelerate the timeline for mission-relevant quantum sensing, computing, and communications, providing a robust, noise-resistant foundation that can bypass current limitations outlined in reports like the Congressional Research Service Defense Primer. KSCCN Mamun, SM (2026) …………………………….1 Quantum Defense Technologies Reimagined: A Geometric, Fractal, and Information-Theoretic Assessment (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132889757/download_file?s=portfolio
This paper provides a rigorous academic review of the seminal work by Mamun and Grok xAI (2025) on advancing quantum matrix computations through the integration of geometric mean subroutines and Chebyshev-accelerated chain multiplications for nonlinear and iterative challenges in machine learning and quantum information. Following a thorough summary of its salient contributions, we compare and contrast the approach with contemporary research in quantum genetic algorithms (QGAs), variational quantum eigensolvers (VQEs), and tensor network methods. Strengths in polylogarithmic complexity and hybrid fault-tolerance are highlighted, alongside limitations in NISQ-era decoherence. Extending the framework, we propose novel integrations with fractal-recursive ontologies for non-integer dimensional simulations, the Principle of Co-Adaptation for bivariational dynamics in adaptive quantum circuits, and Souriau's axiomatic geometric model for positive-type state representations in high-dimensional entanglement. These extensions unify quadratic tensor structures with emergent fractal geometries, enabling scalable applications in quantum gravity simulations and autopoietic cognitive systems. Implications for post-quantum cryptography and geopolitical quantum strategies are discussed, advocating for interdisciplinary policies to harness transformative potential while mitigating risks. Also available on Academia.edu: https://www.academia.edu/attachments/131744773/download_file?s=portfolio