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)
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
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
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
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.
The steady incompressible Navier--Stokes equations pose significant computational challenges due to their nonlinear convective terms and pressure--velocity coupling. Physics-informed neural networks (PINNs) provide a mesh-free framework for approximating such systems, but classical PINNs can experience optimization difficulties in nonlinear flow regimes. In this work, we propose a quantum physics-informed neural network (QPINN) framework with a quantum neural network (QNN)-based trainable embedding for the lid-driven cavity problem. The proposed approach uses a QNN to learn data-adaptive quantum feature maps that encode spatial coordinates before they are processed by a variational quantum circuit within a physics-informed loss formulation. Numerical experiments show that the proposed QNN-TE-QPINN exhibits stable training behavior and competitive solution accuracy compared with classical PINNs and hybrid quantum models using classical embeddings, while requiring significantly fewer trainable parameters. Rather than claiming computational speedup, these results highlight the potential of trainable quantum embeddings for parameter-efficient physics-informed learning. The findings suggest that embedding design plays an important role in quantum-assisted PDE solvers and support further investigation of QNN-based trainable embeddings for nonlinear fluid dynamics benchmarks.
Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods. However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work. This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format. Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost. The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1. The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference. This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations. The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost. Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.
Quantum-AI hybrid algorithms represent the most practical near-term pathway for leveraging quantum computing's probabilistic power alongside classical AI's scalability and optimization strengths. In the noisy intermediate-scale quantum (NISQ) era, these hybrids—primarily variational quantum algorithms (VQAs) such as the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA)—delegate hard subproblems (e.g., high-dimensional sampling, optimization under quantum constraints, or feature extraction in entangled spaces) to quantum processors while classical optimizers or neural networks handle parameter tuning, error mitigation, and large-scale data processing (Ferguson et al., 2026; Acampora, 2026). This exploration builds on the 2025 NSF AI+MPS community paper by integrating accelerated timelines: mass deployment of utility-scale quantum computing by August 2027 and AGI substantially before 2030. Hybrid approaches become critical enablers for MPS grand challenges in Mamun, SM (2026)…………………………………………1 Quantum-AI Hybrid Algorithms: Bridging Classical Intelligence, Quantum Acceleration, and Scientific Discovery in the MPS Domains (Vol-I) chemistry (molecular ground states), materials (inverse design), physics (many-body simulation), and astronomy (anomaly detection under uncertainty). Also available on Academia.edu: https://www.academia.edu/attachments/132469868/download_file?s=portfolio