A QPINN Framework with Quantum Trainable Embeddings for the Lid-Driven Cavity Problem

Abstract: 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.

• The paper introduces a QPINN method that integrates quantum neural network-based trainable embeddings into PINNs to reduce parameters while solving nonlinear Navier–Stokes equations.
• It employs hardware-efficient variational quantum circuits and a composite physics-informed loss function to capture complex flow dynamics and boundary-layer phenomena.
• Empirical results show that QNN-based embeddings achieve competitive accuracy with significantly fewer parameters compared to classical PINNs for lid-driven cavity flow.

QPINN Framework with Quantum Trainable Embeddings for Lid-Driven Cavity Flow

Background and Motivation

Solving nonlinear PDEs, particularly the incompressible Navier-Stokes equations, is central in computational fluid dynamics (CFD) but remains computationally intensive due to nonlinearity and pressure-velocity coupling. PINNs, by embedding governing equations into neural network loss functions, offer mesh-free and continuous surrogate modeling, yet classical PINNs face optimization issues in highly nonlinear regimes, often requiring large parameter counts and extensive training. Recent advances in quantum machine learning (QML) have prompted exploration of hybrid quantum-classical approaches; in the NISQ regime, variational quantum circuits (VQCs) can be integrated for parameter-efficient surrogate modeling.

The lid-driven cavity benchmark is a canonical test for evaluating PDE solvers, presenting rich nonlinear dynamics and boundary-layer phenomena. This paper addresses the computational challenges of this benchmark via a quantum physics-informed neural network (QPINN) using quantum neural network (QNN)-based trainable embeddings, thereby extending previous work on quantum-assisted PINNs to nonlinear CFD contexts (2605.13892).

Methodological Contributions

Quantum Trainable Embedding Network

The QPINN architecture employs a QNN-based embedding module that maps spatial coordinates $(x, y)$ into quantum features. After normalization, coordinates are processed by a parameterized quantum circuit, producing quantum states that incorporate spatial information in a highly nonlinear fashion. The output quantum features are mapped to rotation angles for the subsequent encoding layer, enabling adaptive and domain-sensitive feature representations. Pauli-Z expectation values are used for measurement, resulting in bounded, stable outputs.

Variational Quantum Circuit and Readout

Encoded quantum states are processed by hardware-efficient VQCs with scalable depth and qubit configurations. The VQC parameters are shared across spatial inputs, enforcing a unified latent representation. Physical fields (pressure $p(x, y)$ and stream function $\psi(x, y)$) are extracted via expectation values of Pauli-Z-based observables, either from shared or independent circuits.

Physics-Informed Loss and Optimization

The hybrid quantum-classical model is trained by minimizing a composite physics-informed loss function comprising interior PDE residuals, boundary constraints (no-slip and moving lid conditions), and a reference pressure normalization. All derivatives are computed via automatic differentiation, integrating quantum parameter-shift gradients and classical backpropagation through the embedding network. This allows seamless calculation of spatial and parameter derivatives required for loss evaluation.

Numerical and Empirical Results

Parameter Efficiency and Training Behavior

A classical PINN baseline (6,594 parameters) was compared with QPINN variants: the QNN-TE-QPINN required only 360 parameters (4 qubits, 10 VQC layers), and the FNN-TE-QPINN required 608 parameters. The QNN-TE-QPINN achieved competitive accuracy on the lid-driven cavity problem, with stable training curves and lower loss values than classical PINNs and fixed-embedding QPINNs. This demonstrates substantial parameter efficiency without compromising solution quality. Notably, QNN-based trainable embeddings, when scaled to four or more qubits, outperformed fixed analytical embeddings (e.g., Chebyshev-based) and, under certain configurations, classical PINNs.

Generalization and Error Metrics

Inference results showed that QNN-TE-QPINN reliably captures velocity magnitude distributions comparable to RK45 reference solutions, achieving a mean squared error (MSE) of $6.64 \times 10^{-4}$ and L2 relative error of $9.71 \times 10^{-2}$ on velocity fields. Pressure fields exhibited higher error (MSE $1.61 \times 10^{-1}$, L2 error $5.51 \times 10^{-1}$), reflecting inherent difficulty in pressure recovery under physics-informed learning, especially for incompressible flows.

Reynolds Number Scaling

Across Reynolds numbers from $10$ to $3000$, QNN-TE-QPINN maintained competitive loss values, with improved performance in convection-dominated regimes as qubit count increased. These results indicate suitability for parameter-efficient modeling in moderately turbulent flows.

Embedding Strategy Impact

Adaptive, trainable embeddings demonstrated smoother, more structured quantum feature mappings, supporting better generalization than rigid analytical encodings. The fully quantum embedding (QNN-based) provides a consistent and scalable channel for domain-dependent feature amplification, critical for representing boundary layers and vortex structures.

Theoretical and Practical Implications

The research establishes that trainable quantum embeddings within QPINN architectures substantially reduce trainable parameter counts for nonlinear PDEs such as Navier-Stokes, offering a practical pathway toward quantum-assisted parameter-efficient modeling in CFD. While no computational speedup is claimed with respect to classical hardware, resource-aware quantum embedding strategies enable efficient deployment on NISQ devices, minimizing circuit depth and memory consumption.

The findings highlight the critical role of embedding design in quantum-enhanced PDE solvers—co-training the embedding network jointly with the PDE solver yields adaptive representations aligned with intrinsic fluid dynamic features. The separation between embedding and variational circuit parameters provides modularity, enhancing scalability and optimization in hybrid learning pipelines.

Theoretically, the expressivity of quantum feature maps could facilitate global transformations within the high-dimensional Hilbert space, possibly offering representational advantages in modeling multi-scale nonlinear interactions. Empirically, QNN-based embeddings are shown to be competitive and, in select configurations, superior in loss performance to classical networks with far fewer parameters.

Future Directions

Further investigation into loss balancing, optimization strategies, and architectural scaling is warranted. Addressing gradient pathologies such as barren plateaus is essential for deeper quantum circuits. Future work should explore pressure-velocity coupling mechanisms, more precise pressure loss weighting, and the adoption of mixed PDE formulations to address derivative sensitivity. As quantum hardware matures, integration of deeper circuits with increased qubit counts can be expected, potentially expanding the applicability of QPINN approaches to more turbulent and multi-physics CFD benchmarks.

Conclusion

The proposed QNN-based trainable-embedding QPINN demonstrates parameter-efficient and competitive solution accuracy for the lid-driven cavity Navier-Stokes benchmark. The quantum trainable embedding mechanism, when exploited with moderate qubit counts, achieves robust generalization and reduction in model complexity relative to classical PINNs. These outcomes underscore the importance of embedding architecture in quantum-assisted physics-informed learning and open avenues for resource-aware quantum PDE solvers in fluid dynamics and beyond (2605.13892).