Abstract Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum-classical PINNs (QCPINNs) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy, while using only 10%–30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L 2 error for Helmholtz, Klein–Gordon, and Convection–diffusion equations, with reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics-informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality.
Abstract Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large number of parameters to achieve reasonable accuracy, particularly for complex PDEs. In this paper, we present a quantum-classical PINNs (QCPINNs) that combines quantum and classical components, allowing us to solve PDEs with significantly fewer parameters while maintaining comparable accuracy and convergence to classical PINNs. We systematically evaluated two quantum circuit architectures across various configurations on five benchmark PDEs to identify optimal QCPINN designs. Our results demonstrate that the QCPINN achieves stable convergence and comparable accuracy, while using only 10%–30% of the trainable parameters required by classical PINNs. This approach also results in a significant reduction in the relative L 2 error for Helmholtz, Klein–Gordon, and Convection–diffusion equations, with reduction ranging from 4% to 64% across various fields. These findings demonstrate the potential of parameter efficiency and solution accuracy in physics-informed machine learning, allowing for a substantial decrease in model complexity without compromising solution quality.
Recently, physics-informed neural networks (PINNs) have aroused an upsurge in the field of scientific computing including solving partial differential equations (PDEs), which convert the task of solving PDEs into an optimization challenge by adopting governing equations and definite conditions or observation data as loss functions. Essentially, the underlying logic of PINNs is based on the universal approximation and differentiability properties of classical neural networks (NNs). Recent research has revealed that quantum neural networks (QNNs), known as parameterized quantum circuits, also exhibit universal approximation and differentiability properties. This observation naturally suggests the application of PINNs to QNNs. In this work, we introduce a physics-informed quantum neural network (PI-QNN) by employing the QNN as the function approximator for solving forward and inverse problems of PDEs. The performance of the proposed PI-QNN is evaluated by various forward and inverse PDE problems. Numerical results indicate that PI-QNN demonstrates superior convergence over PINN when solving PDEs with exact solutions that are strongly correlated with trigonometric functions. Moreover, its accuracy surpasses that of PINN by two to three orders of magnitude, while requiring fewer trainable parameters. However, the computational time of PI-QNN exceeds that of PINN due to its operation on classical computers. This limitation may improve with the advent of commercial quantum computers in the future. Furthermore, we briefly investigate the impact of network architecture on PI-QNN performance by examining two different QNN architectures. The results suggest that increasing the number of trainable network layers can enhance the expressiveness of PI-QNN. However, an excessive number of data encoding layers significantly increases computational time, rendering the marginal gains in performance insufficient to compensate for the shortcomings in computational efficiency.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
Physics-informed neural networks (PINNs) often struggle on nonlinear partial differential equations (PDEs) with sharp gradients, stiff dynamics, high-frequency content, or multiscale structure. Such limitations, rooted in spectral bias, ill-conditioned optimization, and unstable convergence, restrict PINN accuracy in regimes where advanced solvers are most needed. In this work, we develop a hybrid quantum-classical physics-informed neural network (HQPINN) that integrates a classical neural-network backbone with a parameterized quantum circuit (PQC) to enrich the solution representation. The framework is benchmarked against a classical PINN on three representative nonlinear PDEs: Burgers' equation, the Allen-Cahn equation, and the Korteweg-de Vries (KdV) equation. The framework is further examined through a systematic sensitivity analysis of qubit count, circuit depth, PQC placement, collocation density, and classical-network width. Across all benchmarks, HQPINNs exhibit smoother training dynamics, reduced loss oscillations, and improved final accuracy, with the largest gains occurring in stiff and multiscale regimes. Relative L2 error decreases by about fourfold for Burgers' equation and fivefold for the Allen-Cahn equation, while improvements for the KdV equation are more moderate. Overall, the results demonstrate that carefully co-designed hybrid quantum-classical architectures can mitigate key limitations of classical PINNs and provide practical design guidance for near-term quantum-enhanced PDE solvers.
Quantum machine learning (QML) has emerged as a promising paradigm for scientific computing, particularly when combined with Physics-Informed Neural Networks (PINNs) to solve partial differential equations. In this work, we introduce a Quantum Physics-Informed Neural Network (QPINN) framework to solve the Helmholtz equation in the context for scattered-wavefield. To overcome the numerical instabilities associated with point-source singularities commonly encountered in classical PINNs, we adopt the Lippmann–Schwinger integral formulation. The proposed hybrid architecture couples a classical neural-network encoder with a variational quantum circuit, enabling the learning of expressive, dynamically generated feature embeddings. Numerical experiments on a heterogeneous velocity model demonstrate that the proposed QPINN achieves high solution accuracy while requiring significantly fewer trainable parameters than analogous classical neural network architectures. Our results highlight the potential of hybrid quantum-classical approaches as efficient and scalable alternatives for wavefield simulation and for solving wave-based inverse problems.
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.
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding governing physical laws directly into the training objective. Recent advances in quantum machine learning have motivated hybrid quantum-classical extensions aimed at enhancing representational capacity while remaining compatible with near-term quantum hardware. In this work, we investigate trainable embedding strategies within quantum-assisted PINNs for solving parabolic PDEs, using one- and two-dimensional heat equations as canonical benchmarks. We introduce two quantum-assisted architectures that differ in their embedding components. In the first approach, a classical feed-forward neural network generates trainable feature maps for quantum data encoding (FNN-TE-QPINN). In the second, the embedding stage is realized entirely by a parameterized quantum circuit (QNN-TE-QPINN), yielding a fully quantum feature map. Our findings emphasize the critical role of embedding design and support hybrid quantum-classical approaches for parabolic PDE modeling in the NISQ era.