Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws into neural architectures. However, these classical approaches often require large number of parameters for solving complex problems or achieving reasonable accuracy. We investigate whether quantum-enhanced architectures can achieve comparable performance while significantly reducing model complexity. We propose a quantum-classical physics-informed neural network (QCPINN) combining quantum and classical components to solve PDEs with fewer parameters while maintaining comparable accuracy and training convergence. Our approach systematically evaluates two quantum circuit paradigms (e.g., continuous-variable (CV) and discrete-variable (DV)) implementations with four circuit topologies (e.g., alternate, cascade, cross-mesh, and layered), two embedding schemes (e.g., amplitude and angle) on five benchmark PDEs (e.g., Helmholtz, lid-driven cavity, wave, Klein-Gordon, and convection-diffusion equations). Results demonstrate that QCPINNs achieve comparable accuracy to classical PINNs while requiring approximately 10\% trainable parameters across different PDEs, and resulting in a further 40\% reduction in relative $L_2$ error for the convection-diffusion equation. DV-based circuits with angle embedding and cascade configurations consistently exhibited enhanced convergence stability across all problem types. Our finding establishes parameter efficiency as a quantifiable quantum advantage in physics-informed machine learning. By significantly reducing model complexity while maintaining solution quality, QCPINNs represent a potential direction for overcoming computational bottlenecks in scientific computing applications where traditional approaches require large parameter spaces.

翻译：物理信息神经网络（PINNs）通过将物理定律嵌入神经架构，已成为求解偏微分方程（PDEs）的有前景的方法。然而，这些经典方法在求解复杂问题或达到合理精度时通常需要大量参数。我们研究了量子增强架构是否能在显著降低模型复杂度的同时，实现可比的性能。我们提出了一种量子-经典物理信息神经网络（QCPINN），结合量子和经典组件来求解偏微分方程，该网络在保持可比精度和训练收敛性的同时，使用更少的参数。我们的方法系统评估了两种量子电路范式（例如连续变量（CV）和离散变量（DV））的实现，涉及四种电路拓扑（例如交替式、级联式、交叉网格式和分层式）、两种嵌入方案（例如幅度嵌入和角度嵌入），并在五个基准偏微分方程（例如亥姆霍兹方程、盖驱动腔流方程、波动方程、克莱因-戈登方程和对流-扩散方程）上进行测试。结果表明，QCPINNs 在不同偏微分方程上实现了与经典 PINNs 可比的精度，同时仅需约 10% 的可训练参数，并且在对流-扩散方程上进一步将相对 $L_2$ 误差降低了 40%。采用角度嵌入和级联配置的基于 DV 的电路在所有问题类型中均表现出增强的收敛稳定性。我们的发现确立了参数效率作为物理信息机器学习中一种可量化的量子优势。通过显著降低模型复杂度同时保持求解质量，QCPINNs 代表了克服科学计算应用中传统方法需要大参数空间所带来的计算瓶颈的潜在方向。