QCPINN: Quantum Classical Physics-Informed Neural Networks for Solving PDEs

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.