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

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 physics-informed neural network (QCPINN) 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 requiring approximately 10% of the trainable parameters used in classical approaches. It also results in a 40% reduction in the relative error L2 for the convection-diffusion equation. These findings demonstrate the potential of parameter efficiency as a measurable quantum advantage in physics-informed machine learning, significantly reducing model complexity while preserving solution quality. This approach presents a promising solution to the computational challenges associated with solving PDEs.