Evaluating the performance of classical, quantum, and hybrid quantum neural networks in solving differential equations

We investigate the performance of classical, quantum, and hybrid quantum neural networks in solving partial differential equations through both supervised and unsupervised learning. The performance of each type of network is evaluated in solving the damped harmonic oscillator, the Einstein field equations, and the time-independent Schrödinger equation. We employed physics-informed supervised neural networks to solve the damped harmonic oscillator and the Einstein field equations, and developed an unsupervised neural network algorithm based on the shooting method for the time-independent Schrödinger equation. For the quantum neural networks, we introduced three variational parameterized quantum feature maps and two quantum circuits for the hybrid quantum neural networks. Each network was examined in three different configurations and evaluated across multiple random seeds. Under favorable parameter initializations, the hybrid quantum neural networks achieve higher accuracy than the classical neural networks in most cases across all tested problems. In contrast, the quantum neural networks attain the best accuracy for the damped harmonic oscillator and also perform well in the Schrödinger equation problem. Both the quantum and the hybrid quantum neural networks require fewer parameters than the classical neural networks and converge faster during optimization. All three models exhibit sensitivity to parameter initialization, with the quantum neural networks displaying the highest variability across different problems.