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

TL;DR

QCPINN introduces a hybrid quantum–classical PINN framework to solve PDEs with dramatically fewer trainable parameters while maintaining competitive accuracy. The study analyzes both continuous-variable and discrete-variable quantum circuits as the core quantum layer, paired with classical pre-/post-processing, and evaluates multiple topologies and embeddings across five benchmark PDEs. DV-Circuit designs with Angle embedding, particularly Cascade and Cross-mesh topologies, achieve the strongest parameter efficiency and robust convergence, yielding substantial reductions in trainable parameters (often >70%) and notable improvements in some L2 errors (up to ~64% in convection-diffusion) compared to classical PINNs. CV-Circuit QCPINN, while offering theoretical advantages, faces stability challenges and generally underperforms DV variants under the tested conditions; overall, the work demonstrates parameter-efficient quantum enhancements for PDE solving and provides open-source tooling for reproducibility and further exploration.

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 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 a 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.QCPINN presents a promising pathway to address the computational challenges associated with solving PDEs.

Figures (12)

• Figure 1: Overview of the proposed QCPINN architecture, illustrating its hybrid structure consisting of three main components: (a) a classical preprocessor for input encoding, (b) the core processing unit using either a quantum neural network (QNN) or classical neural network (NN), and (c) a classical postprocessor for decoding the QNN or NN outputs. The QNN is implemented as either a CV or DV circuit with $k$ layers. The architecture integrates a loss function to concurrently minimize PDE residuals and enforce compliance with initial and boundary conditions, thereby ensuring adherence to physical constraints during training. The various configurations of the CV and DV circuits are detailed in Table \ref{['tab:qcpinn_config']}.
• Figure 2: DV-Circuit topologies evaluated in this study, shown with five qubits and angle embedding as an example. Square boxes denote parameterized single-qubit rotation gates, and lines with dots indicate entangling operations (e.g., CNOT gates). The characteristics of these circuits are shown in Table \ref{['tab:topology_comparison']}.
• Figure 3: Comparison of the loss history of the best models of DV-Circuit QCPINN (Angle-Cascade) with PINN (Model-2) for solving the Helmholtz equation. For loss history of additional models, see Fig. \ref{['fig:loss_history_helmholtz_all']} in the appendix.
• Figure 4: Comparison between the reference solution and model predictions for the Helmholtz equation. This includes the best results from our DV-Circuit QCPINN (Angle-Cascade) alongside PINN (Model-2). The first row shows the PDE solution $u$, while the second row shows the force field $f$.
• Figure 5: Training loss history for the Cavity problem. The plots compare the convergence behavior of the QCPINN (Angle-Cascade) model with the best-performing PINN (Model-1).