将量子电路与柯尔莫哥洛夫-阿诺德网络结合，高效求解复杂偏微分方程。
Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs
- 融合切比雪夫KAN层与参数化量子线路，嵌入物理约束优化训练。
- 在多孔介质渗流问题中实现高精度全局预测与前缘定位。
- 理论证明可指数级加速高频误差收敛，适合高保真模拟场景。
我们提出QCPIKAN，首个用于求解偏微分方程（PDEs）的量子-经典物理信息柯尔莫哥洛夫-阿诺德网络。该框架基于切比雪夫多项式KAN层与参数化量子电路，通过在损失函数中嵌入物理约束，确保模型输出满足物理一致性。理论分析基于逼近论，证明该设计能将高频误差收敛速率提升至指数级，并有效抑制数值色散。我们在多孔介质中的三种典型渗流场景下验证：单相流、组分输运与两相流。相较于现有量子-经典物理信息神经网络，QCPIKAN在全局预测精度、局部误差控制、动态演化跟踪及驱替前缘定位方面均表现更优。本工作为求解复杂偏微分方程提供了高效稳健的新方案。
原文摘要 · Abstract (English)
We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.
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