用于求解模糊微分方程的量子-经典物理信息Kolmogorov-Arnold网络
Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations
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中文总结 AI 辅助
本研究提出量子-经典物理信息Kolmogorov-Arnold网络（QCPIKAN），通过混合函数逼近器求解模糊微分方程，理论分析其误差界更优，数值实验显示其预测精度优于PIKAN。
中文摘要 AI 辅助
本研究提出了一种专门用于求解模糊微分方程的量子-经典物理信息Kolmogorov-Arnold网络（QCPIKAN）。该网络将时空坐标与隶属度作为联合输入，采用ChebyKAN模块和参数化量子电路构建混合函数逼近器，同时逼近与α-截集相关的上下端点函数，并将控制方程、初边值条件及模糊结构约束纳入训练目标。理论层面，为QCPIKAN和PIKAN建立了统一的误差分析框架，其中端点解误差被分解为逼近误差、采样误差、优化误差及模糊结构约束误差。在适定性和残差稳定性假设下，证明当量子纠缠特性带来的表示增益超过额外计算误差时，QCPIKAN具有更小的先验误差界。在理想量子模拟环境中对椭圆型、抛物型和双曲型方程开展数值实验，结果显示QCPIKAN可捕捉到随α增大的解区间整体收缩现象；在多数测试的隶属度水平下，PIKAN的平均相对L2误差约为QCPIKAN的1.1至2.7倍，在模糊对流算例中，PIKAN的平均波前位置误差约为QCPIKAN的1.77倍。尽管如此，两种模型在边界附近、高梯度区域及波前周围仍存在局部模糊结构违反情况。这些结果表明，QCPIKAN提供了一种量子-经典混合的物理信息计算框架，对求解以α-截集表示的模糊偏微分方程具有较高的预测精度。
英文摘要
In this study, we propose a quantum-classical physics-informed Kolmogorov-Arnold network (QCPIKAN) dedicated to the solution of fuzzy differential equations. The network takes the spatiotemporal coordinates and membership level as joint inputs and employs ChebyKAN modules and a parameterized quantum circuit to construct a hybrid function approximator. It simultaneously approximates the lower and upper endpoint functions associated with the α-cuts and incorporates the governing equations, initial-boundary conditions, and fuzzy-structural constraints into the training objective. Theoretically, a unified error-analysis framework is established for QCPIKAN and PIKAN, in which the endpoint-solution error is decomposed into approximation, sampling, optimization, and fuzzy-structure constraint errors. Under the assumptions of well-posedness and residual stability, it is proved that QCPIKAN has a smaller a priori error bound when the representational gain introduced by quantum entanglement features exceeds the additional computational error. Numerical experiments are conducted for elliptic, parabolic, and hyperbolic equations in an ideal quantum-simulation environment. The results show that QCPIKAN captures the overall contraction of the solution interval as increases. At most tested membership levels, the mean relative L2 error of PIKAN is approximately 1.1-2.7 times that of QCPIKAN. In the fuzzy convection example, the mean wavefront-position error of PIKAN is approximately 1.77 times that of QCPIKAN. Nevertheless, both models still exhibit local fuzzy-structure violations near boundaries, in high-gradient regions, and around the wavefront. These results indicate that QCPIKAN provides a quantum-classical hybrid physics-informed computational framework with comparatively high predictive accuracy for solving fuzzy partial differential equations represented by α-cuts.
