š¤ AI Summary
This work addresses the challenges of insufficient accuracy and difficulty in satisfying structural constraints when solving Ī±-cut-based fuzzy partial differential equations. The authors propose QCPIKAN, a quantum-classical hybrid physics-informed network that takes spatiotemporal coordinates and membership degrees as inputs to jointly approximate the upper and lower bounds of Ī±-cuts, while embedding governing equations, initial/boundary conditions, and fuzzy structural constraints for end-to-end training. By innovatively integrating parameterized quantum circuits with ChebyKAN modules, they establish a unified error analysis framework and prove that, under sufficient quantum entanglement gain, QCPIKAN achieves a tighter prior error bound than its classical counterpart. Numerical experiments demonstrate that QCPIKAN reduces the average relative LĀ² error by 1.1ā2.7 times and decreases wavefront location error by approximately 1.77 times across elliptic, parabolic, and hyperbolic fuzzy PDEs, significantly enhancing solution accuracy.
š Abstract
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 Ī±-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 Ī±-cuts.