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arXiv:2608.08782

Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations

Xiang Rao, Yuxuan Shen

Published Aug 9, 2026• 0 citations
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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 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.

DOI
10.48550/arxiv.2608.08782

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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.
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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Physics-informed neural networks (PINNs) and hybrid quantum-classical extensions provide a promising framework for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. In this work, we study embedding strategies for trainable embedding quantum physics-informed neural networks (TE-QPINNs) in the context of nonlinear reaction-diffusion (RD) systems. We introduce an extended TE-QPINN (x-TE-QPINN) architecture that supports both classical and fully quantum embeddings, enabling a controlled comparison between feedforward neural network-based feature maps and parameterized quantum circuit embeddings. The first architecture is the classical embedding feed-forward neural network-based TE-QPINN (FNN-TE-QPINN), while the latter variant is a purely quantum one, referred to as quantum embedding neural network-based TE-QPINN (QNN-TE-QPINN). The proposed framework employs hardware-efficient variational quantum circuits and species-specific readout operators to approximate coupled multi-field dynamics while enforcing governing equations, boundary conditions, and initial conditions through a physics-informed loss function. By isolating the embedding mechanism while keeping the variational ansatz, loss formulation, and optimization procedure fixed, we analyze the impact of embedding design on gradient structure, parameter scaling, and quantum resource requirements. Numerical experiments on one- and two-dimensional RD equations demonstrate that quantum embeddings can replace classical embeddings without degradation in solution accuracy and, in certain regimes, exhibit improved optimization behavior compared to classical PINNs and hybrid quantum models with fixed embeddings. These results provide architectural insight into hybrid quantum PDE solvers and inform the design of resource-efficient quantum physics-informed learning methods.
Partial differential equations (PDEs) form the backbone of simulations of many natural phenomena, for example in climate modeling, material science, and even financial markets. The application of physics-informed neural networks to accelerate the solution of PDEs is promising, but not competitive with numerical solvers yet. Here, we show how quantum computing can improve the ability of physics-informed neural networks to solve partial differential equations. For this, we develop hybrid networks consisting of quantum circuits combined with classical layers and systematically test them on various non linear PDEs and boundary conditions in comparison with purely classical networks. We demonstrate that the advantage of using quantum networks lies in their ability to achieve an accurate approximation of the solution in substantially fewer training epochs, particularly for more complex problems. These findings provide the basis for targeted developments of hybrid quantum neural networks with the goal to significantly accelerate numerical modeling.