Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations
Authors: Xiang Rao, Yuxuan Shen
Organizations: 1. School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · 2. College of Future Technology, Yangtze University, Wuhan 430100, China · 3. School of Computer Science, Yangtze University, Jingzhou 434023, China · 4. State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University), Wuhan 430100, China · 5. Western Research Institute, Yangtze University, Karamay 834000, China
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.
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.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.
Fabio Pereira dos Santos, Renato Portugal, Júlio de Castro Vargas Fernandes +1
Full waveform inversion (FWI) reconstructs heterogeneous material properties from receiver data but remains computationally demanding. Physics-informed neural networks (PINNs) and their domain-decomposed variants (FBPINNs) offer a mesh-free alternative but face convergence challenges when representing complex velocity fields. We present a hybrid quantum-classical FBPINN for acoustic FWI, bringing together quantum computing and classical machine learning, in which the decomposed wavefield network and the global velocity network are implemented as classical-to-quantum pipelines terminating in parameterized quantum circuits (PQCs). The PQCs are realized as differentiable JAX statevector simulators, enabling end-to-end automatic differentiation through the classical PINN, the quantum circuit, and the physics-informed loss. On a geophysical anomaly benchmark, the quantum hybrid reaches a lower L1 velocity error than the primary classical FBPINN baseline in approximately 8x fewer training iterations, despite using approximately 33% fewer trainable parameters, and it outperforms all 15 classical hyperparameter variants tested. A second benchmark (checkerboard) demonstrates the generality of the inversion pipeline, confirming that the quantum hybrid architecture can recover structured spatial variations beyond the localized anomaly benchmark. Our framework is broadly applicable to wave-based inverse problems beyond geophysics, including medical ultrasound tomography and non-destructive evaluation.