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arxiv: 2606.20326 · v2 · pith:DGW5WWTEnew · submitted 2026-06-18 · 💻 cs.LG · physics.comp-ph

Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs

Xiang Rao , Yuxuan Shen This is my paper

Pith reviewed 2026-06-26 17:44 UTC · model grok-4.3

classification 💻 cs.LG physics.comp-ph
keywords Kolmogorov-Arnold networksphysics-informed neural networksquantum-classical hybridpartial differential equationsporous media flowhigh-frequency convergencenumerical dispersionapproximation theory
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The pith

A quantum-classical hybrid Kolmogorov-Arnold network solves PDEs by driving high-frequency errors to exponential convergence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops QCPIKAN, a hybrid model that stacks Chebyshev-polynomial KAN layers with parameterized quantum circuits and trains them under a physics-informed loss that enforces PDE constraints. Approximation theory is used to prove that the combination forces high-frequency components of the solution error to decay exponentially instead of polynomially, while also reducing numerical dispersion. Readers would care because many engineering simulations of fluid flow suffer from slowly decaying high-frequency errors that force finer grids or longer training; an exponential rate would let the same accuracy be reached with less computation. The method is demonstrated on single-phase flow, solute transport, and two-phase flow through porous media, where it outperforms earlier quantum-classical physics-informed networks in global accuracy, local error, and front tracking.

Core claim

QCPIKAN embeds physical constraints into the training loss of a hybrid network built from Chebyshev-polynomial KAN layers and parameterized quantum circuits; approximation theory then shows that the design forces high-frequency error convergence to an exponential rate and suppresses numerical dispersion, delivering superior accuracy on single-phase, component-transport, and two-phase flow problems in porous media.

What carries the argument

The hybrid framework of Chebyshev KAN layers and parameterized quantum circuits jointly optimized under a physics-informed loss function.

If this is right

  • Superior global prediction accuracy compared with existing quantum-classical physics-informed neural networks
  • Better local error control and dynamic evolution tracking
  • Improved displacement front localization in two-phase flows
  • Effective handling of single-phase flow, component transport, and two-phase seepage in porous media
  • Reduced numerical dispersion in the computed solutions

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same hybrid structure could be tested on other classes of PDEs that are dominated by high-frequency features, such as wave propagation or turbulence modeling.
  • If the exponential rate survives on larger quantum hardware, the method might lower the computational cost of ensemble simulations that currently require many repeated PDE solves.
  • The approach might also be applied to inverse problems where the same network must both solve the forward PDE and identify unknown coefficients.
  • Scalability checks on deeper KAN layers or more qubits would show whether the theoretical rate remains intact when the quantum component grows.

Load-bearing premise

The quantum circuits and Chebyshev KAN layers can be jointly optimized under the physics-informed loss without quantum noise or trainability issues destroying the exponential convergence rate.

What would settle it

Measure the decay rate of high-frequency Fourier components in the residual error on a standard linear PDE benchmark and check whether the observed scaling is exponential rather than polynomial.

Figures

Figures reproduced from arXiv: 2606.20326 by Xiang Rao, Yuxuan Shen.

Figure 1
Figure 1. Figure 1: Proposed QCPIKAN architecture. KAN preprocessor. The core idea of KAN originates from the Kolmogorov-Arnold Network architecture proposed by Liu et al. [16]. Its main idea is to replace fixed activation functions on the nodes of traditional neural networks with learnable univariate functions on the edges, thereby enhancing the ability of the network to represent complex nonlinear mappings. Unlike tradition… view at source ↗
Figure 2
Figure 2. Figure 2: Training-loss curves of different methods in example 1. (a) Reference solution [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pressure distributions predicted by different methods and the corresponding error distributions in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bar chart of the Top-K maximum absolute errors. Example 2: transient advection, diffusion and adsorption equation This example solves the concentration-transport problem in a transient chemical-agent migration process. The governing equation is Eq. (28), which integrates convection, diffusion and adsorption effects: b ( ) ( ) cq D c c tt   + =    −    v , (28) [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 5
Figure 5. Figure 5: Training-loss curves of different methods in example 2. (a) Reference solution, t = 0.4 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Concentration distributions predicted by different methods and the corresponding error [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Concentration distributions predicted by different methods and the corresponding error [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Concentration prediction at the fixed point ( [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Concentration-gradient prediction at the fixed point (50, 50) in example 2. Example 3: transient oil-water two-phase seepage When capillary pressure, gravity and source/sink terms are neglected, two-phase incompressible seepage is described by the coupled mass-conservation equations of the water and oil phases: ( ) ( ) 0 w rw w w S k x k S p t     +   −  =    , (29) (1 ) ( ) ( ) 0 w ro w o S k… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces QCPIKAN, a hybrid quantum-classical physics-informed Kolmogorov-Arnold network for PDEs that combines Chebyshev-polynomial KAN layers with parameterized quantum circuits, trained via a physics-informed loss. Theoretical investigations based on approximation theory are claimed to establish exponential convergence of high-frequency errors and mitigation of numerical dispersion. The framework is validated on three seepage scenarios in porous media (single-phase flow, component transport, two-phase flow), with asserted superiority over existing quantum-classical PINNs in global accuracy, local error control, dynamic tracking, and front localization.

Significance. If the exponential convergence result holds for the full hybrid architecture under joint optimization and the empirical claims are supported by rigorous quantitative comparisons, the work would offer a notable contribution to quantum-enhanced solvers for engineering PDEs, particularly in porous media applications. The integration of KANs with quantum circuits is a novel direction, but its impact hinges on whether the theoretical guarantees survive the quantum component's optimization challenges.

major comments (2)
  1. [§4 (Theoretical Analysis)] §4 (Theoretical Analysis): The exponential high-frequency error convergence is derived from approximation theory applied to the Chebyshev KAN layers, but the section provides no error bounds or convergence analysis that incorporates the parameterized quantum circuit parameters or the joint physics-informed optimization dynamics; this leaves the headline claim for the hybrid framework unsupported.
  2. [§5 (Numerical Experiments)] §5 (Numerical Experiments) and associated tables: No quantitative error metrics (L2 norms, convergence rates, or baseline comparisons with error bars) are reported for the three seepage scenarios, making it impossible to assess the claimed superiority over quantum-classical PINNs or to verify mitigation of dispersion.
minor comments (2)
  1. [Abstract and §2 (Method)] Abstract and §2 (Method): The description of the hybrid loss function does not clarify how the quantum circuit measurements are incorporated into the residual terms or whether any classical post-processing is applied.
  2. [§3] Notation in §3: The definition of the quantum circuit ansatz uses symbols that overlap with the KAN layer parameters without explicit disambiguation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below, indicating where revisions will be made to improve clarity and rigor.

read point-by-point responses

  1. Referee: §4 (Theoretical Analysis): The exponential high-frequency error convergence is derived from approximation theory applied to the Chebyshev KAN layers, but the section provides no error bounds or convergence analysis that incorporates the parameterized quantum circuit parameters or the joint physics-informed optimization dynamics; this leaves the headline claim for the hybrid framework unsupported.

    Authors: We acknowledge that the analysis in §4 applies approximation theory directly to the Chebyshev KAN layers and does not derive explicit error bounds that incorporate the parameterized quantum circuit parameters or the joint optimization process. The headline claim for the full hybrid framework therefore requires additional justification. We will revise §4 to either extend the theoretical treatment to the quantum component or to qualify the scope of the existing guarantees so that they align with what is rigorously shown. revision: yes

  2. Referee: §5 (Numerical Experiments) and associated tables: No quantitative error metrics (L2 norms, convergence rates, or baseline comparisons with error bars) are reported for the three seepage scenarios, making it impossible to assess the claimed superiority over quantum-classical PINNs or to verify mitigation of dispersion.

    Authors: The current version of §5 presents results primarily through visualizations and qualitative descriptions. We agree that quantitative metrics are required to substantiate the superiority claims and the mitigation of dispersion. We will add tables reporting L2 norms, observed convergence rates, and direct baseline comparisons (with error bars where appropriate) for all three seepage scenarios. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The paper's central theoretical claim rests on approximation theory investigations establishing exponential high-frequency error convergence for the QCPIKAN hybrid architecture. No equations, parameter fits, self-citations, or ansatzes are provided in the abstract or visible text that reduce the claimed result to a self-definitional input, a fitted quantity renamed as prediction, or a load-bearing self-citation chain. The derivation is presented as grounded in external approximation theory rather than constructed from the model's own outputs or prior author work invoked as uniqueness. This is the common case of a self-contained claim without exhibited circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard approximation-theory results for KANs and the assumption that quantum circuits can be embedded without destroying trainability; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Approximation theory guarantees exponential convergence for the chosen KAN and quantum-circuit combination under the physics-informed loss
    Invoked to prove high-frequency error decay

pith-pipeline@v0.9.1-grok · 5655 in / 1263 out tokens · 24863 ms · 2026-06-26T17:44:43.444930+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

74 extracted references · 67 canonical work pages · 2 internal anchors

  1. [1]

    Raissi, Perdikaris, and Karniadakis, ‘Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations’, 2019

    2019

  2. [2]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P . Perdikaris, S. Wang, and L. Yang, ‘Physics-informed machine learning’, Nat Rev Phys, vol. 3, no. 6, pp. 422–440, May 2021, doi: 10.1038/s42254-021-00314-5

    work page doi:10.1038/s42254-021-00314-5 2021

  3. [3]

    Y . Yang, X. Wang, J. Li, and R. Ge, ‘A data-physic driven method for gear fault diagnosis using PINN and pseudo-dynamic features’, Measurement, vol. 236, p. 115124, Aug. 2024, doi: 10.1016/j.measurement.2024.115124

    work page doi:10.1016/j.measurement.2024.115124 2024

  4. [4]

    Ahmadi Daryakenari, M

    N. Ahmadi Daryakenari, M. De Florio, K. Shukla, and G. E. Karniadakis, ‘AI-aristotle: A physics-informed framework for systems biology gray-box identification’, PLoS Comput Biol, vol. 20, no. 3, p. e1011916, Mar. 2024, doi: 10.1371/journal.pcbi.1011916

    work page doi:10.1371/journal.pcbi.1011916 2024

  5. [5]

    J. Hua, Y . Li, C. Liu, P . Wan, and X. Liu, ‘Physics -informed neural networks with weighted losses by uncertainty evaluation for accurate and stable prediction of manufacturing systems’, IEEE Trans. Neural Netw. Learning Syst., vol. 35, no. 8, pp. 11064–11076, Aug. 2024, doi: 10.1109/TNNLS.2023.3247163

    work page doi:10.1109/tnnls.2023.3247163 2024

  6. [6]

    Xia and Y

    Y . Xia and Y . Meng, ‘Physics-informed neural network (PINN) for solving frictional contact temperature and inversely evaluating relevant input parameters’, Lubricants, vol. 12, no. 2, p. 62, Feb. 2024, doi: 10.3390/lubricants12020062

    work page doi:10.3390/lubricants12020062 2024

  7. [7]

    A. M. Tartakovsky, C. O. Marrero, P . Perdikaris, G. D. Tartakovsky, and D. Barajas-Solano, ‘Physics- informed deep neural networks for learning parameters and constitutive relationships in subsurface flow problems’, Water Resources Research , vol. 56, no. 5, p. e2019WR026731, May 2020, doi: 10.1029/2019WR026731

    work page doi:10.1029/2019wr026731 2020

  8. [8]

    J. M. Hanna, J. V. Aguado, S. Comas-Cardona, R. Askri, and D. Borzacchiello, ‘Residual-based adaptivity for two -phase flow simulation in porous media using physics -informed neural networks’, Computer Methods in Applied Mechanics and Engineering , vol. 396, p. 115100, Jun. 2022, doi: 10.1016/j.cma.2022.115100

    work page doi:10.1016/j.cma.2022.115100 2022

  9. [9]

    Limitations of physics informed machine learning for nonlinear two- phase transport in porous media

    O. Fuks and H. A. Tchelepi, ‘Limitations of physics informed machine learning for nonlinear two-phase transport in porous media’, J Mach Learn Model Comput , vol. 1, no. 1, pp. 19 –37, 2020, doi: 10.1615/JMachLearnModelComput.2020033905

    work page doi:10.1615/jmachlearnmodelcomput.2020033905 2020

  10. [10]

    R. Xu, D. Zhang, M. Rong, and N. Wang, ‘Weak form theory-guided neural network (TgNN-wf) for deep learning of subsurface single- and two-phase flow’, Journal of Computational Physics, vol. 436, p. 110318, Jul. 2021, doi: 10.1016/j.jcp.2021.110318

    work page doi:10.1016/j.jcp.2021.110318 2021

  11. [11]

    M. M. Almajid and M. O. Abu-Al-Saud, ‘Prediction of porous media fluid flow using physics informed neural networks’, Journal of Petroleum Science and Engineering , vol. 208, p. 109205, Jan. 2022, doi: 10.1016/j.petrol.2021.109205

    work page doi:10.1016/j.petrol.2021.109205 2022

  12. [12]

    Lehmann, M

    F. Lehmann, M. Fahs, A. Alhubail, and H. Hoteit, ‘A mixed pressure-velocity formulation to model flow in heterogeneous porous media with physics -informed neural networks’, Advances in Water Resources , vol. 181, p. 104564, Nov. 2023, doi: 10.1016/j.advwatres.2023.104564

    work page doi:10.1016/j.advwatres.2023.104564 2023

  13. [13]

    Fang, ‘A high -efficient hybrid physics -informed neural networks based on convolutional neural network’, IEEE Trans

    Z. Fang, ‘A high -efficient hybrid physics -informed neural networks based on convolutional neural network’, IEEE Trans. Neural Netw. Learning Syst. , vol. 33, no. 10, pp. 5514 –5526, Oct. 2022, doi: 10.1109/TNNLS.2021.3070878

    work page doi:10.1109/tnnls.2021.3070878 2022

  14. [14]

    Yan et al., ‘Physics-informed neural network simulation of two -phase flow in heterogeneous and fractured porous media’, Advances in Water Resources , vol

    X. Yan et al., ‘Physics-informed neural network simulation of two -phase flow in heterogeneous and fractured porous media’, Advances in Water Resources , vol. 189, p. 104731, Jul. 2024, doi: 10.1016/j.advwatres.2024.104731

    work page doi:10.1016/j.advwatres.2024.104731 2024

  15. [15]

    Lin et al., ‘A layer-specific constraint-based enriched physics-informed neural network for solving two-phase flow problems in heterogeneous porous media’, Petroleum Science, vol

    J.-Q. Lin et al., ‘A layer-specific constraint-based enriched physics-informed neural network for solving two-phase flow problems in heterogeneous porous media’, Petroleum Science, vol. 22, no. 11, pp. 4714– 4735, Nov. 2025, doi: 10.1016/j.petsci.2025.07.008

    work page doi:10.1016/j.petsci.2025.07.008 2025

  16. [16]

    KAN: Kolmogorov-Arnold Networks

    Z. Liu et al. , ‘KAN: Kolmogorov -arnold networks’, Feb. 09, 2025, arXiv: arXiv:2404.19756. doi: 10.48550/arXiv.2404.19756

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2404.19756 2025

  17. [17]

    Shukla, J

    K. Shukla, J. D. Toscano, Z. Wang, Z. Zou, and G. E. Karniadakis, ‘A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks’, Computer Methods in Applied Mechanics and Engineering , vol. 431, p. 117290, Nov. 2024, doi: 10.1016/j.cma.2024.117290

    work page doi:10.1016/j.cma.2024.117290 2024

  18. [18]

    C. Zeng, J. Wang, H. Shen, and Q. Wang, ‘KAN versus MLP on irregular or noisy function’, in 2025 15th IEEE International Conference on Pattern Recognition Systems (ICPRS), Valparaiso,Viña del Mar, Chile: IEEE, Dec. 2025, pp. 1–7. doi: 10.1109/ICPRS66293.2025.11302852

    work page doi:10.1109/icprs66293.2025.11302852 2025

  19. [19]

    Shuai and F

    H. Shuai and F. Li, ‘Physics-informed kolmogorov-arnold networks for power system dynamics’, IEEE Open J. Power Energy, vol. 12, pp. 46–58, 2025, doi: 10.1109/OAJPE.2025.3529928

    work page doi:10.1109/oajpe.2025.3529928 2025

  20. [20]

    Jiang, Y

    B. Jiang, Y . Wang, Q. Wang, and H. Geng, ‘A hybrid KAN -ANN based model for interpretable and enhanced short -term load forecasting’, IEEE Trans. Artif. Intell. , pp. 1 –12, 2026, doi: 10.1109/TAI.2026.3656248

    work page doi:10.1109/tai.2026.3656248 2026

  21. [21]

    Lin et al., ‘KAN-UG: Kolmogorov–arnold network-based difference feature enhancement method with uncertainty guidance for change detection’, IEEE Trans

    S. Lin et al., ‘KAN-UG: Kolmogorov–arnold network-based difference feature enhancement method with uncertainty guidance for change detection’, IEEE Trans. Geosci. Remote Sensing , vol. 64, pp. 1 –15, 2026, doi: 10.1109/TGRS.2026.3681936

    work page doi:10.1109/tgrs.2026.3681936 2026

  22. [22]

    Huang, Y .-X

    M.-Z. Huang, Y .-X. Peng, Z.-T. Jiang, P .-N. Sun, and C.-X. Jiang, ‘Physics-informed KAN-coupled FEM for deformation analysis of complex shells’, Thin-Walled Structures , vol. 216, p. 113725, Nov. 2025, doi: 10.1016/j.tws.2025.113725

    work page doi:10.1016/j.tws.2025.113725 2025

  23. [23]

    Jiang, M

    Z.-T. Jiang, M. -Z. Huang, Y . -X. Peng, C. -X. Jiang, and W. -W. Wu, ‘Engineering application and experimental validation of the FEM -PIKAN model in ship hull deformation analysis’, Ocean Engineering, vol. 344, p. 123588, Jan. 2026, doi: 10.1016/j.oceaneng.2025.123588

    work page doi:10.1016/j.oceaneng.2025.123588 2026

  24. [24]

    Y . Gong, Y . He, Y . Mei, X. Zhuang, F. Qin, and T. Rabczuk, ‘Physics -informed kolmogorov-arnold networks for multi-material elasticity problems in electronic packaging’, Applied Mathematical Modelling, vol. 156, p. 116793, Aug. 2026, doi: 10.1016/j.apm.2026.116793

    work page doi:10.1016/j.apm.2026.116793 2026

  25. [25]

    Wan et al., ‘PIKANs: Physics -informed kolmogorov–arnold networks for landslide time -to-failure prediction’, Computers & Geosciences, vol

    J. Wan et al., ‘PIKANs: Physics -informed kolmogorov–arnold networks for landslide time -to-failure prediction’, Computers & Geosciences, vol. 208, p. 106094, Feb. 2026, doi: 10.1016/j.cageo.2025.106094

    work page doi:10.1016/j.cageo.2025.106094 2026

  26. [26]

    X. Rao, Y . Liu, X. He, and H. Hoteit, ‘Physics-informed kolmogorov–arnold networks to model flow in heterogeneous porous media with a mixed pressure-velocity formulation’, Physics of Fluids, vol. 37, no. 7, p. 076654, Jul. 2025, doi: 10.1063/5.0279122

    work page doi:10.1063/5.0279122 2025

  27. [27]

    Imankulov, Y

    T. Imankulov, Y . Kenzhebek, S. D. Bekele, and D. Akhmed-Zaki, ‘Performance and stability of physics- informed kolmogorov-arnold networks for two-phase transport in porous media: A comparative study with physics-informed neural networks’, Results in Engineering , vol. 29, p. 108823, Mar. 2026, doi: 10.1016/j.rineng.2025.108823

    work page doi:10.1016/j.rineng.2025.108823 2026

  28. [28]

    Q. Wang, W. Zha, D. Li, X. Li, L. Shen, and Z. Shi, ‘Parameterized analytical solution of seepage equation for reservoir simulation using physics -informed kolmogorov-arnold network without labels’, Journal of Computational Physics, vol. 548, p. 114599, Mar. 2026, doi: 10.1016/j.jcp.2025.114599

    work page doi:10.1016/j.jcp.2025.114599 2026

  29. [29]

    Quantum Computing in the NISQ era and beyond

    J. Preskill, ‘Quantum computing in the NISQ era and beyond’, Quantum, vol. 2, p. 79, Aug. 2018, doi: 10.22331/q-2018-08-06-79

    work page internal anchor Pith review doi:10.22331/q-2018-08-06-79 2018

  30. [30]

    and Menke, Tim and Mok, Wai-Keong and Sim, Sukin and Kwek, Leong-Chuan and Aspuru-Guzik, Alán , year=

    K. Bharti et al., ‘Noisy intermediate -scale quantum algorithms’, Rev. Mod. Phys., vol. 94, no. 1, p. 015004, Feb. 2022, doi: 10.1103/RevModPhys.94.015004

    work page doi:10.1103/revmodphys.94.015004 2022

  31. [31]

    E. A. R. Guzman and D. Lacroix, ‘Restoring symmetries in quantum computing using classical shadows’, Nov. 08, 2023, arXiv: arXiv:2311.04571. doi: 10.48550/arXiv.2311.04571

    work page doi:10.48550/arxiv.2311.04571 2023

  32. [32]

    Villalba-Díez and J

    J. Villalba-Díez and J. Ordieres -Meré, ‘Quantum-enhanced signal processing via VQE for improved biomechanical feedback control’, Digital Signal Processing , vol. 166, p. 105357, Nov. 2025, doi: 10.1016/j.dsp.2025.105357

    work page doi:10.1016/j.dsp.2025.105357 2025

  33. [33]

    Bayro and H

    A. Bayro and H. Jeong, ‘Enhancing emotional response detection in virtual reality with quantum support vector machine learning’, Computers & Graphics , vol. 128, p. 104196, May 2025, doi: 10.1016/j.cag.2025.104196

    work page doi:10.1016/j.cag.2025.104196 2025

  34. [34]

    Choi et al., ‘Performance evaluation of quantum support vector machine for COVID -19 biomarker analysis’, Computer Methods and Programs in Biomedicine , vol

    J. Choi et al., ‘Performance evaluation of quantum support vector machine for COVID -19 biomarker analysis’, Computer Methods and Programs in Biomedicine , vol. 281, p. 109343, Jul. 2026, doi: 10.1016/j.cmpb.2026.109343

    work page doi:10.1016/j.cmpb.2026.109343 2026

  35. [35]

    Bravo-Prieto, R

    C. Bravo-Prieto, R. LaRose, M. Cerezo, Y . Subasi, L. Cincio, and P . J. Coles, ‘Variational quantum linear solver’, Quantum, vol. 7, p. 1188, Nov. 2023, doi: 10.22331/q-2023-11-22-1188

    work page doi:10.22331/q-2023-11-22-1188 2023

  36. [36]

    Y . Y . Liu et al., ‘Application of a variational hybrid quantum -classical algorithm to heat conduction equation and analysis of time complexity’, Physics of Fluids , vol. 34, no. 11, p. 117121, Nov. 2022, doi: 10.1063/5.0121778

    work page doi:10.1063/5.0121778 2022

  37. [37]

    Rao, ‘Performance study of variational quantum linear solver with an improved ansatz for reservoir flow equations’, Physics of Fluids, vol

    X. Rao, ‘Performance study of variational quantum linear solver with an improved ansatz for reservoir flow equations’, Physics of Fluids, vol. 36, no. 4, p. 047104, Apr. 2024, doi: 10.1063/5.0201739

    work page doi:10.1063/5.0201739 2024

  38. [38]

    M. G. Meena et al., ‘Assessing VQLS for fluid dynamics on a hybrid quantum-HPC stack’, in 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) , Albuquerque, NM, USA: IEEE, Aug. 2025, pp. 484–485. doi: 10.1109/QCE65121.2025.10407

    work page doi:10.1109/qce65121.2025.10407 2025

  39. [39]

    S. K. N. Kannan and S. Kamrava, ‘Quantum computing for modeling fluid flow in subsurface environments’, Advances in Water Resources , vol. 215, p. 105383, Sep. 2026, doi: 10.1016/j.advwatres.2026.105383

    work page doi:10.1016/j.advwatres.2026.105383 2026

  40. [40]

    Rao, ‘The first application of quantum computing algorithm in streamline -based simulation of water-flooding reservoirs’, in ADIPEC, Abu Dhabi, UAE: SPE, Nov

    X. Rao, ‘The first application of quantum computing algorithm in streamline -based simulation of water-flooding reservoirs’, in ADIPEC, Abu Dhabi, UAE: SPE, Nov. 2024, p. D011S003R006. doi: 10.2118/221850-MS

    work page doi:10.2118/221850-ms 2024

  41. [41]

    L. A. Moncayo-Martínez and N. He, ‘Quantum optimisation for supply chain: QUBO formulations and QAOA solutions for facility location and load balancing’, Results in Engineering, vol. 29, p. 108373, Mar. 2026, doi: 10.1016/j.rineng.2025.108373

    work page doi:10.1016/j.rineng.2025.108373 2026

  42. [42]

    Tate and S

    R. Tate and S. Eidenbenz, ‘Theoretical approximation ratios for warm-started QAOA on 3-regular max- cut instances at depth 𝑝 =’, Theoretical Computer Science

  43. [43]

    Havlíček, ‘Supervised learning with quantum-enhanced feature spaces’

    V. Havlíček, ‘Supervised learning with quantum-enhanced feature spaces’

  44. [44]

    Gonon and A

    L. Gonon and A. Jacquier, ‘Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs’ , IEEE Trans. Neural Netw. Learning Syst., vol. 36, no. 6, pp. 11355–11368, Jun. 2025, doi: 10.1109/TNNLS.2025.3552223

    work page doi:10.1109/tnnls.2025.3552223 2025

  45. [45]

    Abbas, D

    A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, ‘The power of quantum neural networks’, Nat Comput Sci, vol. 1, no. 6, pp. 403–409, Jun. 2021, doi: 10.1038/s43588-021-00084-1

    work page doi:10.1038/s43588-021-00084-1 2021

  46. [46]

    doi: 10.1038/s41467-021-21728-w

    M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P . J. Coles, ‘Cost function dependent barren plateaus in shallow parametrized quantum circuits’, Nat Commun , vol. 12, no. 1, p. 1791, Mar. 2021, doi: 10.1038/s41467-021-21728-w

    work page doi:10.1038/s41467-021-21728-w 2021

  47. [47]

    Rahman and D

    R. Rahman and D. C. Nguyen, ‘Quantum convolutional neural networks: A survey on architectures, applications, and future directions’, IEEE Trans. Neural Netw. Learning Syst. , pp. 1 –20, 2026, doi: 10.1109/TNNLS.2026.3677762

    work page doi:10.1109/tnnls.2026.3677762 2026

  48. [48]

    Kyriienko, A

    O. Kyriienko, A. E. Paine, and V. E. Elfving, ‘Solving nonlinear differential equations with differentiable quantum circuits’, Phys. Rev. A, vol. 103, no. 5, p. 052416, May 2021, doi: 10.1103/PhysRevA.103.052416

    work page doi:10.1103/physreva.103.052416 2021

  49. [49]

    S. N. Larki, M. Mosleh, and M. Kheyrandish, ‘Towards quantum audio steganalysis using synergy of quantum fourier transform and quantum neural network’, Engineering Applications of Artificial Intelligence, vol. 159, p. 111595, Nov. 2025, doi: 10.1016/j.engappai.2025.111595

    work page doi:10.1016/j.engappai.2025.111595 2025

  50. [50]

    Tian et al

    J. Tian et al. , ‘Recent advances for quantum neural networks in generative learning’, IEEE Trans. Pattern Anal. Mach. Intell. , vol. 45, no. 10, pp. 12321 –12340, Oct. 2023, doi: 10.1109/TPAMI.2023.3272029

    work page doi:10.1109/tpami.2023.3272029 2023

  51. [51]

    Y . Peng, X. Li, Z. Liang, and Y . Wang, ‘HyQ2: A hybrid quantum neural network for NextG vulnerability detection’, IEEE Trans. Quantum Eng., vol. 5, pp. 1–19, 2024, doi: 10.1109/TQE.2024.3481280

    work page doi:10.1109/tqe.2024.3481280 2024

  52. [52]

    Zheng, Q

    J. Zheng, Q. Gao, M. Ogorzałek, J. Lü, and Y . Deng, ‘A quantum spatial graph convolutional neural network model on quantum circuits’, IEEE Trans. Neural Netw. Learning Syst., vol. 36, no. 3, pp. 5706–5720, Mar. 2025, doi: 10.1109/TNNLS.2024.3382174

    work page doi:10.1109/tnnls.2024.3382174 2025

  53. [53]

    X. Rao, C. Luo, X. He, and K. Hyung, ‘An efficient quantum neural network model for prediction of carbon dioxide CO2 sequestration in saline aquifers’, in ADIPEC, Abu Dhabi, UAE: SPE, Nov. 2024, p. D021S061R005. doi: 10.2118/222257-MS

    work page doi:10.2118/222257-ms 2024

  54. [54]

    H. Cho, J. Kim, K. T. No, and H. Lim, ‘Hybrid quantum neural networks with variational quantum regressor for enhancing QSPR modeling of CO2-capturing amine’, EPJ Quantum Technol., vol. 12, no. 1, p. 79, Dec. 2025, doi: 10.1140/epjqt/s40507-025-00385-8

    work page doi:10.1140/epjqt/s40507-025-00385-8 2025

  55. [55]

    Trahan, M

    C. Trahan, M. Loveland, and S. Dent, ‘Quantum physics -informed neural networks’, Entropy, vol. 26, no. 8, p. 649, Jul. 2024, doi: 10.3390/e26080649

    work page doi:10.3390/e26080649 2024

  56. [56]

    Xiao et al., ‘Physics-informed quantum neural network for solving forward and inverse problems of partial differential equations’, Physics of Fluids , vol

    Y . Xiao et al., ‘Physics-informed quantum neural network for solving forward and inverse problems of partial differential equations’, Physics of Fluids , vol. 36, no. 9, p. 097145, Sep. 2024, doi: 10.1063/5.0226232

    work page doi:10.1063/5.0226232 2024

  57. [57]

    Berger, N

    S. Berger, N. Hosters, and M. Möller, ‘Trainable embedding quantum physics informed neural networks for solving nonlinear PDEs’, Sci Rep, vol. 15, no. 1, p. 18823, May 2025, doi: 10.1038/s41598 - 025-02959-z

    work page doi:10.1038/s41598 2025

  58. [58]

    B. O. Fernandez, C. Leao, D. E. Revelo, and G. Fernandes, ‘Exploring hybrid physics -informed neural networks with quantum layers for solving the wave equation’, in Fifth EAGE Digitalization Conference & Exhibition, Edinburgh, Scotland, United Kingdom,: European Association of Geoscientists & Engineers, 2025, pp. 1–5. doi: 10.3997/2214-4609.202539060

    work page doi:10.3997/2214-4609.202539060 2025

  59. [59]

    N. B. Dehaghani, A. P . Aguiar, and R. Wisniewski, ‘A hybrid quantum-classical physics-informed neural network architecture for solving quantum optimal control problems’ , 2024, arXiv. doi: 10.48550/ARXIV.2404.15015

    work page doi:10.48550/arxiv.2404.15015 2024

  60. [60]

    F. Y . Leong, W.-B. Ewe, S. B. Q. Tran, Z. Zhang, and J. Y . Khoo, ‘Hybrid quantum physics-informed neural network: Towards efficient learning of high -speed flows’, Computers & Fluids , vol. 301, p. 106782, Oct. 2025, doi: 10.1016/j.compfluid.2025.106782

    work page doi:10.1016/j.compfluid.2025.106782 2025

  61. [61]

    Farea, S

    A. Farea, S. Khan, and M. S. Celebi, ‘QCPINN: Quantum-classical physics-informed neural networks for solving PDEs’, Oct. 18, 2025, arXiv: arXiv:2503.16678. doi: 10.48550/arXiv.2503.16678

    work page doi:10.48550/arxiv.2503.16678 2025

  62. [62]

    Z. Song, R. Deaton, B. Gard, and S. H. Bryngelson, ‘Incompressible navier –stokes solve on noisy quantum hardware via a hybrid quantum–classical scheme’, Computers & Fluids, vol. 288, p. 106507, Feb. 2025, doi: 10.1016/j.compfluid.2024.106507

    work page doi:10.1016/j.compfluid.2024.106507 2025

  63. [63]

    P . R. Hegde et al., ‘A hybrid quantum-classical particle-in-cell method for plasma simulations’, Future Generation Computer Systems, vol. 175, p. 108087, Feb. 2026, doi: 10.1016/j.future.2025.108087

    work page doi:10.1016/j.future.2025.108087 2026

  64. [64]

    X. Rao, Y . Liu, and Y . Shen, ‘Quantum-classical physics-informed neural networks for solving reservoir seepage equations’, 2025, arXiv. doi: 10.48550/ARXIV.2512.03923

    work page doi:10.48550/arxiv.2512.03923 2025

  65. [65]

    Lantigua, G

    S. Lantigua, G. Giraldi, and R. Portugal, ‘Classical -quantum hybrid architecture for physics-informed neural networks’, Phys. Rev. A, vol. 113, no. 4, p. 042446, Apr. 2026, doi: 10.1103/fdd3-qz1s

    work page doi:10.1103/fdd3-qz1s 2026

  66. [66]

    Trefethen, Approximation Theory and Approximation Practice

    Lloyd N. Trefethen, Approximation Theory and Approximation Practice . Philadelphia, Pennsylvania, USA: Society for Industrial and Applied Mathematics (SIAM), 2013

    2013

  67. [67]

    S. SS, K. AR, G. R, and A. KP , ‘Chebyshev polynomial-based kolmogorov-arnold networks: An efficient architecture for nonlinear function approximation’ , Jun. 14, 2024, arXiv: arXiv:2405.07200. doi: 10.48550/arXiv.2405.07200

    work page doi:10.48550/arxiv.2405.07200 2024

  68. [68]

    Larocca, N

    M. Larocca, N. Ju, D. García -Martín, P . J. Coles, and M. Cerezo, ‘Theory of overparametrization in quantum neural networks’, Nat Comput Sci , vol. 3, no. 6, pp. 542 –551, Jun. 2023, doi: 10.1038/s43588 - 023-00467-6

    work page doi:10.1038/s43588 2023

  69. [69]

    Mathematical Foundations of Quantum Computing: Operator Theory, Entanglement, and Quantum Algorithms,

    Q. Wang, “Mathematical Foundations of Quantum Computing: Operator Theory, Entanglement, and Quantum Algorithms,” diss., 2025. https://hdl.handle.net/1969.1/1598660

    2025

  70. [70]

    Canuto, M

    C. Canuto, M. Y . Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods: Evolution to complex geometries and applications to fluid dynamics. in Scientific Computation Ser. Berlin, Heidelberg: Springer Berlin / Heidelberg, 2007

    2007

  71. [71]

    Yu et al

    Z. Yu et al. , ‘Non -asymptotic approximation error bounds of parameterized quantum circuits’, in Advances in Neural Information Processing Systems 37 , Vancouver, BC, Canada: Neural Information Processing Systems Foundation, Inc. (NeurIPS), 2024, pp. 99089–99127. doi: 10.52202/079017-3143

    work page doi:10.52202/079017-3143 2024

  72. [72]

    A. R. Barron, ‘Universal approximation bounds for superpositions of a sigmoidal function’, IEEE Trans. Inform. Theory, vol. 39, no. 3, pp. 930–945, May 1993, doi: 10.1109/18.256500

    work page doi:10.1109/18.256500 1993

  73. [73]

    Jacot, F

    A. Jacot, F. Gabriel, and C. Hongler, ‘Neural tangent kernel: Convergence and generalization in neural networks’, 2018

    2018

  74. [74]

    Lazzari and X

    J. Lazzari and X. Liu, ‘Understanding the spectral bias of coordinate based MLPs via training dynamics’ , May 04, 2023, arXiv: arXiv:2301.05816. doi: 10.48550/arXiv.2301.05816

    work page doi:10.48550/arxiv.2301.05816 2023