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A Dual Physics-Informed Kolmogorov-Arnold Neural Network Framework for Continuum Topology Optimization

T0 review · 0 major / 4 minor · reviewed 2026-05-20 · grok-4.3

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Pith's one-line read A pair of Higher-Order ReLU Kolmogorov-Arnold Networks solves PDEs and sensitivities for continuum topology optimization.

desk verdict The dual HRKAN setup splits displacement prediction from sensitivity analysis and shows clear efficiency gains over PINNs in the tested topology optimization cases. read the letter →

arxiv 2605.19536 v1 pith:T2EZ3CY6 submitted 2026-05-19 cs.CE

Junyuan Zhang , Jing Cao , Abdullah Dawar , Kun Cai , Qinghua Qin This is my paper · ORCID
classification cs.CE
keywords continuumtopologyoptimizationphysics-informedneuralnetworksKolmogorov-ArnoldHigher-OrderReLUsensitivityanalysisPDE-constrainedstructuraldesign
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops a dual network approach called DPIKAN-TO that uses one HRKAN to predict displacements by solving the structural PDEs and a second HRKAN to compute sensitivities for updating the material design variables. This addresses the high computational cost, spectral bias, and lack of adaptability seen in earlier physics-informed neural network methods for topology optimization. The key innovation lies in the learnable activation functions that allow the networks to fit complex responses more effectively. Numerical tests confirm it works for linear structures, compliant mechanisms, and fluid-solid interactions. A sympathetic reader would care because this could streamline the design of lightweight or multifunctional components across engineering fields.

What carries the argument

Dual Physics-Informed Kolmogorov-Arnold Networks, specifically the combination of d-HRKAN for PDE solution and s-HRKAN for sensitivity analysis, with learnable activations from Higher-Order ReLU-based KANs.

What would settle it

A direct comparison on the classic MBB beam or cantilever beam problem where DPIKAN-TO either matches or fails to match known optimal topologies while measuring wall-clock time against a baseline PINN-TO implementation.

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Extended reading notes

Core claim

By training a displacement-informed HRKAN on the physical equations and a sensitivity-informed HRKAN on the derivatives needed for optimization, the DPIKAN-TO method finds optimal material distributions in several classes of structural problems while using less computation than PINN alternatives.

Load-bearing premise

Learnable activation functions in the HRKANs provide accurate approximations of structural responses and sensitivities free from spectral bias and excessive computation.

Editorial extensions

If this is right

  • Optimal material layouts emerge for linear structures.
  • Compliant mechanisms are successfully designed.
  • Fluid-solid coupled systems produce valid topologies.
  • The framework applies to new PDE types through its adaptable activations.
  • Overall computational efficiency improves and cost decreases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Designers in aerospace or automotive might adopt it for rapid iteration on optimized parts.
  • It opens a path to optimization under uncertainty or multi-physics without retraining from scratch.
  • Testing on larger scale problems could reveal if the efficiency gains hold.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The manuscript introduces the Dual Physics-Informed Kolmogorov-Arnold Networks-based Topology Optimization (DPIKAN-TO) method. It employs a displacement-informed HRKAN (d-HRKAN) to solve the governing PDEs for structural analysis and a sensitivity-informed HRKAN (s-HRKAN) to compute sensitivities for design-variable updates. The approach replaces standard PINN components with Higher-Order ReLU-based KANs whose learnable activation functions are intended to reduce spectral bias and computational cost. Numerical demonstrations are presented for linear elastic structures, compliant mechanisms, and fluid-solid interaction problems, with the claim that the framework extends readily to new PDE-governed optimization tasks.

Significance. If the reported numerical results hold, the dual-HRKAN construction offers a concrete route to lower-cost, less spectrally biased physics-informed optimization. The explicit quantification of wall-clock reductions versus baseline PINNs and the demonstration across three distinct physics regimes constitute a useful contribution to the growing literature on neural-network topology optimization.

minor comments (4)
  1. [Abstract] Abstract: the statement that DPIKAN-TO 'demonstrates significantly improved computational efficiency' should be accompanied by the concrete wall-clock or iteration-count ratios that appear in the results section.
  2. [Section 3.2] Section 3.2: the definition of the sensitivity loss for s-HRKAN should explicitly state whether the adjoint or direct differentiation route is used; the current wording leaves the exact form of the physics residual ambiguous.
  3. [Figure 5] Figure 5 (compliant-mechanism example): the convergence plot of the objective function lacks a comparison curve for a standard PINN baseline; adding this trace would strengthen the efficiency claim.
  4. [Table 2] Table 2: the reported L2 errors for d-HRKAN are given without the corresponding mesh size or number of collocation points; these parameters are needed to assess whether the accuracy gain is independent of discretization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation for minor revision. We appreciate the recognition that the dual-HRKAN framework offers a concrete route to lower-cost, less spectrally biased physics-informed optimization, along with the explicit quantification of wall-clock reductions and demonstrations across three physics regimes.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained via physics losses and numerical validation

full rationale

The DPIKAN-TO framework defines d-HRKAN to approximate PDE solutions via displacement-informed losses and s-HRKAN to approximate sensitivities via separate physics losses. These are trained independently on the governing equations and boundary conditions of each problem class (linear elasticity, compliant mechanisms, fluid-solid coupling). Optimal layouts are obtained by iterating the two networks with standard design-variable updates. No equation reduces a claimed prediction to a fitted parameter by construction, no uniqueness theorem is imported from self-citation, and the learnable activations are introduced as an empirical improvement whose performance is demonstrated on concrete numerical examples rather than assumed. The central results therefore rest on externally verifiable PDE residuals and topology-optimization convergence metrics rather than on any definitional loop.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities. The core premise rests on the unverified capability of HRKANs to approximate PDE solutions and sensitivities via learnable activations.

assumptions (1)
  • domain assumption Higher-Order ReLU-based Kolmogorov-Arnold Networks with learnable activations can accurately approximate responses of complex structural systems and perform sensitivity analysis.
    This assumption underpins the claim that DPIKAN-TO overcomes PINN limitations and extends to new PDEs.

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Cite this review

Pith. "Pith review of A Dual Physics-Informed Kolmogorov-Arnold Neural Network Framework for Continuum Topology Optimization." pith.science (2026). https://pith.science/paper/T2EZ3CY6

@misc{pith2026260519536,
  author       = {Pith},
  title        = {Pith review of: A Dual Physics-Informed Kolmogorov-Arnold Neural Network Framework for Continuum Topology Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2EZ3CY6}},
  note         = {Machine review of arXiv:2605.19536}
}
read the original abstract

In continuum topology optimization (TO), two essential procedures are involved: structural analysis through the solution of partial differential equations (PDEs) and the subsequent update of design variables. Both procedures can be addressed by training neural networks using the corresponding physical information. Accordingly, Physics-Informed Neural Network (PINN)-based algorithms have been developed for TO. However, PINN-based methods suffer from several notable limitations, including high computational cost, spectral bias, and limited adaptability in solving PDEs.To overcome these challenges, this study proposes a novel algorithm that incorporates two Higher-Order ReLU-based Kolmogorov-Arnold Networks (HRKANs). Specifically, a displacement-informed HRKAN (d-HRKAN) is designed to predict PDE solutions, while a sensitivity-informed HRKAN (s-HRKAN) is developed to perform sensitivity analysis for updating design variables. For convenience, the proposed approach is referred to as the Dual Physics-Informed Kolmogorov-Arnold Networks-based Topology Optimization (DPIKAN-TO) method. By leveraging learnable activation functions, the proposed neural networks can accurately approximate the responses of complex structural systems. Moreover, compared with conventional PINN-based methods, DPIKAN-TO demonstrates significantly improved computational efficiency and reduced computational cost. Numerical examples show that DPIKAN-TO can successfully identify optimal material layouts for linear structures, compliant mechanisms, and fluid-solid coupled systems. Furthermore, owing to the use of learnable activation functions, the proposed framework can be readily extended to structural optimization problems governed by new types of PDEs.

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Works this paper leans on

51 extracted references · 51 canonical work pages

  1. [1]

    Aage, N., Andreassen, E., & Lazarov, B. S. (2015). Topology optimization using PETSc: An easy -to-use, fully parallel, open source topology optimization framework. Structural and Multidisciplinary Optimization , 51 (3), 565 -572, doi:https://doi.org/10.1007/s00158-014-1157-0

    work page doi:10.1007/s00158-014-1157-0 2015

  2. [2]

    S., & Schevenels, M

    Amir, O., Sigmund, O., Lazarov, B. S., & Schevenels, M. (2012). Efficient reanalysis techniques for robust topology ` optimization. Computer Methods in Applied Mechanics Engineering , 245 , 217 -231, doi:https://doi.org/10.1016/j.cma.2012.07.008

    work page doi:10.1016/j.cma.2012.07.008 2012

  3. [3]

    Efficient topology optimization in MAT- LAB using 88 lines of code

    Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B. S., & Sigmund, O. (2011). Efficient topology optimizatio n in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization , 43 , 1 -16, doi:https://doi.org/10.1007/s00158-010-0594-7. Bendsøe, M. P. (1989). Optimal shape design as a material distribution problem. Structural and Multi...

    work page doi:10.1007/s00158-010-0594-7 2011

  4. [4]

    Bourdin, B., & Chambolle, A. (2003). Design -dependent loads in topology optimization. ESAIM: Control, Optimisation and Calculus of Variations, 9, 19-48, doi:https://doi.org/10.1051/cocv:2002070

    work page doi:10.1051/cocv:2002070 2003

  5. [5]

    E., & Tortorelli, D

    Bruns, T. E., & Tortorelli, D. A. (2001). Topology optimization of non-linear elastic structures and compliant mechanisms. Computer Methods in Applied Mechanics Engine ering, 190 (26-27), 3443 -3459, doi: https://doi.org/10.1016/S0045- 7825(00)00278-4

    work page doi:10.1016/s0045- 2001

  6. [6]

    Cavazzuti, M., Baldini, A., Bertocchi, E., Costi, D., Torricelli, E., et al. (2011). High performance automotive chassis design: a topology optimization based approach. Structural and Multidisciplinary Optimization , 44 (1), 45 -56, doi:https://doi.org/10.1007/s00158-010-0578-7

    work page doi:10.1007/s00158-010-0578-7 2011

  7. [7]

    S., & Kumar, P

    Chadha, K. S., & Kumar, P. (2025). PyTOaCNN: Topology optimization using an adaptive convolutional neural network in Python. Soft Computing, 1-21

    work page 2025

  8. [8]

    Chandrasekhar, A., & Suresh, K. (2021). TOuNN: Topology optimization using neural networks. Structural and Multidisciplinary Optimization, 63(3), 1135-1149, doi:https://doi.org/10.1007/s00158-020-02748-4

    work page doi:10.1007/s00158-020-02748-4 2021

Show all 51 references
  1. [9]

    Chen, A., Cai, K., Zhao, Z.-L., Zhou, Y ., Xia, L., et al. (2021). Controlling the maximum first principal stress in topology optimization. Structural and Multidisciplinary Optimization , 63 (1), 327 -339, doi: https://doi.org/10.1007/s00158-020- 02701-5

    2021 doi

  2. [10]

    S., & To, A

    Deng, H., Vulimiri, P. S., & To, A. C. (2022). An efficient 146 -line 3D sensitivity analysis code of stress -based topology optimization written in MATLAB. Optimization and Engineering , 23(3), 1733-1757, doi:https://doi.org/10.1007/s11081- 021-09675-3

    2022 doi

  3. [11]

    Farea, A., & Celebi, M. S. (2025). Learnable activation functions in physics -informed neural networks for solving partial differential equations. Computer Physics Communications, 109753, doi:https://doi.org/10.1016/j.cpc.2025.109753

    2025 doi

  4. [12]

    Frecker, M., Ananthasuresh, G., Nishiwaki, S., Kikuchi, N., & Kota, S. (1997). Topological synthesis of compliant mechanisms using multi-criteria optimization. Journal of Mechanical Design, doi:https://doi.org/10.1115/1.2826242

    1997 doi

  5. [13]

    Gogu, C. (2015). Improving the efficiency of large scale topology optimization through on -the-fly reduced order model construction. International Journal for Numerical Methods in Engineering , 101 (4), 281 -304, doi:https://doi.org/10.1002/nme.4797

    2015 doi

  6. [14]

    K., Prévost, J

    Guest, J. K., Prévost, J. H., & Belytschko, T. (2004). Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Internati onal Journal for Numerical Methods in Engineering , 61(2), 238-254, ` doi:https://doi.org/10.1002/nme.1064

    2004 doi

  7. [15]

    Guo, X., Zhang, W., Zhang, J., & Yuan, J. (2016). Explicit structural topology optimization based on m oving morphable components (MMC) with curved skeletons. Computer Methods in Applied Mechanics Engineering , 310 , 711 -748, doi:https://doi.org/10.1016/j.cma.2016.07.018

    2016 doi

  8. [16]

    B., & Olhoff, N

    Hammer, V . B., & Olhoff, N. (2 000). Topology optimization of continuum structures subjected to pressure loading. Structural and Multidisciplinary Optimization, 19(2), 85-92, doi:https://doi.org/10.1007/s001580050088

    doi

  9. [17]

    A., Koric, S., & Jasiuk, I

    He, J., Abueidda, D., Al-Rub, R. A., Koric, S., & Jasiuk, I. (2023a). A deep learning energy -based method for classical elastoplasticity. International Journal of Plasticity, 162, 103531, doi:https://doi.org/10.1016/j.ijplas.2023.103531

    2023 doi

  10. [18]

    He, J., Chadha, C., Kushwaha, S., Koric, S., Abueidda, D., et al. (2023b). Deep energy method in topology optimization applications. Acta Mechanica, 234(4), 1365-1379, doi:https://doi.org/10.1007/s00707-022-03449-3

    doi

  11. [19]

    Holmberg, E., Torstenfelt, B., & Klarbring, A. (2013). Stress constrained topology optimization. Structural and Multidisciplinary Optimization, 48(1), 33-47, doi:https://doi.org/10.1007/s00158-012-0880-7

    2013 doi

  12. [20]

    Huang, X., & Xie, Y . -M. (2010). A further review of ESO type methods for topology optimization. Structural and Multidisciplinary Optimization, 41(5), 671-683, doi:https://doi.org/10.1007/s00158-010-0487-9

    2010 doi

  13. [21]

    Batch normalization: Accelerating deep network training by reducing internal covariate shift

    Ioffe, S., & Szegedy, C. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International conference on machine learning, 2015 (pp. 448-456): pmlr. doi:https://doi.org/10.48550/arXiv.1502.03167

    2015 doi arXiv

  14. [22]

    Jeong, H., Batuwatta-Gamage, C., Bai, J., Rathnayaka, C., Zhou, Y ., et al. (2025). An advanced p hysics-informed neural network-based framework for nonlinear and complex topology optimization. Engineering Structures , 322 , 119194, doi:https://doi.org/10.1016/j.engstruct.2024.119194

    2025 doi

  15. [23]

    M., Rathnayaka, C., et al

    Jeong, H., Batuwatta-Gamage, C., Bai, J., Xie, Y . M., Rathnayaka, C., et al. (2023). A complete physics-informed neural network-based framework for structural topology optimization. Computer Methods in Applied Mechanics Engineering , 417, 116401, doi:https://doi.org/10.1016/j...

    2023 doi

  16. [24]

    Kumar, P. (2023). TOPress: a MATLAB implementation for topology optimization of structures subjected to design - dependent pressure loads. Structural and Multidisciplina ry Optimization, 66(4), 97, doi: https://doi.org/10.1007/s00158- 023-03533-9

    2023 doi

  17. [25]

    S., & Langelaar, M

    Kumar, P., Frouws, J. S., & Langelaar, M. (2020). Topology optimization of fluidic pressure -loaded structures and compliant mechanisms using the Darcy method. Structural and Multidisciplinary Optimization , 61 (4), 1637 -1655, doi:https://doi.org/10.1007/s00158-019-02442-0

    2020 doi

  18. [26]

    Le, C., Norato, J., Bruns, T., Ha, C., & Torto relli, D. (2010). Stress -based topology optimization for continua. Structural and Multidisciplinary Optimization, 41(4), 605-620, doi:https://doi.org/10.1007/s00158-009-0440-y

    2010 doi

  19. [27]

    Li, W., Bazant, M. Z. , & Zhu, J. (2021). A physics -guided neural network framework for elastic plates: Comparison of governing equations -based and energy -based approaches. Computer Methods in Applied Mechanics Engineering , 383, 113933, doi:https://doi.org/10.1016/j.cma.20...

    2021 doi

  20. [28]

    Liu, K., & Tovar, A. (2014). An efficient 3D topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 50(6), 1175-1196, doi:https://doi.org/10.1007/s00158-014-1107-x

    2014 doi

  21. [29]

    Liu, Z., Wang, Y ., Vaidya, S., Ruehle, F., Halverson, J., et al. (2024). Kan: Kolmogorov -arnold networks. preprint arXiv:2404.19756, doi:https://doi.org/10.48550/arXiv.2404.19756. `

    2024 doi arXiv

  22. [30]

    Lu, L., Pestourie, R., Yao, W., Wang, Z., Verdugo, F., et al. (2021). Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 43(6), B1105-B1132, doi:https://doi.org/10.1137/21M1397908

    2021 doi

  23. [31]

    Luo, Y ., & Bao, J. (2019). A material-field series-expansion method for topology optimization of continuum structures. Computers and Structures, 225, 106122, doi:https://doi.org/10.1016/j.compstruc.2019.106122

    2019 doi

  24. [32]

    Qiu, Q., Zhu, T., Gong, H., Chen, L., & Ning, H. (2024). Relu -kan: New kolmogorov -arnold networks that only need matrix addition, dot multiplication, and relu. preprint arXiv:2406.02075, doi: https://doi.org/10.48550/arXiv.2406.02075

    2024 doi

  25. [33]

    Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707, doi:https://doi.org...

    2019 doi

  26. [34]

    Rawat, S., & Shen, M. (2018). A novel topology des ign approach using an integrated deep learning network architecture. arXiv preprint arXiv:1808.02334, doi:https://doi.org/10.48550/arXiv.1808.02334

    2018 doi arXiv

  27. [35]

    M., Aragon, A

    Sanu, S. M., Aragon, A. M., & Bessa, M. A. (2024). Neural topology optimization: the good, the bad, and the ugly. arXiv preprint arXiv:2407.13954, doi:https://doi.org/10.48550/arXiv.2407.13954

    2024 doi

  28. [36]

    Saxena, A., & Ananthasuresh, G. (2000). On an optimal property of compliant topologies. Structural and Multidisciplinary Optimization, 19(1), 36-49, doi:https://doi.org/10.1007/s001580050084

    2000 doi

  29. [37]

    Schmidt-Hieber, J. (2021). The Kolmogorov –Arnold representation theorem revisited. Neural Networks , 137, 119-126, doi:https://doi.org/10.1016/j.neunet.2021.01.020

    2021 doi

  30. [38]

    Shin, S., Shin, D., & Kang, N. (2023). Topology optimization via machine learning and deep learning: a review. Journal of Computational Design and Engineering, 10(4), 1736-1766, doi:https://doi.org/10.1093/jcde/qwad072

    2023 doi

  31. [39]

    Sigmund, O. (2001). A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 21(2), 120-127, doi:https://doi.org/10.1007/s001580050176

    2001 doi

  32. [40]

    Sigmund, O. (2022). On benchmarking and good scientific practise in topology optimization. Str uctural and Multidisciplinary Optimization, 65(11), 315, doi:https://doi.org/10.1007/s00158-022-03427-2

    2022 doi

  33. [41]

    Singh, N., Kumar, P., & Saxena, A. (2025). Normalized Field Product Approach: A Parameter ‐Free Density Evaluation Method for Close‐To‐Binary Solutions in Topology Optimization With Embedded Length Scale. Internation al Journal for Numerical Methods in Engineering, 126(7), e7673

    2025

  34. [42]

    Sivapuram, R., & Picelli, R. (2018). Topology optimization of binary structures using integer linear programming. Finite Elements in Analysis and Design, 139, 49-61

    2018

  35. [43]

    C., & Yung, S

    So, C. C., & Yung, S. P. Higher -order-ReLU-KANs (HRKANs) for solving physics -informed neural networks (PINNs) more accurately, robustly and faster. In 2025 IE EE World AI IoT Congress (AIIoT), 2025 (pp. 1035 -1042): IEEE. doi:https://doi.org/10.48550/arXiv.2409.14248

    2025 doi

  36. [44]

    Sohail, S. (2024). On training of kolmogorov -arnold networks. arXiv preprint arXiv:.05296 , doi:https://doi.org/10.48550/arXiv.2411.05296

    2024 doi

  37. [45]

    Sosnovik, I., & Oseledets, I. (2019). Neural networks for topology optimization. Russian Journal of Numerical Analysis Mathematical and Modelling, 34(4), 215-223, doi:https://doi.org/10.1515/rnam-2019-0018

    2019 doi

  38. [46]

    Takezawa, A., Nishiwaki, S., & Kitamura, M. (2010). Shape and topology optimization based on the phase field method ` and sensitivity analysis. J ournal of Computational Physics , 229 (7), 2697 -2718, doi:https://doi.org/10.1016/j.jcp.2009.12.017. Van Dijk, N. P., Maute, K., L...

    2010 doi

  39. [47]

    W., Liu, Z., & Hou, T

    Wang, Y ., Siegel, J. W., Liu, Z., & Hou, T. Y . (2024). On the expressiveness and spectral bias of KANs. arXiv preprint arXiv:.01803

    2024

  40. [48]

    Yang, R.-J., & Chen, C.-J. (1996). Stress-based topology optimization. Structural and Multidisciplinary Optimization, 12(2), 98-105, doi:https://doi.org/10.1007/BF01196941

    1996 doi

  41. [49]

    Yin, J., Wang, H., Guo, D., & Li, S. (2022). An efficient topology optimization based on multigrid assisted reanalysis for heat transfer problem. arXiv preprint arXiv:2210.00947, doi:https://doi.org/10.48550/arXiv.2210.00947

    2022 doi

  42. [50]

    Zhang, Z., Li, Y ., Zhou, W., Chen, X., Yao, W., et al. (2021). TONR: An exploration for a novel way combining neural network with topology optimization. Computer Methods in A pplied Mechanics Engineering , 386 , 114083, doi:https://doi.org/10.1016/j.cma.2021.114083

    2021 doi

  43. [51]

    Zheng, W., Wang, Y ., Zheng, Y ., & Da, D. (2020). Efficient topology optimization based on DOF reduction and convergence acceleration methods. Advances in Engineering Software , 149 , 102890, doi:https://doi.org/10.1016/j.advengsoft.2020.102890

    2020 doi

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