Fixed-time zeroing neural networks for solving the singular Yang-Baxter-like matrix equation

Yiqiong Sun; Duanmei Zhou; Yiqiong Sun; Duanmei Zhou

doi:10.3934/math.20261269

AIMS Mathematics

2026, Volume 11, Issue 10: 32288-32317. doi: 10.3934/math.20261269

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Research article

Fixed-time zeroing neural networks for solving the singular Yang-Baxter-like matrix equation

Yiqiong Sun ,
Duanmei Zhou ^,

College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, Jiangxi, China

Received: 04 June 2026 Revised: 31 August 2026 Accepted: 23 September 2026 Published: 08 October 2026
MSC : 15A24, 65F45

Solving singular Yang-Baxter-like matrix equations poses significant challenges due to ill-conditioned matrices and high computational costs. To address these issues, this work investigated the development of zeroing neural networks for singular problems. Initially, we constructed an enhanced improved zeroing neural network as a transitional model by directly integrating Tikhonov regularization into existing frameworks. However, theoretical analysis revealed that this direct regularization approach has limitations, as it only guarantees convergence to a residual limit set rather than the exact solution due to irreducible null spaces. To address this limitation, we proposed the effective zeroing neural network (EZNN). Under the assumption of an invertible auxiliary matrix, the EZNN utilizes full-rank factorization to equivalently transform the high-dimensional singular problem into a low-dimensional full-rank matrix equation. This structural reduction bypasses the singularity issue, avoiding the need for Tikhonov regularization while mitigating memory overflow, thus enabling the precise processing of high-dimensional matrices.
- Yang-Baxter-like matrix equation,
- fixed-time convergence,
- noise robustness,
- zeroing neural network model,
- Tikhonov regularization
Citation: Yiqiong Sun, Duanmei Zhou. Fixed-time zeroing neural networks for solving the singular Yang-Baxter-like matrix equation[J]. AIMS Mathematics, 2026, 11(10): 32288-32317. doi: 10.3934/math.20261269

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Abstract

Solving singular Yang-Baxter-like matrix equations poses significant challenges due to ill-conditioned matrices and high computational costs. To address these issues, this work investigated the development of zeroing neural networks for singular problems. Initially, we constructed an enhanced improved zeroing neural network as a transitional model by directly integrating Tikhonov regularization into existing frameworks. However, theoretical analysis revealed that this direct regularization approach has limitations, as it only guarantees convergence to a residual limit set rather than the exact solution due to irreducible null spaces. To address this limitation, we proposed the effective zeroing neural network (EZNN). Under the assumption of an invertible auxiliary matrix, the EZNN utilizes full-rank factorization to equivalently transform the high-dimensional singular problem into a low-dimensional full-rank matrix equation. This structural reduction bypasses the singularity issue, avoiding the need for Tikhonov regularization while mitigating memory overflow, thus enabling the precise processing of high-dimensional matrices.

References

[1]	S. R. Garcia, A. L. Shoemaker, On the matrix equation $XA+ AX^{\mathrm{T}} = 0$, Linear Algebra Appl., 438 (2013), 2740–2746. https://doi.org/10.1016/j.laa.2012.10.041 doi: 10.1016/j.laa.2012.10.041
[2]	C. Han, B. Zheng, J. Xu, A modified noise-tolerant ZNN model for solving time-varying Sylvester equation with its application to robot manipulator, J. Franklin Ins., 360 (2023), 8633–8650. https://doi.org/10.1016/j.jfranklin.2023.06.045 doi: 10.1016/j.jfranklin.2023.06.045
[3]	J. Hietarinta, Solving the two-dimensional constant quantum Yang-Baxter equation, J. Math. Phys., 34 (1993), 1725–1756. https://doi.org/10.1063/1.530185 doi: 10.1063/1.530185
[4]	J. Ding, H. Tian, Solving the Yang-Baxter-like matrix equation for a class of elementary matrices, Comput. Math. Appl., 72 (2016), 1541–1548. https://doi.org/10.1016/j.camwa.2016.07.015 doi: 10.1016/j.camwa.2016.07.015
[5]	C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19 (1967), 1312. https://doi.org/10.1103/PhysRevLett.19.1312 doi: 10.1103/PhysRevLett.19.1312
[6]	R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys., 70 (1972), 193–228. https://doi.org/10.1016/0003-4916(72)90335-1 doi: 10.1016/0003-4916(72)90335-1
[7]	D. K. Matsumoto, Y. Shibukawa, Quantum Yang-Baxter equation, braided semigroups, and dynamical Yang-Baxter maps, Tokyo J. Math., 38 (2015), 227–237. https://doi.org/10.3836/tjm/1437506246 doi: 10.3836/tjm/1437506246
[8]	Z. Tsuboi, Quantum groups, Yang-Baxter maps and quasi-determinants, Nucl. Phys. B, 926 (2018), 200–238. https://doi.org/10.1016/j.nuclphysb.2017.11.005 doi: 10.1016/j.nuclphysb.2017.11.005
[9]	R. Vieira, A. Lima-Santos, Solutions of the Yang-Baxter equation for (n+1)(2n+1)-vertex models using a differential approach, J. Stat. Mech., 2021 (2021), 053103. https://doi.org/10.1088/1742-5468/abf7be doi: 10.1088/1742-5468/abf7be
[10]	J. H. Przytycki, Knot theory: from fox 3-colorings of links to Yang-Baxter homology and Khovanov homology, In: Knots, low-dimensional topology and applications, Cham: Springer, 2016,115–145. https://doi.org/10.1007/978-3-030-16031-9_5
[11]	Z. K. Lyu, W. R. Xu, A matrix nearness problem constrained on the kernel space of multilevel (R, S $\sigma$)-commutative matrices, Comput. Appl. Math., 44 (2025), 270. https://doi.org/10.1007/s40314-025-03234-7 doi: 10.1007/s40314-025-03234-7
[12]	H. Ren, X. Wang, T. Wang, Commuting solutions of the Yang-Baxter-like matrix equation for a class of rank-two updated matrices, Comput. Math. Appl., 76 (2018), 1085–1098. https://doi.org/10.1016/j.camwa.2018.05.042 doi: 10.1016/j.camwa.2018.05.042
[13]	D. Shen, M. Wei, Z. Jia, On commuting solutions of the Yang-Baxter-like matrix equation, J. Math. Anal. Appl., 462 (2018), 665–696. https://doi.org/10.1016/j.jmaa.2018.02.030 doi: 10.1016/j.jmaa.2018.02.030
[14]	D. Shen, M. Wei, All solutions of the Yang-Baxter-like matrix equation for diagonalizable coefficient matrix with two different eigenvalues, Appl. Math. Lett., 101 (2020), 106048. https://doi.org/10.1016/j.aml.2019.106048 doi: 10.1016/j.aml.2019.106048
[15]	L. Jin, S. Li, H. M. La, X. Zhang, B. Hu, Dynamic task allocation in multi-robot coordination for moving target tracking: a distributed approach, Automatica, 100 (2019), 75–81. https://doi.org/10.1016/j.automatica.2018.11.001 doi: 10.1016/j.automatica.2018.11.001
[16]	G. Wang, D. Li, X. Chen, H. Huang, Two modified Newton-Raphson iteration algorithms for Yang-Baxter-like matrix equation with step-size analyses, Proceedings of 35th Youth Academic Annual Conference of Chinese Association of Automation (YAC), 2020,259–263. https://doi.org/10.1109/YAC51587.2020.9337656
[17]	L. Jin, S. Li, L. Xiao, R. Lu, B. Liao, Cooperative motion generation in a distributed network of redundant robot manipulators with noises, IEEE Trans. Syst. Man Cyb., 48 (2018), 1715–1724. https://doi.org/10.1109/TSMC.2017.2693400 doi: 10.1109/TSMC.2017.2693400
[18]	L. Jin, Y. Zhang, Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulator motion generation, IEEE Trans. Neur. Net. Lear., 26 (2015), 1525–1531. https://doi.org/10.1109/TNNLS.2014.2342260 doi: 10.1109/TNNLS.2014.2342260
[19]	Y. Zhang, S. S. Ge, Design and analysis of a general recurrent neural network model for time-varying matrix inversion, IEEE Trans. Neural Networ., 16 (2005), 1477–1490. https://doi.org/10.1109/TNN.2005.857946 doi: 10.1109/TNN.2005.857946
[20]	Y. Zhang, D. Jiang, J. Wang, A recurrent neural network for solving Sylvester equation with time-varying coefficients, IEEE Trans. Neural Networ., 13 (2002), 1053–1063. https://doi.org/10.1109/TNN.2002.1031938 doi: 10.1109/TNN.2002.1031938
[21]	R. Manherz, B. Jordan, S. Hakimi, Analog methods for computation of the generalized inverse, IEEE Trans. Automat. Contr., 13 (1968), 582–585. https://doi.org/10.1109/TAC.1968.1098973 doi: 10.1109/TAC.1968.1098973
[22]	M. Hüsken, P. Stagge, Recurrent neural networks for time series classification, Neurocomputing, 50 (2003), 223–235. https://doi.org/10.1016/S0925-2312(01)00706-8 doi: 10.1016/S0925-2312(01)00706-8
[23]	T. Shi, Y. Tian, Z. Sun, K. Liu, L. Jin, J. Yu, Noise-tolerant neural algorithm for online solving Yang-Baxter-type matrix equation in the presence of noises: a control-based method, Neurocomputing, 424 (2021), 84–96. https://doi.org/10.1016/j.neucom.2020.10.110 doi: 10.1016/j.neucom.2020.10.110
[24]	V. N. Katsikis, S. D. Mourtas, P. S. Stanimirović, Y. Zhang, Solving complex-valued time-varying linear matrix equations via QR decomposition with applications to robotic motion tracking and on angle-of-arrival localization, IEEE Trans. Neur. Net. Lear., 33 (2022), 3415–3424. https://doi.org/10.1109/TNNLS.2021.3052896 doi: 10.1109/TNNLS.2021.3052896
[25]	P. S. Stanimirović, V. N. Katsikis, S. Li, Hybrid GNN-ZNN models for solving linear matrix equations, Neurocomputing, 316 (2018), 124–134. https://doi.org/10.1016/j.neucom.2018.07.058 doi: 10.1016/j.neucom.2018.07.058
[26]	M. Kornilova, V. Kovalnogov, R. Fedorov, M. Zamaleev, V. N. Katsikis, S. D. Mourtas, et al., Zeroing neural network for pseudoinversion of an arbitrary time-varying matrix based on singular value decomposition, Mathematics, 10 (2022), 1208. https://doi.org/10.3390/math10081208 doi: 10.3390/math10081208
[27]	V. N. Katsikis, S. D. Mourtas, P. S. Stanimirović, S. Li, X. Cao, Time-varying mean–variance portfolio selection problem solving via LVI-PDNN, Comput. Oper. Res., 138 (2022), 105582. https://doi.org/10.1016/j.cor.2021.105582 doi: 10.1016/j.cor.2021.105582
[28]	L. Xiao, Y. Zhang, Z. Hu, J. Dai, Performance benefits of robust nonlinear zeroing neural network for finding accurate solution of Lyapunov equation in presence of various noises, IEEE Trans. Ind. Inform., 15 (2019), 5161–5171. https://doi.org/10.1109/TII.2019.2900659 doi: 10.1109/TII.2019.2900659
[29]	J. Zhao, Y. Zhang, A novel zeroing neural network with integral upper-bound function for solving time-varying Sylvester equation, Comput. Appl. Math., 44 (2025), 374. https://doi.org/10.1007/s40314-025-03340-6 doi: 10.1007/s40314-025-03340-6
[30]	T. Huang, S. X. Miao, An improved zeroing neural network model for solving the time-varying Yang-Baxter-like matrix equation, J. Math. Anal. Appl., 545 (2025), 129095. https://doi.org/10.1016/j.jmaa.2024.129095 doi: 10.1016/j.jmaa.2024.129095
[31]	T. Huang, S. X. Miao, An improved noise-tolerant zeroing neural network model for solving the time-varying Yang-Baxter-like matrix equation, Comput. Appl. Math., 45 (2026), 180. https://doi.org/10.1007/s40314-025-03576-2 doi: 10.1007/s40314-025-03576-2
[32]	W. Jiang, C. L. Lin, V. N. Katsikis, S. D. Mourtas, P. S. Stanimirović, T. E. Simos, Zeroing neural network approaches based on direct and indirect methods for solving the Yang-Baxter-like matrix equation, Mathematics, 10 (2022), 1950. https://doi.org/10.3390/math10111950 doi: 10.3390/math10111950
[33]	H. N. Nguyen, S. Olaru, P. O. Gutman, M. Hovd, Constrained control of uncertain, time-varying linear discrete-time systems subject to bounded disturbances, IEEE Trans. Automat. Contr., 60 (2015), 831–836. https://doi.org/10.1109/TAC.2014.2346872 doi: 10.1109/TAC.2014.2346872
[34]	Y. Orlov, Finite time stability and robust control synthesis of uncertain switched systems, SIAM J. Control Optim., 43 (2004), 1253–1271. https://doi.org/10.1137/S0363012903425593 doi: 10.1137/S0363012903425593
[35]	A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Automat. Contr., 57 (2012), 2106–2110. https://doi.org/10.1109/TAC.2011.2179869 doi: 10.1109/TAC.2011.2179869
[36]	B. Qiu, Y. Zhang, Z. Yang, New discrete-time ZNN models for least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient, IEEE Trans. Neur. Net. Lear., 29 (2018), 5767–5776. https://doi.org/10.1109/TNNLS.2018.2805810 doi: 10.1109/TNNLS.2018.2805810
[37]	H. Zhang, L. Wan, Zeroing neural network methods for solving the Yang-Baxter-like matrix equation, Neurocomputing, 383 (2020), 409–418. https://doi.org/10.1016/j.neucom.2019.11.101 doi: 10.1016/j.neucom.2019.11.101
[38]	L. Xiao, Y. Zhang, J. Dai, K. Chen, S. Yang, W. Li, et al., A new noise-tolerant and predefined-time ZNN model for time-dependent matrix inversion, Neural Networks, 117 (2019), 124–134. https://doi.org/10.1016/j.neunet.2019.05.005 doi: 10.1016/j.neunet.2019.05.005
[39]	L. Lu, Constructing solutions of the Yang-Baxter-like matrix equation for singular matrices, J. Comput. Appl. Math., 436 (2024), 115403. https://doi.org/10.1016/j.cam.2023.115403 doi: 10.1016/j.cam.2023.115403

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