Abstract: This study proposes a fast zeroing neural network algorithm to solve time-varying Laplacian linear systems arising from the modeling of superstructure quadrilateral dynamic resistor networks on hammock surfaces. By incorporating the intrinsic structure of the underlying special-form matrices into the core neurodynamic design, the proposed algorithm enables efficient real-time computation of electric potentials under dynamic conditions. The Lyapunov-based analysis proves global exponential convergence. Numerical simulations on resistor networks of various scales demonstrate high computational efficiency and verify convergence to solutions from arbitrary initial conditions. Furthermore, by integrating the proposed algorithm as a potential field solver with a directional potential field path planning algorithm and exploiting the natural descent property of resistor network node potentials, we propose a fast path planning algorithm for robotic navigation on hammock surfaces. Compared with conventional path planning approaches, the proposed algorithm achieves higher computational efficiency in the aforementioned hammock surface path planning task, and this advantage becomes increasingly pronounced as the scale increases. The proposed algorithm is also applied to dynamic path planning tasks, further validating its potential in robotics and control applications. Finally, we present two conjectures.

用于吊床表面超结构四边形动态电阻网络的神经网络算法及其在人工智能中的应用

[1]Aybars U, 2008. Path planning on a cuboid using genetic algorithms. Inform Sci, 178(16):3275-3287.

[2]Bapst V, Keck T, Grabska-Barwińska A, et al., 2020. Unveiling the predictive power of static structure in glassy systems. Nat Phys, 16(4):448-454.

[3]Bianconi G, 2021. Higher-Order Networks. Cambridge University Press, Cambridge, UK.

[4]Bommes D, Lévy B, Pietroni N, et al., 2013. Quad-mesh generation and processing: a survey. Comput Graph Forum, 32(6):51-76.

[5]Campen M, 2017. Partitioning surfaces into quadrilateral patches: a survey. Comput Graph Forum, 36(8):567-588.

[6]Chair N, 2014. The effective resistance of the N-cycle graph with four nearest neighbors. J Stat Phys, 154(4):1177-1190.

[7]Chen JQ, Ji WY, Tan ZZ, 2024. Electrical properties of a 2 × n non-regular hammock network. Indian J Phys, 98(8):2851-2860.

[8]Darwish M, Boysan H, Liewald C, et al., 2018. A resistor network simulation model for laser-scanning photo-current microscopy to quantify low conductance regions in organic thin films. Org Electron, 62:474-480.

[9]Essam JW, Izmailyan NS, Kenna R, et al., 2015. Comparison of methods to determine point-to-point resistance in nearly rectangular networks with application to a ‘hammock’ network. R Soc Open Sci, 2(4):140420.

[10]Fu Z, Zhao YZ, Qian ZY, et al., 2006. Wall-climbing robot path planning for testing cylindrical oilcan weld based on Voronoi diagram. Proc IEEE/RSJ Int Conf on Intelligent Robots and Systems, p.2749-2753.

[11]Gao Z, Liu Q, Hesthaven JS, et al., 2021. Non-intrusive reduced order modeling of convection dominated flows using artificial neural networks with application to Rayleigh-Taylor instability. Commun Comput Phys, 30(1):97-123.

[12]Garcia SR, Yih S, 2018. Supercharacters and the discrete Fourier, cosine, and sine transforms. Commun Algebr, 46(9):3745-3765.

[13]Greiner W, 1998. Classical Electrodynamics. Springer, New York, USA.

[14]Hadad Y, Soric JC, Khanikaev AB, et al., 2018. Self-induced topological protection in nonlinear circuit arrays. Nat Electron, 1(3):178-182.

[15]Han Y, Cheng GF, Qiu BB, 2025. A triple-integral noise-resistant RNN for time-dependent constrained nonlinear optimization applied to manipulator control. IEEE Trans Ind Electron, 72(6):6124-6133.

[16]Hum SV, Du BZ, 2017. Equivalent circuit modeling for reflectarrays using Floquet modal expansion. IEEE Trans Antenn Propag, 65(3):1131-1140.

[17]Izmailian NS, Kenna R, Wu FY, 2013. The two-point resistance of a resistor network: a new formulation and application to the cobweb network. J Phys A Math Theor, 47(3):035003.

[18]Jeon SJ, Lee JS, Kim YJ, et al., 2021. Electrical analysis method for gamma-ray imaging system based on resistive network readout. IEEE Trans Nucl Sci, 68(9):2392-2399.

[19]Ji YX, Ni LT, Zhao C, et al., 2023. TriPField: a 3D potential field model and its applications to local path planning of autonomous vehicles. IEEE Trans Intell Transp Syst, 24(3):3541-3554.

[20]Jiang XY, Zhang GJ, Zheng YP, et al., 2024. Explicit potential function and fast algorithm for computing potentials in α×β conic surface resistor network. Expert Syst Appl, 238:122157.

[21]Jiang XY, Meng DX, Zheng YP, et al., 2025a. A dynamic cobweb resistance network solution based on a structured zeroing neural network and its applications. Sci Rep, 15(1):5222.

[22]Jiang XY, Dai JW, Zheng YP, et al., 2025b. Modeling of omnidirectional antenna space radiation shaped resistor networks and efficient path planning. Appl Phys Rev, 12(4):041405.

[23]Jiang XY, Guan Y, Zheng YP, et al., 2026. Robot path planning based on potential function of a resistor network. Appl Math Model, 151:116534.

[24]Jiang ZL, Zhou YF, Jiang XY, et al., 2023. Analytical potential formulae and fast algorithm for a horn torus resistor network. Phys Rev E, 107(4):044123.

[25]Jin L, Ying LF, Lu HY, et al., 2021. Saturation-allowed neural dynamics applied to perturbed time-dependent system of linear equations and robots. IEEE Trans Ind Electron, 68(10):9844-9854.

[26]Jin L, Li YT, Zhang XY, et al., 2024. Fuzzy k-winner-take-all network for competitive coordination in multirobot systems. IEEE Trans Fuzzy Syst, 32(4):2005-2016.

[27]Koul S, Horiuchi TK, 2019. Waypoint path planning with synaptic-dependent spike latency. IEEE Trans Circ Syst I Reg Pap, 66(4):1544-1557.

[28]Kulathunga G, 2022. A reinforcement learning based path planning approach in 3D environment. Proc Comput Sci, 212:152-160.

[29]Lian YF, Xiao XT, Zhang JL, et al., 2024. Neural dynamics for cooperative motion control of omnidirectional mobile manipulators in the presence of noises: a distributed approach. IEEE/CAA J Autom Sin, 11(7):1605-1620.

[30]Liu LX, Wang X, Yang X, et al., 2023. Path planning techniques for mobile robots: review and prospect. Expert Syst Appl, 227:120254.

[31]Lüpke F, Eschbach M, Heider T, et al., 2017. Electrical resistance of individual defects at a topological insulator surface. Nat Commun, 8(1):15704.

[32]Mazaheri H, Goli S, Nourollah A, 2024. Path planning in three-dimensional space based on butterfly optimization algorithm. Sci Rep, 14(1):2332.

[33]Millán AP, Sun HL, Giambagli L, et al., 2025. Topology shapes dynamics of higher-order networks. Nat Phys, 21(3):353-361.

[34]Novoselov KS, Geim AK, Morozov SV, et al., 2004. Electric field effect in atomically thin carbon films. Science, 306(5696):666-669.

[35]Novoselov KS, Geim AK, Morozov SV, et al., 2005. Two-dimensional gas of massless Dirac fermions in graphene. Nature, 438(7065):197-200.

[36]Pan ZH, Zhang CX, Xia YQ, et al., 2022. An improved artificial potential field method for path planning and formation control of the multi-UAV systems. IEEE Trans Circ Syst II Exp Briefs, 69(3):1129-1133.

[37]Patle BK, Babu LG, Pandey A, et al., 2019. A review: on path planning strategies for navigation of mobile robot. Def Technol, 15(4):582-606.

[38]Pei SC, Yeh MH, 2001. The discrete fractional cosine and sine transforms. IEEE Trans Signal Process, 49(6):1198-1207.

[39]Qiu BB, Guo JJ, Li XD, et al., 2022. Discrete-time advanced zeroing neurodynamic algorithm applied to future equality-constrained nonlinear optimization with various noises. IEEE Trans Cybern, 52(5):3539-3552.

[40]Qiu BB, Guo JJ, Mao MZ, et al., 2024. A fuzzy-enhanced robust DZNN model for future multiconstrained nonlinear optimization with robotic manipulator control. IEEE Trans Fuzzy Syst, 32(1):160-173.

[41]Rhazaoui K, Cai Q, Adjiman CS, et al., 2013. Towards the 3D modeling of the effective conductivity of solid oxide fuel cell electrodes: I. Model development. Chem Eng Sci, 99:161-170.

[42]Shi Y, Zhang YN, 2020. New discrete-time models of zeroing neural network solving systems of time-variant linear and nonlinear inequalities. IEEE Trans Syst Man Cybern Syst, 50(2):565-576.

[43]Shi Y, Jin L, Li S, et al., 2022. Novel discrete-time recurrent neural networks handling discrete-form time-variant multi-augmented Sylvester matrix problems and manipulator application. IEEE Trans Neur Netw Learn Syst, 33(2):587-599.

[44]Sun ZB, Tang SJ, Jin L, et al., 2023a. Nonconvex activation noise-suppressing neural network for time-varying quadratic programming: application to omnidirectional mobile manipulator. IEEE Trans Ind Inform, 19(11):10786-10798.

[45]Sun ZB, Tang SJ, Zhang JL, et al., 2023b. Nonconvex noise-tolerant neural model for repetitive motion of omnidirectional mobile manipulators. IEEE/CAA J Autom Sin, 10(8):1766-1768.

[46]Tan Z, Tan ZZ, Zhou L, et al., 2018. Electrical properties of an m × n hammock network. Commun Theor Phys, 69(5):610-616.

[47]Tan ZZ, 2015. Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries. Phys Rev E, 91(5):052122.

[48]Tan ZZ, Tan Z, 2020. The basic principle of m × n resistor networks. Commun Theor Phys, 72(5):055001.

[49]Wang XZ, Che ML, Wei YM, 2019. Neural networks based approach solving multi-linear systems with M-tensors. Neurocomputing, 351:33-42.

[50]Wei P, Wang XZ, Wei YM, 2023. Neural network models for time-varying tensor complementarity problems. Neurocomputing, 523:18-32.

[51]Wen X, Don WS, Gao Z, et al., 2020. An edge detector based on artificial neural network with application to hybrid compact-WENO finite difference scheme. J Sci Comput, 83(3):49.

[52]Willke P, Kotzott T, Pruschke T, et al., 2017. Magnetotransport on the nano scale. Nat Commun, 8(1):15283.

[53]Wu FY, 2004. Theory of resistor networks: the two-point resistance. J Phys A Math Gen, 37(26):6653-6673.

[54]Wu WQ, Zheng B, 2020. Improved recurrent neural networks for solving Moore-Penrose inverse of real-time full-rank matrix. Neurocomputing, 418:221-231.

[55]Xu GY, Eleftheriades GV, Hum SV, 2021. Analysis and design of general printed circuit board metagratings with an equivalent circuit model approach. IEEE Trans Antenn Propag, 69(8):4657-4669.

[56]Yang LW, Li P, Qian S, et al., 2023. Path planning technique for mobile robots: a review. Machines, 11(10):980.

[57]Zhang D, Yang B, Tan JP, et al., 2021. Impact damage localization and mode identification of CFRPs panels using an electric resistance change method. Compos Struct, 276:114587.

[58]Zhang YN, Jiang DC, Wang J, 2002. A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans Neur Netw, 13(5):1053-1063.

[59]Zhang ZJ, Sun XL, Liu YQ, et al., 2025. A meta-interactive neural network for solving time-varying quadratic programming problems. Nat Commun, 16(1):10284.

[60]Zhao H, 2025. A scientist question: research on the impact of super structured quadrilateral meshes on convergence and accuracy of finite element analysis. https://arxiv.org/abs/2507.17184

[61]Zhao WJ, Zheng YP, Jiang XY, et al., 2023. Two optimized novel potential formulas and numerical algorithms for m × n cobweb and fan resistor networks. Sci Rep, 13(1):12417.

[62]Zhu ZX, Yin Y, Lyu H, 2023. Automatic collision avoidance algorithm based on route-plan-guided artificial potential field method. Ocean Eng, 271:113737.

Similar articles