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Supervised stochastic Levenberg–Marquardt intelligent netwoks for dynamics of convective Eyring–Powell magneto-nanofluid model

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Abstract

The presented work examines the dynamics of convective Eyring–Powell magneto-nanofluid model (CEP-MNFM) with a stretching cylinder by using stupendous knacks of supervised stochastic Levenberg–Marquardt intelligent networks (SSLMINs). The partial differential equations governing the CEP-MNFM are reduced into coupled ODEs by incorporating the similarity transformations. The dataset of the proposed SSLMINs approach is generated with state-of-the-art Adam numerical method for seven different scenarios of CEP-MNFM including variation of radiation, Brownian diffusivity, and thermophoresis parameter, as well as, Biot, Schmidh, and Prandtl numbers. The reference dataset is further utilized for numerical calculation of various physical quantities on CEP-MNFM by applying the AI based methods via SSLMINs. The precision and accuracy of the designed SSLMINs approach is efficaciously substantiated through the negligible level of mean squared error with magnitude around 10–8 to 10–10, histograms with maximum instances error range 10–5, very near to the optimum correlation/regression measures.

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Data Availability Statement

No data are associated with this manuscript.

Abbreviations

\(u\) , \(v\) , \(x\), \(r\) :

Components of velocity

\(\rho_{f}\) :

Liquid density

\(l\) :

Characteristic length

\(\nu\) :

Kinematic viscosity

\(\beta\), \(c\) :

Material variables

\(R_{1}\) :

Radius of cylinder

\(\sigma\) :

Liquid electrical conductivity

\(B_{0}\) :

Magnetic field potency

\(c_{p}\) :

Specific heat

\(k^{ * }\) :

Mean-absorption coefficient

\(\left( {D_{B} ,\,D_{T} } \right)\) :

Coefficient of Brownian diffusion & thermophoretic

\(\sigma^{ * }\) :

Stefan-Boltzmann constant

\(Q_{0}\) :

Heat sink/source coefficient

\({\left(\rho c\right)}_{f}\) :

Liquid heat capacity

\({\left(\rho c\right)}_{p}\) :

Nano-particles effective heat capacity

\({S}_{2}\) :

Solutal stratification parameter

\({S}_{1}\) :

Thermal stratification parameter

\({\gamma }_{2}\) :

Solutal biot number

\({\gamma }_{1}\) :

Thermal biot number

\(Re\) :

Reynolds number

\(\hbar_{f} ,\) \(\hbar_{\theta }\), \(\hbar_{\varphi }\) :

Auxiliary constraints

EPFM:

Eyring–Powell fluid model

SSLMINs:

Supervised stochastic Levenberg–Marquardt intelligent networks

\(f,g\) :

Dimensionless velocities

\(\theta \) :

Dimensionless temperature

\(\phi \) :

Dimensionless concentration

\({q}_{r}\) :

Radiative heat flux

ANNs:

Artificial neural networks

\(C\) :

Concentration

\(T\) :

Temperature

\(\left( {h_{2} ,\,h_{1} } \right)\) :

Dimensional constants

\({U}_{w}\) :

Stretching velocity

\({U}_{o}\) :

Reference velocity

\({C}_{f}\left(x\right)\) :

Convective liquid concentration

\({T}_{f}\left(x\right)\) :

Convective liquid temperature

\(a,b,c,d\) :

Dimensional constants

\({C}_{\infty }\left(x\right)\) :

Wall mass transport coefficients

\({T}_{\infty }\left(x\right)\) :

Wall heat transport coefficients

\(\tau = \tfrac{{\left( {\rho c} \right)_{{_{p} }} }}{{\left( {\rho c} \right)_{{_{f} }} }}\) :

Ratio of heat capacity

\(\varepsilon ,\lambda \) :

Material variables

\(S\) :

Suction/injection

\(\gamma \) :

Curvature parameter

\({H}_{a}\) :

Hartmann number

\({P}_{r}\) :

Prandtl number

\({N}_{b}\) :

Brownian diffusion factor

\({N}_{t}\) :

Thermophoretic factor

\(Sc\) :

Schmidt number

\(R\) :

Radiation variable

\(\delta \) :

Heat source

ODE:

Ordinary differential equation

CEP-MNFM:

Convective Eyring–Powell magneto-nanofluid model

EP:

Eyring-Powell

\({D}_{B}\) :

Coefficient of Brownian diffusion

\({D}_{T}\) :

Coefficient of Brownian thermophoretic

ɳ :

Dimensionless variable

References

  1. S.U. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles (No. ANL/MSD/CP-84938; CONF-951135–29) (1995), Argonne National Lab. (ANL), Argonne, IL (United States).

  2. O.D. Makinde, I.L. Animasaun, Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution. J. Mol. Liq. 221, 733–743 (2016)

    Article  CAS  Google Scholar 

  3. L. Yang, W. Ji, M. Mao, J.N. Huang, An updated review on the properties, fabrication and application of hybrid-nanofluids along with their environmental effects. J. Clean. Prod. 257, 120408 (2020)

    Article  CAS  Google Scholar 

  4. M. Waqas, Z. Asghar, W.A. Khan, Thermo-solutal Robin conditions significance in thermally radiative nanofluid under stratification and magnetohydrodynamics. Eur. Phys. J. Spec. Top. 230(5), 1307–1316 (2021)

    Article  CAS  Google Scholar 

  5. A.H. Pordanjani, S. Aghakhani, M. Afrand, M. Sharifpur, J.P. Meyer, H. Xu, H.M. Ali, N. Karimi, G. Cheraghian, Nanofluids: physical phenomena, applications in thermal systems and the environment effects-a critical review. J. Clean. Prod. 320, 128573 (2021)

    Article  CAS  Google Scholar 

  6. M. Imran, M. Naveed, B. Iftikhar, Z. Abbas, Heat transfer analysis in a curvilinear flow of hybrid nanoliquid across a curved oscillatory stretched surface with nonlinear thermal radiation. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 20, 220–0600 (2023). https://doi.org/10.1002/zamm.202200600

    Article  Google Scholar 

  7. M. Ali, F. Sultan, W.A. Khan, M. Shahzad, H. Arif, Important features of expanding/contracting cylinder for cross magneto-nanofluid flow. Chaos Solit. Fract. 133, 109656 (2020)

    Article  MathSciNet  Google Scholar 

  8. K.S. Tshivhi, O.D. Makinde, Magneto-nanofluid coolants past heated shrinking/stretching surfaces: dual solutions and stability analysis. Results Eng. 10, 100229 (2021)

    Article  CAS  Google Scholar 

  9. A.S. Alshomrani, M.Z. Ullah, S.S. Capizzano, W.A. Khan, M. Khan, Interpretation of chemical reactions and activation energy for unsteady 3D flow of Eyring–Powell magneto-nanofluid. Arab. J. Sci. Eng. 44, 579–589 (2019)

    Article  CAS  Google Scholar 

  10. Y. Akbar, S. Huang, M.U. Ashraf, K.S. Nisar, M.M. Alam, Electrothermal analysis for reactive Powell Eyring nanofluid flow regulated by peristaltic pumping with mass transfer. Case Stud. Thermal Eng. 44, 102828 (2023)

    Article  Google Scholar 

  11. A. Shafiq, A.B. Çolak, T.N. Sindhu, Significance of EMHD graphene oxide (GO) water ethylene glycol nanofluid flow in a Darcy-Forchheimer medium by machine learning algorithm. Eur. Phys. J. Plus 138(3), 213 (2023)

    Article  CAS  Google Scholar 

  12. N.M. Sakan, I. Traparic, V.A. Sreckovic, M. Ivkovic, The usage of perceptron, feed and deep feed forward artificial neural networks on the spectroscopy data: astrophysical & fusion plasmas. Contrib. Astron. Obs. Skalnaté Pleso 52(3), 97–104 (2022)

    Google Scholar 

  13. A. Jahani, S. Allahverdi, M. Saffariha, A. Alitavoli, S. Ghiyasi, Environmental modeling of landscape aesthetic value in natural urban parks using artificial neural network technique. Model. Earth Syst. Environ. 8, 1–10 (2021)

    Google Scholar 

  14. Z. Fu, Y. Shen, S. Wang, W. Jiang, J. Li, G. Bin, B. Hu, Asymmetric Bouc-Wen hysteresis modeling for MFC actuator via hybrid APSO-TRR identification algorithm. Sens. Actuators A 346, 113830 (2022)

    Article  CAS  Google Scholar 

  15. C. Ma, J. Mou, F. Yang, H. Yan, A fractional-order hopfield neural network chaotic system and its circuit realization. Eur. Phys. J. Plus 135(1), 100 (2020)

    Article  ADS  Google Scholar 

  16. Y. Liu, L. Liu, F. Lombardi, J. Han, An energy-efficient and noise-tolerant recurrent neural network using stochastic computing. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 27(9), 2213–2221 (2019)

    Article  Google Scholar 

  17. I. Khan et al., Design of neural network With Levenberg-Marquardt and Bayesian Regularization backpropagation for solving pantograph delay differential equations. IEEE Access 8, 137918–137933 (2020)

    Article  Google Scholar 

  18. E. Fol, R. Tomás, G. Franchetti, Supervised learning-based reconstruction of magnet errors in circular accelerators. Eur. Phys. J. Plus 136(4), 365 (2021)

    Article  Google Scholar 

  19. A. Jafarian, M. Mokhtarpour, D. Baleanu, Artificial neural network approach for a class of fractional ordinary differential equation. Neural Comput. Appl. 28(4), 765–773 (2017)

    Article  Google Scholar 

  20. A. Jafarian, S.M. Nia, A.K. Golmankhaneh, D. Baleanu, On artificial neural networks approach with new cost functions. Appl. Math. Comput. 339, 546–555 (2018)

    MathSciNet  Google Scholar 

  21. A. Jafarian, D. Baleanu, Application of ANNs approach for wave-like and heat-like equations. Open Phys. 15(1), 1086–1094 (2017)

    Article  Google Scholar 

  22. R.P. Sharma, J.K. Madhukesh, S. Shukla, B.C. Prasannakumara, Numerical and Levenberg–Marquardt backpropagation neural networks computation of ternary nanofluid flow across parallel plates with Nield boundary conditions. Eur. Phys. J. Plus 138(1), 63 (2023)

    Article  CAS  Google Scholar 

  23. F. Nasirzadehroshenin, M. Sadeghzadeh, A. Khadang, H. Maddah, M.H. Ahmadi, H. Sakhaeinia, L. Chen, Modeling of heat transfer performance of carbon nanotube nanofluid in a tube with fixed wall temperature by using ANN–GA. Eur. Phys. J. Plus 135(2), 217 (2020)

    Article  CAS  Google Scholar 

  24. M. Umar, F. Amin, H.A. Wahab, D. Baleanu, Unsupervised constrained neural network modeling of boundary value corneal model for eye surgery. Appl. Soft Comput. 85, 105826 (2019)

    Article  Google Scholar 

  25. M. Umar et al., Stochastic numerical technique for solving HIV infection model of CD4+ T cells. Eur. Phys. J. Plus 135(6), 403 (2020)

    Article  Google Scholar 

  26. A. Mehmood et al., Design of neuro-computing paradigms for nonlinear nanofluidic systems of MHD Jeffery-Hamel flow. J. Taiwan Inst. Chem. Eng. 91, 57–85 (2018)

    Article  CAS  Google Scholar 

  27. J. Shlomi, P. Battaglia, J.R. Vlimant, Graph neural networks in particle physics. Mach. Learn. Sci. Technol. 2(2), 021001 (2020)

    Article  Google Scholar 

  28. A.Y.T. Wang, R.J. Murdock, S.K. Kauwe, A.O. Oliynyk, A. Gurlo, J. Brgoch, K.A. Persson, T.D. Sparks, Machine learning for materials scientists: an introductory guide toward best practices. Chem. Mater. 32(12), 4954–4965 (2020)

    Article  CAS  Google Scholar 

  29. A. Mehmood et al., Design of nature-inspired heuristic paradigm for systems in nonlinear electrical circuits. Neural Comput. Appl. 32, 1–17 (2019)

    Google Scholar 

  30. I. Ahmad et al., Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels. Neural Comput. Appl. 31(12), 9041–9059 (2019)

    Article  Google Scholar 

  31. J. Rabault, M. Kuchta, A. Jensen, U. Réglade, N. Cerardi, Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control. J. Fluid Mech. 865, 281–302 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  32. C.H. Martin, M.W. Mahoney, Implicit self-regularization in deep neural networks: evidence from random matrix theory and implications for learning. J. Mach. Learn. Res. 22(1), 7479–7551 (2021)

    MathSciNet  Google Scholar 

  33. G. Rajput, V. Logashree, K.N. Biyani, S.K. Vishvakarma, Clock gating-based effectual realization of stochastic hyperbolic tangent function for deep neural hardware accelerators. Circuit Syst. Signal Proc. 42, 1–23 (2023)

    Google Scholar 

  34. F. Shahzad, S.A. Shehzad, W.A. Khan, M. Waqas, M. Manzur, M. Zubair, Convective stratified flow of magnetized Eyring–Powell (EP) nanofluid by a stretching cylinder. Appl. Nanosci. 10, 5401–5408 (2020)

    Article  ADS  CAS  Google Scholar 

  35. J. Buongiorno, Convective transport in nanofluids. J. Heat Transfer. 128, 240–250 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Islamabad Campus, Pakistan

    Zahoor Shah

  2. Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin, 64002, Taiwan, ROC

    Muhammad Asif Zahoor Raja

  3. Yuan Ze University, AI Center, 320, Taoyuan, Taiwan

    Muhammad Shoaib

  4. Department of Mathematics, Mohi-Ud-Din Islamic University, Nerian Sharif, AJK, Pakistan

    Faisal Shahzad

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Correspondence to Muhammad Asif Zahoor Raja.

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Shah, Z., Raja, M.A.Z., Shoaib, M. et al. Supervised stochastic Levenberg–Marquardt intelligent netwoks for dynamics of convective Eyring–Powell magneto-nanofluid model. Eur. Phys. J. Plus 139, 173 (2024). https://doi.org/10.1140/epjp/s13360-023-04852-y

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