{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,24]],"date-time":"2026-02-24T05:49:32Z","timestamp":1771912172335,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T00:00:00Z","timestamp":1755475200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main goal of this study is to create a computational solver that analyzes the effects of magnetohydrodynamics (MHD) on heat radiation in Cu\u2013water-based Prandtl nanofluid flow using artificial neural networks. Copper nanoparticles are utilized to boost the water-based fluid\u2019s thermal effect. This study primarily focuses on heat transfer over a horizontal sheet, exploring different scenarios by varying key factors such as the magnetic field and thermal radiation properties. The mathematical model is formulated using partial differential equations (PDEs), which are then transformed into a corresponding set of ordinary differential equations (ODEs) through appropriate similarity transformations. The bvp4c solver is then used to simulate the numerical behavior. The effects of relevant parameters on the temperature, velocity, skin friction, and local Nusselt number profiles are examined. It is discovered that the parameters of the Prandtl fluid have a considerable impact. The local skin friction and the local Nusselt number are improved by increasing these parameters. The dataset is split into 70% training, 15% validation, and 15% testing. The ANN model successfully predicts skin friction and Nusselt number profiles, showing good agreement with numerical simulations. This hybrid framework offers a robust predictive approach for heat management systems in industrial applications. This study provides important insights for researchers and engineers aiming to comprehend flow characteristics and their behavior and to develop accurate predictive models.<\/jats:p>","DOI":"10.3390\/sym17081347","type":"journal-article","created":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T11:31:37Z","timestamp":1755516697000},"page":"1347","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Numerical\u2013ANN Framework for Thermal Analysis of MHD Water-Based Prandtl Nanofluid Flow over a Stretching Sheet Using Bvp4c"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7092-970X","authenticated-orcid":false,"given":"Syed Asif Ali","family":"Shah","sequence":"first","affiliation":[{"name":"Department of Modern Mechanics, School of Engineering Science, University of Science and Technology of China, Hefei 230026, China"},{"name":"Jinhua Hangda Beidou Applied Technology Co., Ltd., Jinhua 321004, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3110-414X","authenticated-orcid":false,"given":"Fehaid Salem","family":"Alshammari","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Ridyadh 11564, Saudi Arabia"}]},{"given":"Muhammad Fawad","family":"Malik","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan"}]},{"given":"Saira","family":"Batool","sequence":"additional","affiliation":[{"name":"Centre for Integrated Mountain Research, University of the Punjab, Lahore 54590, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,18]]},"reference":[{"key":"ref_1","unstructured":"Choi, S.U.S., and Eastman, J.A. (1995). Enhancing Thermal Conductivity of Fluids with Nanoparticles, Argonne National Lab. (ANL). CONF-951135-29."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Qureshi, M.A. (2020). Numerical simulation of heat transfer flow subject to MHD of Williamson nanofluid with thermal radiation. Symmetry, 13.","DOI":"10.3390\/sym13010010"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"12983","DOI":"10.1007\/s10973-024-13391-2","article-title":"Significance of modified Fourier heat flux on Maxwell hybrid (Cu-Al2O3\/H2O) nanofluid transport past an inclined stretching cylinder","volume":"149","author":"Raghu","year":"2024","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"122330","DOI":"10.1016\/j.applthermaleng.2023.122330","article-title":"Highly efficient pool boiling heat transfer on surfaces with zoned rose-petal-inspired hierarchical structures","volume":"241","author":"Long","year":"2024","journal-title":"Appl. Therm. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"100826","DOI":"10.1016\/j.csite.2020.100826","article-title":"Thermal radiation impact on MHD heat transfer natural convective nano fluid flow over an impulsively started vertical plate","volume":"24","author":"Kumar","year":"2021","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"105956","DOI":"10.1016\/j.csite.2025.105956","article-title":"Impact on induced magnetic field over a second-grade hybrid nanofluid in unsteady thermal systems","volume":"69","author":"Gajjela","year":"2025","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"102247","DOI":"10.1016\/j.csite.2022.102247","article-title":"The numerical simulation of nanoparticle size and thermal radiation with the magnetic field effect based on tangent hyperbolic nanofluid flow","volume":"37","author":"Kumar","year":"2022","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1908","DOI":"10.1007\/s11630-021-1497-1","article-title":"A neural regression model for predicting thermal conductivity of CNT nanofluids with multiple base fluids","volume":"30","author":"Zou","year":"2021","journal-title":"J. Therm. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106376","DOI":"10.1016\/j.icheatmasstransfer.2022.106376","article-title":"Heat transport of nanofluid flow through a porous channel with thermal radiation effects","volume":"138","author":"Waqas","year":"2022","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3413","DOI":"10.1109\/TII.2023.3293863","article-title":"A Physical-constrained decomposition method of infrared thermography: Pseudo restored heat flux approach based on Ensemble Bayesian Variance Tensor Fraction","volume":"20","author":"Wang","year":"2023","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1166\/jon.2023.1939","article-title":"Convective conditions on 3D magnetohydrodynamic (MHD) non-Newtonian nanofluid flow with nonlinear thermal radiation and heat absorption: A numerical analysis","volume":"12","author":"Tarakaramu","year":"2023","journal-title":"J. Nanofluids"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"121158","DOI":"10.1016\/j.energy.2021.121158","article-title":"Thermal conductivity and phase change characteristics of hierarchical porous diamond\/erythritol composite phase change materials","volume":"233","author":"Yan","year":"2021","journal-title":"Energy"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"107863","DOI":"10.1016\/j.rinp.2024.107863","article-title":"Mathematical analysis of heat and mass transfer efficiency of bioconvective Casson nanofluid flow through conical gap among the rotating surfaces under the influences of thermal radiation and activation energy","volume":"63","author":"Basit","year":"2024","journal-title":"Results Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1007\/s10973-023-12782-1","article-title":"Optimization of RSM and sensitivity analysis in MHD ternary nanofluid flow between parallel plates with quadratic radiation and activation energy","volume":"149","author":"Vinutha","year":"2024","journal-title":"J. Therm. Anal. Calorim."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Shoaib, M., Raja, M.A.Z., Sabir, M.T., Islam, S., Shah, Z., Kumam, P., and Alrabaiah, H. (2020). Numerical investigation for rotating flow of MHD hybrid nanofluid with thermal radiation over a stretching sheet. Sci. Rep., 10.","DOI":"10.1038\/s41598-020-75254-8"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1108\/HFF-03-2020-0126","article-title":"MHD flow and heat transfer of hybrid nanofluid over a permeable moving surface in the presence of thermal radiation","volume":"31","author":"Zainal","year":"2021","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"106299","DOI":"10.1016\/j.icheatmasstransfer.2022.106299","article-title":"Significance of bio-convection, MHD, thermal radiation and activation energy across Prandtl nanofluid flow: A case of stretching cylinder","volume":"137","author":"Shah","year":"2022","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_18","first-page":"101078","article-title":"Neural network algorithms of a curved riga sensor in a ternary hybrid nanofluid with chemical reaction and Arrhenius kinetics","volume":"17","author":"Ramesh","year":"2024","journal-title":"J. Radiat. Res. Appl. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"100796","DOI":"10.1016\/j.rineng.2022.100796","article-title":"Comprehensive analysis of thermal radiation impact on an unsteady MHD nanofluid flow across an infinite vertical flat plate with ramped temperature with heat consumption","volume":"17","author":"Reddy","year":"2023","journal-title":"Results Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104129","DOI":"10.1016\/j.csite.2024.104129","article-title":"Chemical reaction, slip effects, and non-linear thermal radiation on unsteady MHD Jeffreys nanofluid flow over a stretching sheet","volume":"55","author":"Krishna","year":"2024","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"8447","DOI":"10.1016\/j.egyr.2021.10.121","article-title":"Modeling and optimization the effective parameters of nanofluid heat transfer performance using artificial neural network and genetic algorithm method","volume":"7","author":"Pourpasha","year":"2021","journal-title":"Energy Rep."},{"key":"ref_22","first-page":"1036","article-title":"Three-dimensional fluid-solid coupling heat transfer simulation based on the multireference frame for a side-blown aluminum annealing furnace","volume":"13","author":"Qiu","year":"2019","journal-title":"Eng. Appl. Comput. Fluid Mech."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Akbar, A., Ullah, H., Raja, M.A.Z., Nisar, K.S., Islam, S., and Shoaib, M. (2022). A design of neural networks to study mhd and heat transfer in two phase model of nano-fluid flow in the presence of thermal radiation. Waves Random Complex Media, 1\u201324.","DOI":"10.1080\/17455030.2022.2152905"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Roopa, G.S., Vishalakshi, C.S., Madhukesh, J.K., and Ramesh, G.K. (2025). Thermal investigation of Tiwari\u2013Das and Xue nanofluid model on space and thermal-dependent heat source\/sink over a spinning disk: A numerical study. Numer. Heat Transf. Part Appl., 1\u201316.","DOI":"10.1080\/10407782.2025.2522315"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Rehman, K.U., \u00c7olak, A.B., and Shatanawi, W. (2022). Artificial neural networking (ANN) model for convective heat transfer in thermally magnetized multiple flow regimes with temperature stratification effects. Mathematics, 10.","DOI":"10.3390\/math10142394"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"733","DOI":"10.1016\/j.cjph.2021.07.023","article-title":"Falkner\u2013Skan flow of aqueous magnetite\u2013graphene oxide nanoliquid driven by a wedge","volume":"77","author":"Ramesh","year":"2022","journal-title":"Chin. J. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"131261","DOI":"10.1016\/j.jhydrol.2024.131261","article-title":"Physics-informed neural network for diffusive wave model","volume":"637","author":"Hou","year":"2024","journal-title":"J. Hydrol."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Nasir, S., Berrouk, A.S., Gul, T., and Ali, A. (2023). Develop the artificial neural network approach to predict thermal transport analysis of nanofluid inside a porous enclosure. Sci. Rep., 13.","DOI":"10.1038\/s41598-023-48412-x"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"100936","DOI":"10.1016\/j.padiff.2024.100936","article-title":"Predicting heat transfer performance in transient flow of CNT nanomaterials with thermal radiation past a heated spinning sphere using an artificial neural network: A machine learning approach","volume":"12","author":"Mishra","year":"2024","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"105825","DOI":"10.1016\/j.csite.2025.105825","article-title":"An artificial neural network approach to comparative aspects: A predictive analysis of magnetic dipole on the heat transfer of maxwell hybrid nano coolants flow in an inclined cylinder","volume":"68","author":"Aruna","year":"2025","journal-title":"Case Stud. Therm. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"e202400050","DOI":"10.1002\/zamm.202400050","article-title":"Analysis of chemical characteristics of engine-oil-based Prandtl hybrid nanofluid flow","volume":"104","author":"Awan","year":"2024","journal-title":"ZAMM-J. Appl. Math. Mech.\/Z. Angew. Math. Mech."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2450421","DOI":"10.1142\/S0217984924504219","article-title":"Thermal characterization of Sutterby nanofluid flow under Riga plate: Tiwari and Das model","volume":"39","author":"Shah","year":"2024","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"106069","DOI":"10.1016\/j.icheatmasstransfer.2022.106069","article-title":"Simulation of the dynamics of colloidal mixture of water with various nanoparticles at different levels of partial slip: Ternary-hybrid nanofluid","volume":"135","author":"Cao","year":"2022","journal-title":"Int. Commun. Heat Mass Transf."},{"key":"ref_34","unstructured":"Cengel, Y.A., and Ghajar, A.J. (2014). Heat and Mass Transfer (in SI Units), Mcgraw-Hill Education-Europe."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"e202301055","DOI":"10.1002\/zamm.202301055","article-title":"Mixed convected synchronization of gyrotactic microorganism flow of an Eyring\u2013Powell nanofluid over a riga plate","volume":"104","author":"Awan","year":"2024","journal-title":"ZAMM-J. Appl. Math. Mech.\/Z. Angew. Math. Mech."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1134\/S1810232818020078","article-title":"Finite element analysis of MHD flow of micropolar fluid over a shrinking sheet with a convective surface boundary condition","volume":"27","author":"Gupta","year":"2018","journal-title":"J. Eng. Thermophys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1347\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:29:47Z","timestamp":1760034587000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1347"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,18]]},"references-count":36,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081347"],"URL":"https:\/\/doi.org\/10.3390\/sym17081347","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,18]]}}}