{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:17:11Z","timestamp":1757618231183,"version":"3.44.0"},"reference-count":85,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T00:00:00Z","timestamp":1748217600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T00:00:00Z","timestamp":1748217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007504","name":"Istanbul Technical University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007504","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper, a feed-forward artificial neural network (FFNN) is proposed to analyze the behaviour characterized by nonlinear advection-diffusion-reaction (ADR) equations. This approach uses a trial function that satisfies the initial and boundary conditions and depends on a neural network constructed to approximate the solution of the problem. Since the trial function contains unknown parameters, the solution process must be minimized by using efficient optimization techniques to obtain these parameters. Therefore, in this paper, the gradient descent (GD) and particle swarm optimization (PSO) techniques are proposed to address the minimization issue. The results obtained by combining artificial neural network (ANN) method with the optimization techniques have been compared and the advantages and disadvantages of the problems have been discussed. The results revealed that the proposed ANN techniques have produced accurate and reliable solutions by comparing the exact and available literature. Furthermore, these techniques are economical in terms of computational memory.<\/jats:p>","DOI":"10.1007\/s40314-025-03215-w","type":"journal-article","created":{"date-parts":[[2025,5,26]],"date-time":"2025-05-26T05:06:10Z","timestamp":1748235970000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neural network based techniques for steep behaviour represented by nonlinear advection\u2013diffusion-reaction models"],"prefix":"10.1007","volume":"44","author":[{"given":"Seda","family":"Gulen","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0508-2917","authenticated-orcid":false,"given":"Murat","family":"Sari","sequence":"additional","affiliation":[]},{"given":"Pelin","family":"Celenk","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,26]]},"reference":[{"issue":"3","key":"3215_CR1","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1023\/A:1012784129883","volume":"14","author":"LP Aarts","year":"2001","unstructured":"Aarts LP, Van Der Veer P (2001) Neural network method for solving partial differential equations. Neural Process Lett 14(3):261-271S","journal-title":"Neural Process Lett"},{"issue":"21","key":"3215_CR2","doi-asserted-by":"publisher","first-page":"2689","DOI":"10.3390\/electronics10212689","volume":"10","author":"MGM Abdolrasol","year":"2021","unstructured":"Abdolrasol MGM, Hussain SMS, Ustun TS, Sarker MR, Hannan MA, Mohamed R, Ali JA, Mekhilef S, Milad A (2021) Artificial neural networks based optimization techniques: a review. Electronics 10(21):2689","journal-title":"Electronics"},{"doi-asserted-by":"crossref","unstructured":"Ablowitz MJ, Fuchssteiner B, Kruskal M (1987) Topics in soliton theory and exactly solvable nonlinear equations: proceedings of the conference on nonlinear evolution equations, solitons and the inverse scattering transform","key":"3215_CR3","DOI":"10.1142\/9789814542210"},{"issue":"4","key":"3215_CR4","doi-asserted-by":"publisher","first-page":"1121","DOI":"10.1007\/s11075-018-0646-4","volume":"82","author":"N Alinia","year":"2019","unstructured":"Alinia N, Zarebnia M (2019) A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation. Numer Algor 82(4):1121\u20131142","journal-title":"Numer Algor"},{"issue":"2","key":"3215_CR5","doi-asserted-by":"publisher","DOI":"10.5812\/jjhs.67544","volume":"10","author":"M Alizamir","year":"2018","unstructured":"Alizamir M, Sobhanardakani S (2018) An artificial neural network - particle swarm optimization (ANN- PSO) approach to predict heavy metals contamination in groundwater resources. Jundishapur J Health Sci 10(2):e67544","journal-title":"Jundishapur J Health Sci"},{"issue":"5","key":"3215_CR6","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/S0016-0032(03)00036-X","volume":"340","author":"H Alli","year":"2003","unstructured":"Alli H, Ucar A, Demir Y (2003) The solutions of vibration control problems using artificial neural networks. J Franklin Inst 340(5):307\u2013325","journal-title":"J Franklin Inst"},{"key":"3215_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109827","volume":"424","author":"E Bachini","year":"2021","unstructured":"Bachini E, Farthing MW, Putti M (2021) Intrinsic finite element method for advection-diffusion-reaction equations on surfaces. J Comput Phys 424:109827","journal-title":"J Comput Phys"},{"issue":"3","key":"3215_CR8","doi-asserted-by":"publisher","first-page":"660","DOI":"10.1016\/j.chaos.2006.06.080","volume":"36","author":"B Batiha","year":"2008","unstructured":"Batiha B, Noorani MSM, Hashim I (2008) Application of variational iteration method to the generalized Burgers-Huxley equation. Chaos, Solitons Fractals 36(3):660\u2013663","journal-title":"Chaos, Solitons Fractals"},{"issue":"9","key":"3215_CR9","doi-asserted-by":"publisher","first-page":"898","DOI":"10.1016\/j.jfranklin.2009.05.003","volume":"346","author":"RS Beidokhti","year":"2009","unstructured":"Beidokhti RS, Malek A (2009) Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques. J Franklin Inst 346(9):898\u2013913","journal-title":"J Franklin Inst"},{"key":"3215_CR10","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.neucom.2018.06.056","volume":"317","author":"J Berg","year":"2018","unstructured":"Berg J, Nystr\u00f6m K (2018) A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317:28\u201341","journal-title":"Neurocomputing"},{"key":"3215_CR11","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.cam.2019.01.028","volume":"356","author":"AG Bratsos","year":"2019","unstructured":"Bratsos AG, Khaliq AQM (2019) An exponential time differencing method of lines for Burgers-Fisher and coupled Burger\u2019s equations. J Comput Appl Math 356:182\u2013197","journal-title":"J Comput Appl Math"},{"key":"3215_CR12","doi-asserted-by":"publisher","first-page":"1567","DOI":"10.1007\/s10915-019-01099-7","volume":"81","author":"KP Chang","year":"2019","unstructured":"Chang KP, Libertini JM, Seay S (2019) Comparing gradient descent with automatic differentiation and particle swarm optimization techniques for estimating tumor blood flow parameters in contrast-enhanced imaging. J Sci Comput 81:1567\u20131576","journal-title":"J Sci Comput"},{"unstructured":"Clow B, White T (2004). An evolutionary race: a comparison of genetic algorithms and particle swarm optimization used for training neural networks. In: Proceedings of the International Conference on Artificial Intelligence. pp 582\u2013588","key":"3215_CR13"},{"unstructured":"Divband M (2010) A comparison of particle swarm optimization and gradient descent in training wavelet neural network to predict DGPS corrections. In: Proceedings of the World Congress on Engineering and Computer Science Vol I WCECS 2010","key":"3215_CR14"},{"key":"3215_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2021.107100","volume":"102","author":"QH Doan","year":"2021","unstructured":"Doan QH, Le T, Thai DK (2021) Optimization strategies of neural networks for impact damage classification of RC panels in a small dataset. Appl Soft Comput 102:107100","journal-title":"Appl Soft Comput"},{"issue":"1","key":"3215_CR16","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1108\/HFF-05-2017-0198","volume":"29","author":"U Erdogan","year":"2018","unstructured":"Erdogan U, Sari M, Kocak H (2018) Efficient numerical treatment of nonlinearities in the advection\u2013diffusion\u2013reaction equations. Int J Numer Methods Heat Fluid Flow 29(1):132\u2013145","journal-title":"Int J Numer Methods Heat Fluid Flow"},{"issue":"3","key":"3215_CR17","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1007\/s10614-020-10070-w","volume":"58","author":"S Eskiizmirliler","year":"2021","unstructured":"Eskiizmirliler S, G\u00fcnel K, Polat R (2021) On the solution of the black\u2013scholes equation using feed-forward neural networks. Comput Econ 58(3):915\u2013941","journal-title":"Comput Econ"},{"issue":"2","key":"3215_CR18","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1109\/TPWRS.2005.846049","volume":"20","author":"AA Esmin","year":"2005","unstructured":"Esmin AA, Lambert-Torres G, De Souza AZ (2005) A hybrid particle swarm optimization applied to loss power minimization. IEEE Trans Power Syst 20(2):859\u2013866","journal-title":"IEEE Trans Power Syst"},{"issue":"3","key":"3215_CR19","doi-asserted-by":"publisher","first-page":"1412","DOI":"10.11591\/ijpeds.v9.i3.pp1412-1422","volume":"9","author":"Z Frijet","year":"2018","unstructured":"Frijet Z, Zribi A, Chtourou M (2018) An adaptive neural network controller based on PSO and gradient descent method for PMSM speed drive. Int J Power Electron Drive Syst (IJPEDS) 9(3):1412\u20131422","journal-title":"Int J Power Electron Drive Syst (IJPEDS)"},{"issue":"5","key":"3215_CR20","doi-asserted-by":"publisher","first-page":"2747","DOI":"10.1016\/j.apm.2012.06.010","volume":"37","author":"I Gasser","year":"2013","unstructured":"Gasser I, Rybicki M (2013) Modelling and simulation of gas dynamics in an exhaust pipe. Appl Math Model 37(5):2747\u20132764","journal-title":"Appl Math Model"},{"issue":"5","key":"3215_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40819-018-0565-z","volume":"4","author":"K Gilani","year":"2018","unstructured":"Gilani K, Saeed U (2018) CAS wavelet Picard technique for Burger\u2019s\u2013Huxley and Burgers equation. Int J Appl Comput Math 4(5):1\u201314","journal-title":"Int J Appl Comput Math"},{"issue":"1","key":"3215_CR22","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1137\/S0036141003427373","volume":"36","author":"JB Greer","year":"2004","unstructured":"Greer JB, Bertozzi AL (2004) Traveling wave solutions of fourth order PDEs for image processing. SIAM J Math Anal 36(1):38\u201368","journal-title":"SIAM J Math Anal"},{"doi-asserted-by":"crossref","unstructured":"Gudise, V. G., Venayagamoorthy, G. K. (2003) Comparison of particle swarm optimization and backpropagation as training algorithms for neural networks. In: Proceedings of the IEEE Swarm Intelligence Symposium (SIS), 110\u2013117.","key":"3215_CR23","DOI":"10.1109\/SIS.2003.1202255"},{"key":"3215_CR24","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/s10665-022-10249-0","volume":"138","author":"S Gulen","year":"2023","unstructured":"Gulen S (2023) An efficient hybrid method based on cubic B-spline and fourth-order compact finite difference for solving nonlinear advection-diffusion-reaction equations. J Eng Math 138:13","journal-title":"J Eng Math"},{"issue":"2","key":"3215_CR25","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1515\/ijnsns-2017-0277","volume":"21","author":"S Gulen","year":"2020","unstructured":"Gulen S, Ozis T (2020) Solution of a moving boundary problem for soybean hydration by numerical approximation. Int J Nonlinear Sci Numer. Simul 21(2):115\u2013122","journal-title":"Int J Nonlinear Sci Numer. Simul"},{"issue":"2","key":"3215_CR26","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1002\/mma.7821","volume":"45","author":"S Gulen","year":"2022","unstructured":"Gulen S, Sari M (2022) A Fr\u00e9chet derivative-based novel approach to option pricing models in illiquid markets. Math Methods Appl Sci 45(2):899\u2013913","journal-title":"Math Methods Appl Sci"},{"issue":"8","key":"3215_CR27","doi-asserted-by":"publisher","first-page":"760","DOI":"10.3390\/math7080760","volume":"7","author":"S Gulen","year":"2019","unstructured":"Gulen S, Popescu C, Sari M (2019) A new approach for the black-scholes model with linear and nonlinear volatilities. Mathematics 7(8):760","journal-title":"Mathematics"},{"issue":"4","key":"3215_CR28","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1016\/j.dsp.2005.12.004","volume":"16","author":"Q Guo","year":"2006","unstructured":"Guo Q, Yu H, Xu A (2006) A hybrid PSO-GD based intelligent method for machine diagnosis. Dig Signal Proc 16(4):402\u2013418","journal-title":"Dig Signal Proc"},{"key":"3215_CR29","first-page":"172","volume":"281","author":"Y Guo","year":"2016","unstructured":"Guo Y, Shi YF, Li YM (2016) A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers\u2019 equation. Appl Math Comput 281:172\u2013185","journal-title":"Appl Math Comput"},{"issue":"1","key":"3215_CR30","first-page":"2207","volume":"16","author":"T Hai","year":"2022","unstructured":"Hai T, Li H, Band SS, Shadkani S, Samadianfard S, Hashemi S, Chau KW, Mousavi A (2022) Comparison of the efficacy of particle swarm optimization and stochastic gradient descent algorithms on multi-layer perceptron model to estimate longitudinal dispersion coefficients in natural streams. Eng Appl Comput Fluid Mech 16(1):2207\u20132221","journal-title":"Eng Appl Comput Fluid Mech"},{"key":"3215_CR31","first-page":"296","volume":"258","author":"DA Hammad","year":"2015","unstructured":"Hammad DA, El-Azab MS (2015) 2N order compact finite difference scheme with collocation method for solving the generalized Burger\u2019s\u2013Huxley and Burger\u2019s\u2013 Fisher equations. Appl Math Comput 258:296\u2013311","journal-title":"Appl Math Comput"},{"issue":"11\u201312","key":"3215_CR32","doi-asserted-by":"publisher","first-page":"1404","DOI":"10.1016\/j.mcm.2005.08.017","volume":"43","author":"I Hashim","year":"2006","unstructured":"Hashim I, Noorani MSM, Al-Hadidi MS (2006) Solving the generalized Burgers-Huxley equation using the Adomian decomposition method. Math Comput Model 43(11\u201312):1404\u20131411","journal-title":"Math Comput Model"},{"issue":"19","key":"3215_CR33","doi-asserted-by":"publisher","first-page":"2812","DOI":"10.3923\/jas.2007.2812.2817","volume":"7","author":"M Hayati","year":"2007","unstructured":"Hayati M, Karami B (2007) Feedforward neural network for solving partial differential equations. J Appl Sci 7(19):2812\u20132817","journal-title":"J Appl Sci"},{"issue":"6","key":"3215_CR34","doi-asserted-by":"publisher","first-page":"3042","DOI":"10.3390\/app12063042","volume":"12","author":"Y He","year":"2022","unstructured":"He Y, Xue G, Chen W, Tian Z (2022) Three-dimensional inversion of semi- airborne transient electromagnetic data based on a particle swarm optimization-gradient descent algorithm. Appl Sci 12(6):3042","journal-title":"Appl Sci"},{"key":"3215_CR35","doi-asserted-by":"publisher","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","volume":"117","author":"AL Hodgkin","year":"1952","unstructured":"Hodgkin AL (1952) A quantitative description of ion currents and its application and excitation in nerve membranes. J Physiol 117:500\u2013544","journal-title":"J Physiol"},{"issue":"6","key":"3215_CR36","doi-asserted-by":"publisher","first-page":"3050","DOI":"10.1137\/07069208X","volume":"30","author":"C Hogea","year":"2008","unstructured":"Hogea C, Davatzikos C, Biros G (2008) Brain-Tumor interaction biophysical models for medical image registration. SIAM J Sci Comput 30(6):3050\u20133072","journal-title":"SIAM J Sci Comput"},{"issue":"4","key":"3215_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40314-021-01505-7","volume":"40","author":"M Hussain","year":"2021","unstructured":"Hussain M (2021) Hybrid radial basis function methods of lines for the numerical solution of viscous Burgers\u2019 equation. Comput Appl Math 40(4):1\u201349","journal-title":"Comput Appl Math"},{"doi-asserted-by":"crossref","unstructured":"Jafari, R., Yu, W. (2015, October). Artificial neural network approach for solving strongly degenerate parabolic and burgers-fisher equations. In: 2015 12th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)\u00a0(pp. 1\u20136). IEEE.","key":"3215_CR38","DOI":"10.1109\/ICEEE.2015.7357914"},{"key":"3215_CR39","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1016\/j.cpc.2014.11.004","volume":"188","author":"R Jiwari","year":"2015","unstructured":"Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the Burgers\u2019 equation. Comput Phys Commun 188:59\u201367","journal-title":"Comput Phys Commun"},{"issue":"12","key":"3215_CR40","first-page":"6680","volume":"219","author":"R Jiwari","year":"2013","unstructured":"Jiwari R, Mittal RC, Sharma KK (2013) A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers\u2019 equation. Appl Math Comput 219(12):6680\u20136691","journal-title":"Appl Math Comput"},{"key":"3215_CR41","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1108\/EC-04-2018-0189","volume":"36","author":"R Jiwari","year":"2019","unstructured":"Jiwari R, Kumar S, Mittal RC (2019) Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers\u2019 type problems. Eng Comput 36:1142","journal-title":"Eng Comput"},{"issue":"2","key":"3215_CR42","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1002\/nme.1620290203","volume":"29","author":"K Kakuda","year":"1990","unstructured":"Kakuda K, Tosaka N (1990) The generalized boundary element approach to Burgers\u2019 equation. Int J Numer Meth Eng 29(2):245\u2013261","journal-title":"Int J Numer Meth Eng"},{"unstructured":"Karaboga D (2005) An idea based on honey bee swarm for numerical optimization. Technical report, Technical Report-tr06, Erciyes University, Engineering Faculty, Computer Engineering Department","key":"3215_CR43"},{"doi-asserted-by":"crossref","unstructured":"Khan, J. A., Zahoor, R. M. A., & Qureshi, M. (2009). Swarm intelligence for the solution of problems in differential qquations. In: 2009 Second International Conference on Environmental and Computer Science, Dubai, United Arab Emirates. pp. 141\u2013147","key":"3215_CR44","DOI":"10.1109\/ICECS.2009.85"},{"issue":"2","key":"3215_CR45","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1016\/j.cam.2007.11.007","volume":"222","author":"AH Khater","year":"2008","unstructured":"Khater AH, Temsah RS, Hassan M (2008) A Chebyshev spectral collocation method for solving Burgers\u2019-type equations. J Comput Appl Math 222(2):333\u2013350","journal-title":"J Comput Appl Math"},{"issue":"5","key":"3215_CR46","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1007\/s40009-015-0359-3","volume":"38","author":"M Kumar","year":"2015","unstructured":"Kumar M, Yadav N (2015) Numerical solution of Bratu\u2019s problem using multilayer perceptron neural network method. Natl Acad Sci Lett 38(5):425\u2013428","journal-title":"Natl Acad Sci Lett"},{"key":"3215_CR47","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1007\/s40819-023-01557-9","volume":"9","author":"H Kumar","year":"2023","unstructured":"Kumar H, Yadav N (2023) A deep learning algorithm for solving generalized Burgers- Fisher and Burger\u2019s equation. Int J Appl Comput Math 9:76","journal-title":"Int J Appl Comput Math"},{"key":"3215_CR48","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2022.105289","volume":"115","author":"H Kumar","year":"2022","unstructured":"Kumar H, Yadav N, Nagar AK (2022) Numerical solution of generalized Burger- Huxley & Huxley\u2019s equation using Deep Galerkin neural network method. Eng Appl Artif Intell 115:105289","journal-title":"Eng Appl Artif Intell"},{"issue":"12","key":"3215_CR49","doi-asserted-by":"publisher","first-page":"1706","DOI":"10.1080\/10236198.2019.1689236","volume":"25","author":"A Kumar Verma","year":"2019","unstructured":"Kumar Verma A, Kayenat S (2019) On the stability of Micken\u2019s type NSFD schemes for generalized Burgers Fisher equation. J Differ Equ Appl 25(12):1706\u20131737","journal-title":"J Differ Equ Appl"},{"issue":"5","key":"3215_CR50","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","volume":"9","author":"IE Lagaris","year":"1998","unstructured":"Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Networks 9(5):987\u20131000","journal-title":"IEEE Trans Neural Networks"},{"issue":"5","key":"3215_CR51","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1109\/72.870037","volume":"11","author":"IE Lagaris","year":"2000","unstructured":"Lagaris IE, Likas AC, Papageorgiou DG (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Networks 11(5):1041\u20131049","journal-title":"IEEE Trans Neural Networks"},{"key":"3215_CR52","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2020.106896","volume":"114","author":"PW Li","year":"2021","unstructured":"Li PW (2021) Space-time generalized finite difference nonlinear model for solving unsteady Burgers\u2019 equations. Appl Math Lett 114:106896","journal-title":"Appl Math Lett"},{"issue":"5\u20136","key":"3215_CR53","doi-asserted-by":"publisher","first-page":"835","DOI":"10.1016\/j.neunet.2005.06.007","volume":"18","author":"D Liu","year":"2005","unstructured":"Liu D, Xiong X, Hou ZG, DasGupta B (2005) Identification of motifs with insertions and deletions in protein sequences using self-organizing neural networks. Neural Netw 18(5\u20136):835\u2013842","journal-title":"Neural Netw"},{"issue":"2","key":"3215_CR54","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/s10483-019-2429-8","volume":"40","author":"Z Liu","year":"2019","unstructured":"Liu Z, Yang Y, Cai Q (2019) Neural network as a function approximator and its application in solving differential equations. Appl Math Mech 40(2):237\u2013248","journal-title":"Appl Math Mech"},{"issue":"3","key":"3215_CR55","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0121728","volume":"10","author":"SA Malik","year":"2015","unstructured":"Malik SA, Qureshi IM, Amir M, Malik AN, Haq I (2015) Numerical solution to generalized Burgers\u2019-Fisher equation using exp-function method hybridized with heuristic computation. PLoS ONE 10(3):e0121728","journal-title":"PLoS ONE"},{"issue":"4","key":"3215_CR56","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/BF02478259","volume":"5","author":"WS McCulloch","year":"1943","unstructured":"McCulloch WS, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5(4):115\u2013133","journal-title":"Bull Math Biophys"},{"issue":"8","key":"3215_CR57","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1109\/TNN.2009.2020735","volume":"20","author":"KS McFall","year":"2009","unstructured":"McFall KS, Mahan JR (2009) Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions. IEEE Trans Neural Networks 20(8):1221\u20131233","journal-title":"IEEE Trans Neural Networks"},{"issue":"5","key":"3215_CR58","doi-asserted-by":"publisher","first-page":"1756","DOI":"10.1016\/j.chaos.2006.03.031","volume":"33","author":"M Moghimi","year":"2007","unstructured":"Moghimi M, Hejazi FS (2007) Variational iteration method for solving generalized Burger-Fisher and Burger equations. Chaos, Solitons Fractals 33(5):1756\u20131761","journal-title":"Chaos, Solitons Fractals"},{"key":"3215_CR59","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1007\/s10064-014-0638-0","volume":"74","author":"ET Mohamad","year":"2015","unstructured":"Mohamad ET, Jahed Armaghani D, Momeni E et al (2015) Prediction of the unconfined compressive strength of soft rocks: a PSO-based ANN approach. Bull Eng Geol Env 74:745\u2013757","journal-title":"Bull Eng Geol Env"},{"issue":"15","key":"3215_CR60","doi-asserted-by":"publisher","first-page":"2639","DOI":"10.3390\/math10152639","volume":"10","author":"N Ndou","year":"2022","unstructured":"Ndou N, Dlamini P, Jacobs BA (2022) Enhanced unconditionally positive finite difference method for advection-diffusion-reaction equations. Mathematics 10(15):2639","journal-title":"Mathematics"},{"key":"3215_CR61","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/S0933-3657(03)00050-2","volume":"30","author":"M Ohlsson","year":"2004","unstructured":"Ohlsson M (2004) WeAidU-a decision support system for myocardial perfusion images using artificial neural networks. Artif Intell Med 30:49\u201360","journal-title":"Artif Intell Med"},{"issue":"3","key":"3215_CR62","doi-asserted-by":"publisher","first-page":"2147","DOI":"10.1007\/s11063-021-10508-8","volume":"53","author":"S Panghal","year":"2021","unstructured":"Panghal S, Kumar M (2021) Approximate analytic solution of Burger Huxley equation using feed-forward artificial neural network. Neural Process Lett 53(3):2147\u20132163","journal-title":"Neural Process Lett"},{"issue":"10","key":"3215_CR63","doi-asserted-by":"publisher","first-page":"1441","DOI":"10.1016\/j.ins.2008.11.034","volume":"179","author":"R Pasti","year":"2009","unstructured":"Pasti R, Castro LN (2009) Bio-inspired and gradient-based algorithms to train MLPs: the influence of diversity. Inf Sci 179(10):1441\u20131453","journal-title":"Inf Sci"},{"issue":"1","key":"3215_CR64","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.100.016002","volume":"100","author":"ML Piscopo","year":"2019","unstructured":"Piscopo ML, Spannowsky M, Waite P (2019) Solving differential equations with neural networks: applications to the calculation of cosmological phase transitions. Phys Rev D 100(1):016002","journal-title":"Phys Rev D"},{"issue":"1","key":"3215_CR65","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S0893-6080(96)00075-5","volume":"10","author":"RG Rosandich","year":"1997","unstructured":"Rosandich RG (1997) HAVNET: a new neural network architecture for pattern recognition. Neural Netw 10(1):139\u2013151","journal-title":"Neural Netw"},{"key":"3215_CR66","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/j.neucom.2014.11.058","volume":"155","author":"K Rudd","year":"2015","unstructured":"Rudd K, Ferrari S (2015) A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks. Neurocomputing 155:277\u2013285","journal-title":"Neurocomputing"},{"issue":"2","key":"3215_CR67","first-page":"475","volume":"208","author":"M Sari","year":"2009","unstructured":"Sari M, Gurarslan G (2009) A sixth-order compact finite difference scheme to the numerical solutions of Burgers\u2019 equation. Appl Math Comput 208(2):475\u2013483","journal-title":"Appl Math Comput"},{"issue":"1","key":"3215_CR68","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1002\/num.20421","volume":"26","author":"M Sari","year":"2010","unstructured":"Sari M, Gurarslan G, Dag I (2010) A compact finite difference method for the solution of the generalized Burgers-Fisher equation. Numer Methods Partial Differ Equ: Int J 26(1):125\u2013134","journal-title":"Numer Methods Partial Differ Equ: Int J"},{"issue":"5","key":"3215_CR69","doi-asserted-by":"publisher","first-page":"1313","DOI":"10.1002\/num.20585","volume":"27","author":"M Sari","year":"2011","unstructured":"Sari M, Gurarslan G, Zeytinoglu A (2011) High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation. Numer Methods Partial Differ Equ 27(5):1313\u20131326","journal-title":"Numer Methods Partial Differ Equ"},{"key":"3215_CR70","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1016\/j.cam.2018.05.063","volume":"344","author":"M Seydaoglu","year":"2018","unstructured":"Seydaoglu M (2018) An accurate approximation algorithm for Burgers\u2019 equation in the presence of small viscosity. J Comput Appl Math 344:473\u2013481","journal-title":"J Comput Appl Math"},{"issue":"13","key":"3215_CR71","doi-asserted-by":"publisher","first-page":"3831","DOI":"10.1007\/s12648-022-02304-4","volume":"96","author":"MA Shallal","year":"2022","unstructured":"Shallal MA, Taqi AH, Jumaa BF, Rezazadeh H, Inc M (2022) Numerical solutions to the 1-D Burgers\u2019 equation by a cubic hermite finite element method. Indian J Phys 96(13):3831\u20133836","journal-title":"Indian J Phys"},{"issue":"11\u201312","key":"3215_CR72","doi-asserted-by":"publisher","first-page":"2943","DOI":"10.1016\/j.mcm.2011.07.016","volume":"54","author":"L Shao","year":"2011","unstructured":"Shao L, Feng X, He Y (2011) The local discontinuous Galerkin finite element method for Burger\u2019s equation. Math Comput Model 54(11\u201312):2943\u20132954","journal-title":"Math Comput Model"},{"issue":"1","key":"3215_CR73","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.asoc.2008.02.003","volume":"9","author":"Y Shirvany","year":"2009","unstructured":"Shirvany Y, Hayati M, Moradian R (2009) Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations. Appl Soft Comput 9(1):20\u201329","journal-title":"Appl Soft Comput"},{"issue":"1","key":"3215_CR74","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s00366-020-01037-4","volume":"38","author":"S Shukla","year":"2020","unstructured":"Shukla S, Kumar M (2020) Error analysis and numerical solution of Burgers\u2013 Huxley equation using 3-scale Haar wavelets. Eng Comput 38(1):3\u201311","journal-title":"Eng Comput"},{"key":"3215_CR75","volume":"399","author":"BK Singh","year":"2021","unstructured":"Singh BK, Gupta M (2021) A new efficient fourth order collocation scheme for solving Burgers\u2019 equation. Appl Math Comput 399:126011","journal-title":"Appl Math Comput"},{"issue":"3","key":"3215_CR76","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/j.neunet.2005.01.003","volume":"18","author":"AH Tan","year":"2005","unstructured":"Tan AH, Pan H (2005) Predictive neural networks for gene expression data analysis. Neural Netw 18(3):297\u2013306","journal-title":"Neural Netw"},{"issue":"6","key":"3215_CR77","first-page":"2853","volume":"27","author":"H Tunc","year":"2020","unstructured":"Tunc H, Sari M (2020) Simulations of nonlinear advection-diffusion models through various finite element techniques. Scientia Iranica 27(6):2853\u20132870","journal-title":"Scientia Iranica"},{"key":"3215_CR78","first-page":"eae495","volume-title":"Encyclopedia of aerospace engineering","author":"G Venter","year":"2010","unstructured":"Venter G (2010) Review of optimization techniques. In: Blockley R, Shyy W (eds) Encyclopedia of aerospace engineering. John Wiley & Sons Ltd, Chichester, p eae495"},{"key":"3215_CR79","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.61.956","volume":"61","author":"NK Vitanov","year":"2000","unstructured":"Vitanov NK (2000) Upper bound on the heat transport in a layer of fluid of infinite Prandtl number, rigid lower boundary, and stress-free upper boundary. Phys Rev E 61:956959","journal-title":"Phys Rev E"},{"issue":"1","key":"3215_CR80","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s00521-015-2046-1","volume":"28","author":"N Yadav","year":"2017","unstructured":"Yadav N, Yadav A, Kumar M, Kim JH (2017) An efficient algorithm based on artificial neural networks and particle swarm optimization for solution of nonlinear Troesch\u2019s problem. Neural Comput Appl 28(1):171\u2013178","journal-title":"Neural Comput Appl"},{"issue":"1","key":"3215_CR81","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1109\/TNN.2004.836196","volume":"16","author":"ZR Yang","year":"2005","unstructured":"Yang ZR, Thomson R (2005) Bio-basis function neural network for prediction of protease cleavage sites in proteins. IEEE Trans Neural Networks 16(1):263\u2013274","journal-title":"IEEE Trans Neural Networks"},{"issue":"12","key":"3215_CR82","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0188746","volume":"12","author":"F Ye","year":"2017","unstructured":"Ye F (2017) Particle swarm optimization-based automatic parameter selection for deep neural networks and its applications in large-scale and high-dimensional data. PLoS ONE 12(12):e0188746","journal-title":"PLoS ONE"},{"key":"3215_CR83","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108271","volume":"133","author":"X Zhang","year":"2022","unstructured":"Zhang X, Xu X (2022) A moving finite element method for solving two-dimensional coupled Burgers\u2019 equations at high reynolds numbers. Appl Math Lett 133:108271","journal-title":"Appl Math Lett"},{"issue":"3","key":"3215_CR84","doi-asserted-by":"publisher","first-page":"1046","DOI":"10.1016\/j.apm.2011.07.059","volume":"36","author":"T Zhao","year":"2012","unstructured":"Zhao T, Li C, Zang Z, Wu Y (2012) Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation. Appl Math Model 36(3):1046\u20131056","journal-title":"Appl Math Model"},{"issue":"13","key":"3215_CR85","doi-asserted-by":"publisher","first-page":"5207","DOI":"10.3390\/en16135207","volume":"16","author":"E Zulu","year":"2023","unstructured":"Zulu E, Hara R, Kita H (2023) An efficient hybrid particle swarm and gradient descent method for the estimation of the hosting capacity of photovoltaics by distribution networks. Energies 16(13):5207","journal-title":"Energies"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03215-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03215-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03215-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T18:26:33Z","timestamp":1757183193000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03215-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,26]]},"references-count":85,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["3215"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03215-w","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,5,26]]},"assertion":[{"value":"21 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 May 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"300"}}