{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T11:28:18Z","timestamp":1764070098355,"version":"3.45.0"},"reference-count":23,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T00:00:00Z","timestamp":1764028800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Serbian Ministry of Science, Technological Development, and Innovations","award":["451-03-137\/2025-03\/200122"],"award-info":[{"award-number":["451-03-137\/2025-03\/200122"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>The FitzHugh\u2013Nagumo (FHN) equation in one dimension is solved in this paper using an improved physics-informed neural network (PINN) approach. Examining test problems with known analytical solutions and the explicit finite difference method (EFDM) allowed for the demonstration of the PINN\u2019s effectiveness. Our study presents an improved PINN formulation tailored to the FitzHugh\u2013Nagumo reaction\u2013diffusion system. The proposed framework is efficiently designed, validated, and systematically optimized, demonstrating that a careful balance among model complexity, collocation density, and training strategy enables high accuracy within limited computational time. Despite the very strong agreement that both methods provide, we have demonstrated that the PINN results exhibit a closer agreement with the analytical solutions for Test Problem 1, whereas the EFDM yielded more accurate results for Test Problem 2. This study is crucial for evaluating the PINN\u2019s performance in solving the FHN equation and its application to nonlinear processes like pulse propagation in optical fibers, drug delivery, neural behavior, geophysical fluid dynamics, and long-wave propagation in oceans, highlighting the potential of PINNs for complex systems. Numerical models for this class of nonlinear partial differential equations (PDEs) may be developed by existing and future model creators of a wide range of various nonlinear physical processes in the physical and engineering sectors using the concepts of the solution methods employed in this study.<\/jats:p>","DOI":"10.3390\/computation13120275","type":"journal-article","created":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T10:46:18Z","timestamp":1764067578000},"page":"275","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Improved Physics-Informed Neural Network Approach for Solving the FitzHugh\u2013Nagumo Equation"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8974-2267","authenticated-orcid":false,"given":"Milo\u0161","family":"Ivanovi\u0107","sequence":"first","affiliation":[{"name":"Faculty of Science, University of Kragujevac, R. Domanovi\u0107a 12, 34000 Kragujevac, Serbia"}]},{"given":"Matija","family":"Savovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Medical Sciences, University of Kragujevac, Svetozara Markovi\u0107a 69, 34000 Kragujevac, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9038-2393","authenticated-orcid":false,"given":"Svetislav","family":"Savovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Science, University of Kragujevac, R. Domanovi\u0107a 12, 34000 Kragujevac, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2024.09.014","article-title":"Six decades of the FitzHugh\u2013Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines","volume":"1096","author":"Gelens","year":"2024","journal-title":"Phys. Rep."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"014207","DOI":"10.1103\/PhysRevE.105.014207","article-title":"Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection","volume":"105","author":"Zemskov","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2463183","DOI":"10.1080\/27684830.2025.2463183","article-title":"Application of Haar-scale-3 wavelet collocation method to the solution of Fitzhugh\u2013Nagumo non-linear partial differential equation","volume":"12","author":"Arora","year":"2025","journal-title":"Res. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100764","DOI":"10.1016\/j.padiff.2024.100764","article-title":"Analysis of the nonlinear Fitzhugh\u2013Nagumo equation and its derivative based on the Rabotnov fractional exponential function","volume":"11","author":"Aychluh","year":"2024","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Rufai, M.A., Kosti, A.A., Anastassi, Z.A., and Carpentieri, B. (2024). A new two-step hybrid block method for the FitzHugh\u2013Nagumo model equation. Mathematics, 12.","DOI":"10.3390\/math12010051"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ahmad, I., Ahsan, M., Din, Z.U., Masood, A., and Kumam, P. (2019). An efficient local formulation for time-dependent PDEs. Mathematics, 7.","DOI":"10.20944\/preprints201901.0330.v1"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1201","DOI":"10.3934\/math.2025057","article-title":"A collocation procedure for the numerical treatment of FitzHugh\u2013Nagumo equation using a kind of Chebyshev polynomials","volume":"10","author":"Alqubori","year":"2025","journal-title":"AIMS Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"180","DOI":"10.32323\/ujma.561873","article-title":"Compact finite differences method for FitzHugh-Nagumo equation","volume":"2","author":"Akkoyunlu","year":"2019","journal-title":"Univers. J. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"100477","DOI":"10.1016\/j.rinam.2024.100477","article-title":"Numerical analysis of the stochastic FitzHugh\u2013Nagumo model driven by multiplicative noise based on the spectral Galerkin method","volume":"23","author":"Yang","year":"2024","journal-title":"Results Appl. Math."},{"key":"ref_10","first-page":"1384","article-title":"Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses","volume":"33","author":"Dehghan","year":"2010","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s40314-021-01729-7","article-title":"Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations","volume":"41","author":"Ramos","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"469","DOI":"10.4208\/jcm.1901-m2017-0263","article-title":"A fully discrete implicit-explicit finite element method for solving the FitzHugh-Nagumo model","volume":"38","author":"Cai","year":"2020","journal-title":"J. Comput. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"21040","DOI":"10.3934\/math.2025940","article-title":"A sixth-order compact finite difference framework for solving nonlinear reaction-diffusion equations: Application to FitzHugh-Nagumo model","volume":"10","author":"Fu","year":"2025","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Savovi\u0107, S., Ivanovi\u0107, M., and Min, R. (2023). A Comparative study of the explicit finite difference method and physics-informed neural networks for solving the Burgers\u2019 equation. Axioms, 12.","DOI":"10.3390\/axioms12100982"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Savovi\u0107, S., Ivanovi\u0107, M., Drlja\u010da, B., and Simovi\u0107, A. (2024). Numerical solution of the Sine-Gordon equation by physics-informed neural networks and two different finite difference methods. Axioms, 13.","DOI":"10.3390\/axioms13120872"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"179","DOI":"10.54254\/2753-8818\/10\/20230339","article-title":"Dynamics of the FitzHugh-Nagumo equation with numerical methods","volume":"10","author":"Fu","year":"2023","journal-title":"Theor. Nat. Sci."},{"key":"ref_18","unstructured":"Raissi, M., Perdikaris, P., and Karniadakis, G.E. (2017). Physics informed deep learning (part I): Data-driven solutions of nonlinear partial differential equations. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bono, F.M., Radicioni, L., and Cinquemani, S. (2023, January 12\u201317). A comparison between regular and physics-informed neural networks based on a numerical multibody model: A test case for the synthesis of mechanisms. Proceedings of the SPIE Smart Structures + Nondestructive Evaluation, Long Beach, CA, USA.","DOI":"10.1117\/12.2657981"},{"key":"ref_20","unstructured":"Lu, L., Meng, X., Mao, Z., and Karniadakis, G.E. (2019). Deepxde: A deep learning library for solving differential equations. arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Markidis, S. (2021). The old and the new: Can physics-informed deep-learning replace traditional linear solvers?. Front. Big Data, 4.","DOI":"10.3389\/fdata.2021.669097"},{"key":"ref_22","first-page":"e03011","article-title":"A hybrid firefly-Bayesian Neural Network approach for survival and reliability estimation in non-homogeneous Poisson processes","volume":"30","author":"Mushref","year":"2025","journal-title":"Sci. Afr."},{"key":"ref_23","first-page":"25","article-title":"Estimating parameters of Software Reliability Growth Models using Artificial Neural Networks optimized by the Artificial Bee Colony algorithm based on a novel NHPP","volume":"12","author":"HHussein","year":"2025","journal-title":"Math. Model. Eng. Probl."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/12\/275\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,25]],"date-time":"2025-11-25T11:25:26Z","timestamp":1764069926000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/12\/275"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,25]]},"references-count":23,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["computation13120275"],"URL":"https:\/\/doi.org\/10.3390\/computation13120275","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,25]]}}}