{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T21:37:30Z","timestamp":1770327450299,"version":"3.49.0"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,8]],"date-time":"2025-01-08T00:00:00Z","timestamp":1736294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"High-Level Talent Research Start-up Project Funding of Henan Academy of Sciences","award":["241819246"],"award-info":[{"award-number":["241819246"]}]},{"name":"High-Level Talent Research Start-up Project Funding of Henan Academy of Sciences","award":["FZZS-2024-0003"],"award-info":[{"award-number":["FZZS-2024-0003"]}]},{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"publisher","award":["241819246"],"award-info":[{"award-number":["241819246"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education of the Russian Federation","doi-asserted-by":"publisher","award":["FZZS-2024-0003"],"award-info":[{"award-number":["FZZS-2024-0003"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Nonlinear differential equations and systems play a crucial role in modeling systems where time-dependent factors exhibit nonlinear characteristics. Due to their nonlinear nature, solving such systems often presents significant difficulties and challenges. In this study, we propose a method utilizing Physics-Informed Neural Networks (PINNs) to solve the nonlinear energy supply\u2013demand (ESD) system. We design a neural network with four outputs, where each output approximates a function that corresponds to one of the unknown functions in the nonlinear system of differential equations describing the four-dimensional ESD problem. The neural network model is then trained, and the parameters are identified and optimized to achieve a more accurate solution. The solutions obtained from the neural network for this problem are equivalent when we compare and evaluate them against the Runge\u2013Kutta numerical method of order 5(4) (RK45). However, the method utilizing neural networks is considered a modern and promising approach, as it effectively exploits the superior computational power of advanced computer systems, especially in solving complex problems. Another advantage is that the neural network model, after being trained, can solve the nonlinear system of differential equations across a continuous domain. In other words, neural networks are not only trained to approximate the solution functions for the nonlinear ESD system but can also represent the complex dynamic relationships between the system\u2019s components. However, this approach requires significant time and computational power due to the need for model training. Furthermore, as this method is evaluated based on experimental results, ensuring the stability and convergence speed of the model poses a significant challenge. The key factors influencing this include the manner in which the neural network architecture is designed, such as the selection of hyperparameters and appropriate optimization functions. This is a critical and highly complex task, requiring experimentation and fine-tuning, which demand substantial expertise and time.<\/jats:p>","DOI":"10.3390\/computation13010013","type":"journal-article","created":{"date-parts":[[2025,1,8]],"date-time":"2025-01-08T04:54:08Z","timestamp":1736312048000},"page":"13","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Solving Nonlinear Energy Supply and Demand System Using Physics-Informed Neural Networks"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-2701-4775","authenticated-orcid":false,"given":"Van Truong","family":"Vo","sequence":"first","affiliation":[{"name":"Scientific Research Department, Irkutsk National Research Technical University, 664074 Irkutsk, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2307-0891","authenticated-orcid":false,"given":"Samad","family":"Noeiaghdam","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3131-1325","authenticated-orcid":false,"given":"Denis","family":"Sidorov","sequence":"additional","affiliation":[{"name":"Scientific Research Department, Irkutsk National Research Technical University, 664074 Irkutsk, Russia"},{"name":"Applied Mathematics Department, Melentiev Energy Systems Institute, Siberian Branch of Russian Academy of Sciences, 664003 Irkutsk, Russia"},{"name":"School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5032-0665","authenticated-orcid":false,"given":"Aliona","family":"Dreglea","sequence":"additional","affiliation":[{"name":"Scientific Research Department, Irkutsk National Research Technical University, 664074 Irkutsk, Russia"},{"name":"School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China"}]},{"given":"Liguo","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,8]]},"reference":[{"key":"ref_1","unstructured":"Chicone, C. (1999). Ordinary Differential Equations with Applications, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wong, P.J.Y. (2023). Applications of Partial Differential Equations, MDPI.","DOI":"10.3390\/books978-3-0365-9565-8"},{"key":"ref_3","unstructured":"Zachmanoglou, E.C., and Thoe, D.W. (1986). Introduction to Partial Differential Equations with Applications, Dover Publications, Inc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1016\/j.renene.2022.08.151","article-title":"A multi-criteria approach to designing and managing a renewable energy community","volume":"199","author":"Tomin","year":"2022","journal-title":"Renew. Energy"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sidorov, D., Tao, Q., Muftahov, I., Zhukov, A., Karamov, D., Dreglea, A., and Liu, F. (2019, January 27\u201330). Energy balancing using charge\/discharge storages control and load forecasts in a renewable-energy-based grids. Proceedings of the 2019 Chinese Control Conference (CCC), Guangzhou, China.","DOI":"10.23919\/ChiCC.2019.8865777"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.chaos.2007.01.125","article-title":"A new four-dimensional energy resources system and its linear feedback control","volume":"39","author":"Sun","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.chaos.2005.10.085","article-title":"An energy resources demand\u2013supply system and its dynamical analysis","volume":"32","author":"Sun","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1943","DOI":"10.1016\/j.chaos.2007.06.117","article-title":"Adaptive control and synchronization of a four-dimensional energy resources system with unknown parameters","volume":"39","author":"Sun","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Vuik, C., Vermolen, F.J., van Gijzen, M.B., and Vuik, M.J. (2023). Numerical Methods for Ordinary Differential Equations, Delft University of Technology (TU Delft).","DOI":"10.5074\/T.2023.001"},{"key":"ref_10","unstructured":"Lyengar, S.R.K., and Jain, R.K. (2009). Numerical Methods, New Age International Publishers."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","article-title":"Approximation by superpositions of a sigmoidal function","volume":"2","author":"Cybenko","year":"1989","journal-title":"Math. Control Signal Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1016\/0893-6080(90)90005-6","article-title":"Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks","volume":"3","author":"Hornik","year":"1990","journal-title":"Neural Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1109\/72.712178","article-title":"Artificial Neural Networks for Solving Ordinary and Partial Differential Equations","volume":"9","author":"Lagaris","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"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","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_16","unstructured":"Raissi, M., Yazdani, A., and Karniadakis, G.E. (2018). Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"110983","DOI":"10.1016\/j.jcp.2022.110983","article-title":"A Neural Network Multigrid Solver for the Navier-Stokes Equations","volume":"460","author":"Margenberg","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hu, B., and McDaniel, D. (2023). Applying Physics-Informed Neural Networks to Solve Navier\u2013Stokes Equations for Laminar Flow Around a Particle. Math. Comput. Appl., 28.","DOI":"10.3390\/mca28050102"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Farkane, A., Ghogho, M., Oudani, M., and Boutayeb, M. (2023). EPINN-NSE: Enhanced Physics-Informed Neural Networks for Solving Navier-Stokes Equations. arXiv.","DOI":"10.1002\/fld.5250"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"075117","DOI":"10.1063\/5.0095270","article-title":"Physics-Informed Neural Networks for Solving Reynolds-Averaged Navier\u2013Stokes Equations","volume":"34","author":"Eivazi","year":"2022","journal-title":"Phys. Fluids"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1137\/19M1274067","article-title":"DeepXDE: A Deep Learning Library for Solving Differential Equations","volume":"63","author":"Lu","year":"2021","journal-title":"SIAM Rev."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Sidorov, D., Tynda, A., Muftahov, I., Dreglea, A., and Liu, F. (2020). Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation. Mathematics, 8.","DOI":"10.3390\/math8081257"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"111260","DOI":"10.1016\/j.jcp.2022.111260","article-title":"A-PINN: Auxiliary Physics-Informed Neural Networks for Forward and Inverse Problems of Nonlinear Integro-Differential Equations","volume":"462","author":"Yuan","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"108242","DOI":"10.1016\/j.cnsns.2024.108242","article-title":"Machine Learning for Nonlinear Integro-Differential Equations with Degenerate Kernel Scheme","volume":"138","author":"Li","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","unstructured":"Matthews, J., and Bihlo, A. (2024). PinnDE: Physics-Informed Neural Networks for Solving Differential Equations. arXiv."},{"key":"ref_26","unstructured":"Baty, H., and Baty, L. (2023). Solving Differential Equations Using Physics-Informed Deep Learning: A Hands-On Tutorial with Benchmark Tests. arXiv."},{"key":"ref_27","unstructured":"Uriarte, C. (2024). Solving Partial Differential Equations Using Artificial Neural Networks. arXiv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012008","DOI":"10.1088\/1742-6596\/2308\/1\/012008","article-title":"Neural Networks in Solving Differential Equations","volume":"2308","author":"Gorikhovskii","year":"2022","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_29","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Learning, MIT Press."},{"key":"ref_30","unstructured":"Chollet, F. (2021). Deep Learning with Python, Manning Publications. [2nd ed.]."},{"key":"ref_31","unstructured":"G\u00e9ron, A. (2019). Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, O\u2019Reilly Media. [2nd ed.]."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Nielsen, M. (2024, October 15). Neural Networks and Deep Learning. Available online: http:\/\/neuralnetworksanddeeplearning.com\/.","DOI":"10.1049\/PBTE106E_ch2"},{"key":"ref_33","unstructured":"Nguyen, T.T. (2024, October 01). Basic Deep Learning. Available online: https:\/\/nttuan8.com\/sach-deep-learning-co-ban\/."},{"key":"ref_34","unstructured":"(2024, September 05). SciPy Reference. Available online: https:\/\/docs.scipy.org\/doc\/scipy\/reference\/integrate.html."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0771-050X(80)90013-3","article-title":"A Family of Embedded Runge-Kutta Formulae","volume":"6","author":"Dormand","year":"1980","journal-title":"J. Comput. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1090\/S0025-5718-1986-0815836-3","article-title":"Some Practical Runge-Kutta Formulas","volume":"46","author":"Shampine","year":"1986","journal-title":"Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"112126","DOI":"10.1016\/j.est.2024.112126","article-title":"Investigating the Deviation Between Prediction Accuracy Metrics and Control Performance Metrics in the Context of an Ice-Based Thermal Energy Storage System","volume":"91","author":"Wang","year":"2024","journal-title":"J. Energy Storage"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/1\/13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:24:37Z","timestamp":1759919077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/1\/13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,8]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["computation13010013"],"URL":"https:\/\/doi.org\/10.3390\/computation13010013","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,8]]}}}