{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T09:58:56Z","timestamp":1771235936732,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T00:00:00Z","timestamp":1766188800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Conservatively perturbed equilibrium (CPE) experiments yield transient concentration extrema that surpass steady-state equilibrium values. A physics-informed neural network (PINN) framework is introduced to simulate these over-equilibrium dynamics in linear chemical reaction networks without reliance on extensive time-series data. The PINN incorporates the reaction kinetics, stoichiometric invariants, and equilibrium constraints directly into its loss function, ensuring that the learned solution strictly satisfies physical conservation laws. Applied to three- and four-species reversible mechanisms (both acyclic and cyclic), the PINN surrogate matches conventional ODE integration results, reproducing the characteristic early concentration extrema (maxima or minima) in unperturbed species and the subsequent relaxation to equilibrium. It captures the timing and magnitude of these extrema with high accuracy while inherently preserving total mass. Through the physics-informed approach, the model achieves accurate results with minimal data and a compact network architecture, highlighting its parameter efficiency.<\/jats:p>","DOI":"10.3390\/e28010009","type":"journal-article","created":{"date-parts":[[2025,12,26]],"date-time":"2025-12-26T03:06:02Z","timestamp":1766718362000},"page":"9","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Physics-Informed Neural Network (PINN) Approach to Over-Equilibrium Dynamics in Conservatively Perturbed Linear Equilibrium Systems"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0714-1119","authenticated-orcid":false,"given":"Abhishek","family":"Dutta","sequence":"first","affiliation":[{"name":"Department of Chemical Engineering, Izmir Institute of Technology, Izmir 35430, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-6776-9074","authenticated-orcid":false,"given":"Bitan","family":"Mukherjee","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Jadavpur University, Kolkata 700032, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-1471-8183","authenticated-orcid":false,"given":"Sk Aftab","family":"Hosen","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, Jadavpur University, Kolkata 700032, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6068-3013","authenticated-orcid":false,"given":"Meltem","family":"Turan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ege University, Izmir 35180, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6826-6185","authenticated-orcid":false,"given":"Denis","family":"Constales","sequence":"additional","affiliation":[{"name":"Department of Electronics and Information Systems, Ghent University, Building S-8, Krijgslaan 281, B-9000 Ghent, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8970-1943","authenticated-orcid":false,"given":"Gregory","family":"Yablonsky","sequence":"additional","affiliation":[{"name":"McKelvey School of Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in Saint Louis, St. Louis, MO 63130, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.ces.2018.11.010","article-title":"Conservatively Perturbed Equilibrium (CPE) in Chemical Kinetics","volume":"196","author":"Yablonsky","year":"2019","journal-title":"Chem. Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Xi, Y., Liu, X., Constales, D., and Yablonsky, G. (2020). Perturbed and Unperturbed: Analyzing the Conservatively Perturbed Equilibrium (Linear Case). Entropy, 22.","DOI":"10.3390\/e22101160"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"116008","DOI":"10.1016\/j.ces.2020.116008","article-title":"Experimental Verification of Conservatively Perturbed Equilibrium for a Complex Non-Linear Chemical Reaction","volume":"229","author":"Peng","year":"2021","journal-title":"Chem. Eng. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1515\/jnet-2022-0054","article-title":"Conservatively Perturbed Equilibrium in Multi-Route Catalytic Reactions","volume":"48","author":"Trishch","year":"2023","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1515\/jnet-2021-0036","article-title":"Over-Equilibrium as a Result of Conservatively-Perturbed Equilibrium (Acyclic and Cyclic Mechanisms)","volume":"47","author":"Trishch","year":"2022","journal-title":"J. Non-Equilib. Thermodyn."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed Machine Learning","volume":"3","author":"Karniadakis","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"109652","DOI":"10.1016\/j.cep.2023.109652","article-title":"Physics-Informed Neural Network to Predict Kinetics of Biodiesel Production in Microwave Reactors","volume":"196","author":"Bibeau","year":"2024","journal-title":"Chem. Eng. Process. Process Intensif."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"8098","DOI":"10.1021\/acs.jpca.1c05102","article-title":"Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics","volume":"125","author":"Ji","year":"2021","journal-title":"J. Phys. Chem. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"8534","DOI":"10.1021\/acs.jpca.2c06513","article-title":"Multiscale Physics-Informed Neural Networks for Stiff Chemical Kinetics","volume":"126","author":"Weng","year":"2022","journal-title":"J. Phys. Chem. A"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"De Florio, M., Schiassi, E., and Furfaro, R. (2022). Physics-informed neural networks and functional interpolation for stiff chemical kinetics. Chaos: Interdiscip. J. Nonlinear Sci., 32.","DOI":"10.1063\/5.0086649"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"110768","DOI":"10.1016\/j.jcp.2021.110768","article-title":"When and Why PINNs Fail to Train: A Neural Tangent Kernel Perspective","volume":"449","author":"Wang","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"18178","DOI":"10.1021\/acs.iecr.3c02383","article-title":"The application of physics-informed machine learning in multiphysics modeling in chemical engineering","volume":"62","author":"Wu","year":"2023","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108195","DOI":"10.1016\/j.compchemeng.2023.108195","article-title":"Physics-informed recurrent neural networks and hyper-parameter optimization for dynamic process systems","volume":"173","author":"Asrav","year":"2023","journal-title":"Comput. Chem. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"164977","DOI":"10.1016\/j.cej.2025.164977","article-title":"A novel conceptual framework for droplet\/particle size distribution in suspension polymerization using Physics-Informed Neural Network (PINN)","volume":"519","author":"Turan","year":"2025","journal-title":"Chem. Eng. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s00366-024-02038-3","article-title":"Multiple scattering simulation via physics-informed neural networks","volume":"41","author":"Nair","year":"2025","journal-title":"Eng. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"114823","DOI":"10.1016\/j.cma.2022.114823","article-title":"Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems","volume":"393","author":"Yu","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"106998","DOI":"10.1016\/j.neunet.2024.106998","article-title":"An extrapolation-driven network architecture for physics-informed deep learning","volume":"183","author":"Wang","year":"2025","journal-title":"Neural Netw."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1038\/s41524-022-00712-y","article-title":"Physics-informed deep learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium","volume":"8","author":"Li","year":"2022","journal-title":"NPJ Comput. Mater."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"103005","DOI":"10.1016\/j.jprocont.2023.103005","article-title":"Physics-informed recurrent neural network modeling for predictive control of nonlinear processes","volume":"128","author":"Zheng","year":"2023","journal-title":"J. Process Control"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ngo, S.I., and Lim, Y.I. (2021). Solution and parameter identification of a fixed-bed reactor model for catalytic CO2 methanation using physics-informed neural networks. Catalysts, 11.","DOI":"10.3390\/catal11111304"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"108801","DOI":"10.1016\/j.compchemeng.2024.108801","article-title":"Physics-informed and data-driven modeling of an industrial wastewater treatment plant with actual validation","volume":"189","author":"Koksal","year":"2024","journal-title":"Comput. Chem. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"161284","DOI":"10.1016\/j.cej.2025.161284","article-title":"Conservatively perturbed equilibrium and perturbation: Linear case","volume":"510","author":"Sachs","year":"2025","journal-title":"Chem. Eng. J."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,26]],"date-time":"2025-12-26T03:11:41Z","timestamp":1766718701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,20]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010009"],"URL":"https:\/\/doi.org\/10.3390\/e28010009","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,20]]}}}