{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T18:11:09Z","timestamp":1790014269696,"version":"4.0.1"},"reference-count":43,"publisher":"Public Library of Science (PLoS)","issue":"9","license":[{"start":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T00:00:00Z","timestamp":1789948800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62020106003"],"award-info":[{"award-number":["62020106003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>\n                    The solution of partial differential equations (PDEs) remains a significant problem in scientific computing. Although physics-informed neural networks (PINNs) provide a mesh-free paradigm, pointwise constraints offer limited supervision between collocation points and uniform training can under-resolve regions with greater learning difficulty. To address these coupled limitations, this paper introduces\n                    <jats:bold>Co<\/jats:bold>\n                    gnitive\n                    <jats:bold>R<\/jats:bold>\n                    egion\n                    <jats:bold>O<\/jats:bold>\n                    ptimized\n                    <jats:bold>P<\/jats:bold>\n                    hysics-\n                    <jats:bold>I<\/jats:bold>\n                    nformed\n                    <jats:bold>N<\/jats:bold>\n                    eural\n                    <jats:bold>N<\/jats:bold>\n                    etworks (CoRoPINN). Rather than treating region optimization and cognitive learning as independent additions, CoRoPINN forms a closed feedback loop. Gradient-stability statistics calibrate a globally stable baseline radius, PDE residuals estimate pointwise learning difficulty, and the same difficulty signal jointly adjusts sample-specific neighborhoods and cognitive weights. This coupling expands continuous physical constraints around under-resolved collocation points while progressively reallocating optimization from easier to more difficult regions. We evaluate CoRoPINN on three one-dimensional PDEs with five backbone architectures, Poisson equations in two, three, and five dimensions, and Burgers2D. Across the five backbones, CoRoPINN achieves the lowest geometric-mean relative mean absolute error (rMAE) on all three one-dimensional benchmarks, reducing the aggregate error relative to the second-best method by 21.0%, 61.6%, and 52.9% on 1D-Reaction, 1D-Wave, and Convection, respectively. It also ranks first in both metrics for 2D and 5D Poisson, remains second in 3D, and attains the lowest rMAE and relative root mean square error (rMSE) among all four methods on Burgers2D. These results establish a clear overall accuracy advantage for CoRoPINN among the evaluated baselines.\n                  <\/jats:p>","DOI":"10.1371\/journal.pone.0358646","type":"journal-article","created":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T17:29:28Z","timestamp":1790011768000},"page":"e0358646","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["CoRoPINN: Cognitive region optimized physics-informed neural networks"],"prefix":"10.1371","volume":"21","author":[{"given":"Jingcong","family":"Li","sequence":"first","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Hubei Minzu University","place":["Enshi, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4678-6219","authenticated-orcid":true,"given":"Yu","family":"Liao","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Hubei Minzu University","place":["Enshi, China"]},{"name":"College of Electronics and Information Engineering, Sichuan University","place":["Chengdu, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ximeng","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Hubei Minzu University","place":["Enshi, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Deng","sequence":"additional","affiliation":[{"name":"College of Intelligent Systems Science and Engineering, Hubei Minzu University","place":["Enshi, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"340","published-online":{"date-parts":[[2026,9,21]]},"reference":[{"key":"pone.0358646.ref001","volume-title":"Partial differential equations","author":"LC Evans","year":"2022"},{"key":"pone.0358646.ref002","volume-title":"Nonlinear partial differential equations with applications","author":"T Roub\u00ed\u010dek","year":"2005"},{"key":"pone.0358646.ref003","volume-title":"Partial differential equations and the finite element method","author":"P Solin","year":"2006"},{"issue":"1","key":"pone.0358646.ref004","first-page":"1","article-title":"Finite element method: an overview","volume":"10","author":"V Jagota","year":"2013","journal-title":"Walailak J Sci Technol"},{"key":"pone.0358646.ref005","volume-title":"Numerical partial differential equations: finite difference methods","author":"JW Thomas","year":"2013"},{"key":"pone.0358646.ref006","doi-asserted-by":"crossref","unstructured":"Barth T, Ohlberger M. Finite volume methods: foundation and analysis. In: Encyclopedia of computational mechanics. vol 1. John Wiley & Sons; 2004. p. 1\u201360.","DOI":"10.1002\/9781119176817.ecm2010"},{"key":"pone.0358646.ref007","unstructured":"Hao Z, Liu S, Zhang Y, Ying C, Feng Y, Su H, et al. Physics-informed machine learning: A survey on problems, methods and applications. arXiv preprint. 2022;arXiv:2211.08064."},{"issue":"6","key":"pone.0358646.ref008","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","author":"GE Karniadakis","year":"2021","journal-title":"Nat Rev Phys"},{"key":"pone.0358646.ref009","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":"M Raissi","year":"2019","journal-title":"J Computat Phys"},{"issue":"3","key":"pone.0358646.ref010","doi-asserted-by":"crossref","DOI":"10.1007\/s10915-022-01939-z","article-title":"Scientific Machine Learning Through Physics\u2013Informed Neural Networks: Where we are and What\u2019s Next","volume":"92","author":"S Cuomo","year":"2022","journal-title":"J Sci Comput"},{"key":"pone.0358646.ref011","doi-asserted-by":"crossref","first-page":"23761","DOI":"10.52202\/075280-1032","article-title":"Separable physics-informed neural networks","volume":"36","author":"J Cho","year":"2023","journal-title":"Adv Neural Inform Process Syst"},{"key":"pone.0358646.ref012","doi-asserted-by":"crossref","first-page":"106750","DOI":"10.1016\/j.neunet.2024.106750","article-title":"Deep fuzzy physics-informed neural networks for forward and inverse PDE problems","volume":"181","author":"W Wu","year":"2025","journal-title":"Neural Netw"},{"key":"pone.0358646.ref013","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":"J Yu","year":"2022","journal-title":"Comput Methods Appl Mech Eng"},{"key":"pone.0358646.ref014","unstructured":"Yao J, Su C, Hao Z, Liu S, Su H, Zhu J. MultiAdam: Parameter-wise scale-invariant optimizer for multiscale training of physics-informed neural networks. In: International conference on machine learning. PMLR; 2023. p. 39702\u201321."},{"key":"pone.0358646.ref015","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":"S Wang","year":"2022","journal-title":"J Computat Phys"},{"key":"pone.0358646.ref016","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.neucom.2022.05.015","article-title":"Self-adaptive loss balanced Physics-informed neural networks","volume":"496","author":"Z Xiang","year":"2022","journal-title":"Neurocomputing"},{"key":"pone.0358646.ref017","unstructured":"Kharazmi E, Zhang Z, Karniadakis GE. Variational physics-informed neural networks for solving partial differential equations. arXiv preprint. 2019;arXiv:1912.00873."},{"key":"pone.0358646.ref018","doi-asserted-by":"crossref","first-page":"115671","DOI":"10.1016\/j.cma.2022.115671","article-title":"A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks","volume":"403","author":"C Wu","year":"2023","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"pone.0358646.ref019","doi-asserted-by":"crossref","DOI":"10.1137\/18M1225409","article-title":"Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations","volume":"42","author":"L Yang","year":"2020","journal-title":"SIAM J Sci Comput"},{"key":"pone.0358646.ref020","doi-asserted-by":"crossref","first-page":"110494","DOI":"10.52202\/079017-3508","article-title":"RoPINN: Region optimized physics-informed neural networks","volume":"37","author":"H Wu","year":"2024","journal-title":"Adv Neural Inform Process Syst"},{"key":"pone.0358646.ref021","first-page":"14759","volume-title":"CoPINN: Cognitive Physics-Informed Neural Networks","author":"S Duan","year":"2025"},{"key":"pone.0358646.ref022","doi-asserted-by":"crossref","unstructured":"Jiang L, Meng D, Zhao Q, Shan S, Hauptmann A. Self-paced curriculum learning. In: Proceedings of the AAAI conference on artificial intelligence. vol. 29. 2015. p. 2694\u2013700.","DOI":"10.1609\/aaai.v29i1.9608"},{"key":"pone.0358646.ref023","volume-title":"Simulation and the Monte Carlo method","author":"RY Rubinstein","year":"2008"},{"key":"pone.0358646.ref024","first-page":"271","volume-title":"A review of trust region algorithms for optimization","author":"YX Yuan","year":"2000"},{"issue":"6","key":"pone.0358646.ref025","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1007\/s11263-022-01611-x","article-title":"Curriculum Learning: A Survey","volume":"130","author":"P Soviany","year":"2022","journal-title":"Int J Comput Vis"},{"issue":"9","key":"pone.0358646.ref026","first-page":"4555","article-title":"A Survey on Curriculum Learning","volume":"44","author":"X Wang","year":"2021","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"pone.0358646.ref027","unstructured":"Hardt M, Recht B, Singer Y. Train faster, generalize better: Stability of stochastic gradient descent. In: International conference on machine learning. PMLR; 2016. p. 1225\u201334."},{"key":"pone.0358646.ref028","doi-asserted-by":"crossref","first-page":"15446","DOI":"10.52202\/068431-1124","article-title":"Stability analysis and generalization bounds of adversarial training","volume":"35","author":"J Xiao","year":"2022","journal-title":"Adv Neural Inform Process Syst"},{"key":"pone.0358646.ref029","first-page":"26548","article-title":"Characterizing possible failure modes in physics-informed neural networks","volume":"34","author":"A Krishnapriyan","year":"2021","journal-title":"Adv Neural Inform Process Syst"},{"key":"pone.0358646.ref030","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/BF01589116","article-title":"On the limited memory BFGS method for large scale optimization","volume":"45","author":"DC Liu","year":"1989","journal-title":"Math Programm"},{"key":"pone.0358646.ref031","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","article-title":"DGM: A deep learning algorithm for solving partial differential equations","volume":"375","author":"J Sirignano","year":"2018","journal-title":"J Comput Phys"},{"issue":"4","key":"pone.0358646.ref032","doi-asserted-by":"crossref","first-page":"2341","DOI":"10.1137\/120880811","article-title":"Stochastic First- and Zeroth-Order Methods for Nonconvex Stochastic Programming","volume":"23","author":"S Ghadimi","year":"2013","journal-title":"SIAM J Optim"},{"key":"pone.0358646.ref033","unstructured":"Rathore P, Lei W, Frangella Z, Lu L, Udell M. Challenges in training PINNs: A loss landscape perspective. arXiv preprint. 2024;arXiv:2402.01868."},{"key":"pone.0358646.ref034","unstructured":"Mojgani R, Balajewicz M, Hassanzadeh P. Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks. arXiv preprint. arXiv:2205.02902. 2022."},{"key":"pone.0358646.ref035","doi-asserted-by":"crossref","first-page":"76721","DOI":"10.52202\/079017-2442","article-title":"PINNacle: A comprehensive benchmark of physics-informed neural networks for solving PDEs","volume":"37","author":"Z Hao","year":"2024","journal-title":"Adv Neural Inform Process Syst"},{"key":"pone.0358646.ref036","doi-asserted-by":"crossref","unstructured":"Bu J, Karpatne A. Quadratic residual networks: A new class of neural networks for solving forward and inverse problems in physics involving PDEs. In: Proceedings of the 2021 SIAM International Conference on Data Mining (SDM). SIAM; 2021. p. 675\u201383.","DOI":"10.1137\/1.9781611976700.76"},{"issue":"3","key":"pone.0358646.ref037","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1109\/TAI.2022.3192362","article-title":"Learning in Sinusoidal Spaces With Physics-Informed Neural Networks","volume":"5","author":"JC Wong","year":"2022","journal-title":"IEEE Trans Artif Intell"},{"key":"pone.0358646.ref038","unstructured":"Zhao Z, Ding X, Prakash BA. PINNsFormer: A transformer-based framework for physics-informed neural networks. arXiv preprint. arXiv:2307.11833. 2023."},{"key":"pone.0358646.ref039","unstructured":"Liu Z, Wang Y, Vaidya S, Ruehle F, Halverson J, Soljacic M, et al. KAN: Kolmogorov\u2013Arnold networks. In: International conference on learning representations. vol. 2025. 2025. p. 70367\u2013413."},{"key":"pone.0358646.ref040","article-title":"PyTorch: An imperative style, high-performance deep learning library","volume":"32","author":"A Paszke","year":"2019","journal-title":"Adv Neural Inform Process Syst"},{"issue":"3","key":"pone.0358646.ref041","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.5194\/gmd-7-1247-2014","article-title":"Root mean square error (RMSE) or mean absolute error (MAE)? \u2013 Arguments against avoiding RMSE in the literature","volume":"7","author":"T Chai","year":"2014","journal-title":"Geosci Model Dev"},{"key":"pone.0358646.ref042","first-page":"1","article-title":"Root-mean-square error (RMSE) or mean absolute error (MAE): when to use them or not","volume":"2022","author":"TO Hodson","year":"2022","journal-title":"Geosci Model Dev"},{"key":"pone.0358646.ref043","doi-asserted-by":"crossref","unstructured":"Wang S, Sankaran S, Wang H, Perdikaris P. An expert\u2019s guide to training physics-informed neural networks. arXiv preprint. arXiv:2308.08468. 2023.","DOI":"10.1016\/j.cma.2024.116813"}],"container-title":["PLOS One"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0358646","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T17:29:38Z","timestamp":1790011778000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pone.0358646"}},"subtitle":[],"editor":[{"given":"Vikash Kumar","family":"Sinha","sequence":"first","affiliation":[{"name":"Srinath University","place":["INDIA"]}],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,9,21]]},"references-count":43,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9,21]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pone.0358646","relation":{},"ISSN":["1932-6203"],"issn-type":[{"value":"1932-6203","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,21]]}}}