{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,2]],"date-time":"2026-11-02T08:28:23Z","timestamp":1793608103372,"version":"4.5.0"},"reference-count":41,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2027,2,1]],"date-time":"2027-02-01T00:00:00Z","timestamp":1801440000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42130409"],"award-info":[{"award-number":["42130409"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U24A20606"],"award-info":[{"award-number":["U24A20606"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42476193"],"award-info":[{"award-number":["42476193"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42450178"],"award-info":[{"award-number":["42450178"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Neural Networks"],"published-print":{"date-parts":[[2027,2]]},"DOI":"10.1016\/j.neunet.2026.109698","type":"journal-article","created":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T15:43:38Z","timestamp":1790869418000},"page":"109698","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"PA","title":["Physics-informed hybrid neural networks with data constraints: A case study on lorenz63"],"prefix":"10.1016","volume":"206","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5136-6772","authenticated-orcid":false,"given":"Yao","family":"Xiao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Youmin","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3333-2231","authenticated-orcid":false,"given":"Yi","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3457-0832","authenticated-orcid":false,"given":"Zheqi","family":"Shen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.neunet.2026.109698_bib0001","series-title":"IGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium","first-page":"3987","article-title":"Towards physically-consistent, data-driven models of convection","author":"Beucler","year":"2020"},{"key":"10.1016\/j.neunet.2026.109698_bib0002","doi-asserted-by":"crossref","unstructured":"Beucler, T., Pritchard, M., Rasp, S., Ott, J., Baldi, P., & Gentine, P. (2021, January 27). Enforcing analytic constraints in neural-networks emulating physical systems. https:\/\/doi.org\/10.1103\/PhysRevLett.126.098302.","DOI":"10.1103\/PhysRevLett.126.098302"},{"issue":"23","key":"10.1016\/j.neunet.2026.109698_bib0003","doi-asserted-by":"crossref","first-page":"3409","DOI":"10.1175\/1520-0469(1998)055<3409:AEOTLA>2.0.CO;2","article-title":"An extension of the Lyapunov analysis for the predictability problem","volume":"55","author":"Boffetta","year":"1998","journal-title":"J. Atmos. Sci."},{"issue":"12","key":"10.1016\/j.neunet.2026.109698_bib0004","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1007\/s10409-021-01148-1","article-title":"Physics-informed neural networks (PINNs) for fluid mechanics: a review","volume":"37","author":"Cai","year":"2021","journal-title":"Acta Mechanica Sinica"},{"issue":"3","key":"10.1016\/j.neunet.2026.109698_bib0005","doi-asserted-by":"crossref","first-page":"373","DOI":"10.5194\/npg-27-373-2020","article-title":"Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: Reservoir computing, artificial neural network, and long short-term memory network","volume":"27","author":"Chattopadhyay","year":"2020","journal-title":"Nonlinear Processes in Geophysics"},{"key":"10.1016\/j.neunet.2026.109698_bib0006","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2022.114909","article-title":"CAN-PINN: A fast physics-informed neural network based on coupled-automatic\u2013numerical differentiation method","volume":"395","author":"Chiu","year":"2022","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"3","key":"10.1016\/j.neunet.2026.109698_bib0007","doi-asserted-by":"crossref","first-page":"88","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":"Cuomo","year":"2022","journal-title":"Journal of Scientific Computing"},{"issue":"6","key":"10.1016\/j.neunet.2026.109698_bib0008","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3530911","article-title":"CoPhy -PGNN: Learning physics-guided neural networks with competing loss functions for solving eigenvalue problems","volume":"13","author":"Elhamod","year":"2022","journal-title":"ACM Transactions on Intelligent Systems and Technology"},{"issue":"3","key":"10.1016\/j.neunet.2026.109698_bib0009","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1007\/s00220-012-1438-7","article-title":"On the recurrence and robust properties of lorenz\u201963 model","volume":"313","author":"Gianfelice","year":"2012","journal-title":"Communications in Mathematical Physics"},{"key":"10.1016\/j.neunet.2026.109698_bib0010","unstructured":"Hafeez, H. Y., Ndikilar, C. E., & Isyaku, S. (2015). Analytical study of the van der pol equation in the autonomous regime. #i{Progress in Physics}."},{"issue":"7775","key":"10.1016\/j.neunet.2026.109698_bib0011","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1038\/s41586-019-1559-7","article-title":"Deep learning for multi-year ENSO forecasts","volume":"573","author":"Ham","year":"2019","journal-title":"Nature"},{"issue":"8","key":"10.1016\/j.neunet.2026.109698_bib0012","doi-asserted-by":"crossref","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","article-title":"Long short-term memory","volume":"9","author":"Hochreiter","year":"1997","journal-title":"Neural Computation"},{"issue":"1","key":"10.1016\/j.neunet.2026.109698_bib0013","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0096-3003(97)10060-1","article-title":"An efficient numerical scheme for burgers' equation","volume":"95","author":"Hon","year":"1998","journal-title":"Applied Mathematics & Computation"},{"key":"10.1016\/j.neunet.2026.109698_bib0014","doi-asserted-by":"crossref","unstructured":"Hou, Y., Chen, J., Ma, D., & Chan, P. W. (2025, February 4). Steady-state linear response matrix of the lorenz-63 system. https:\/\/doi.org\/10.22541\/essoar.173869623.33180280\/v1.","DOI":"10.22541\/essoar.173869623.33180280\/v1"},{"issue":"6","key":"10.1016\/j.neunet.2026.109698_bib0015","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":"Nature Reviews Physics"},{"key":"10.1016\/j.neunet.2026.109698_bib0016","doi-asserted-by":"crossref","DOI":"10.1016\/j.neunet.2024.106152","article-title":"Data-driven learning of chaotic dynamical systems using discrete-temporal sobolev networks","volume":"173","author":"Kennedy","year":"2024","journal-title":"Neural Networks"},{"key":"10.1016\/j.neunet.2026.109698_bib0017","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113547","article-title":"hp-VPINNs: Variational physics-informed neural networks with domain decomposition","volume":"374","author":"Kharazmi","year":"2021","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"10.1016\/j.neunet.2026.109698_bib0018","article-title":"DPM: A novel training method for physics-informed neural networks in extrapolation","author":"Kim","year":"2020","journal-title":"arXiv"},{"key":"10.1016\/j.neunet.2026.109698_bib0019","unstructured":"Kingma, D. P., & Ba, J. (2017, January 30). Adam: A method for stochastic optimization. Computer Science. https:\/\/doi.org\/10.48550\/arXiv.1412.6980."},{"key":"10.1016\/j.neunet.2026.109698_bib0020","article-title":"Characterizing possible failure modes in physics-informed neural networks","author":"Krishnapriyan","year":"2021","journal-title":"arXiv"},{"issue":"7","key":"10.1016\/j.neunet.2026.109698_bib0021","article-title":"Comparative study of Lorenz model based ensemble forecasting and single forecasting","volume":"67","author":"Liang","year":"2018","journal-title":"Acta Physica Sinica"},{"key":"10.1016\/j.neunet.2026.109698_bib0022","first-page":"1","article-title":"Forecasting the indian ocean dipole with deep learning techniques","author":"Liu","year":"2021","journal-title":"Geophysical Research Letters"},{"issue":"2","key":"10.1016\/j.neunet.2026.109698_bib0024","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"Journal of the Atmospheric Sciences"},{"key":"10.1016\/j.neunet.2026.109698_bib0025","article-title":"Physics-informed neural networks with hard constraints for inverse design","author":"Lu","year":"2021","journal-title":"arXiv"},{"key":"10.1016\/j.neunet.2026.109698_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111722","article-title":"Self-Adaptive physics-informed neural networks using a soft attention mechanism","volume":"474","author":"McClenny","year":"2023","journal-title":"Journal of Computational Physics"},{"key":"10.1016\/j.neunet.2026.109698_bib0027","article-title":"Estimates on the generalization error of physics informed neural networks (PINNs) for approximating a class of inverse problems for PDEs","author":"Mishra","year":"2023","journal-title":"arXiv"},{"key":"10.1016\/j.neunet.2026.109698_bib0028","article-title":"Embedding hard physical constraints in neural network coarse-graining of 3D turbulence","author":"Mohan","year":"2020","journal-title":"arXiv"},{"issue":"2","key":"10.1016\/j.neunet.2026.109698_bib0029","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/S0167-2789(02)00534-1","article-title":"Modelling the dynamics of nonlinear time series using canonical variate analysis","volume":"170","author":"Pilgram","year":"2002","journal-title":"Physica D: Nonlinear Phenomena"},{"key":"10.1016\/j.neunet.2026.109698_bib0030","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":"Journal of Computational Physics"},{"key":"10.1016\/j.neunet.2026.109698_bib0031","unstructured":"Ren, X., Zhou, P., Meng, X., Huang, X., Wang, Y., Wang, W., et al. (2023, March 19). PanGu-{\\Sigma}: Towards trillion parameter language model with sparse heterogeneous computing. arXiv. Retrieved from http:\/\/arxiv.org\/abs\/2303.10845."},{"issue":"1","key":"10.1016\/j.neunet.2026.109698_bib0033","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1038\/s41598-023-49977-3","article-title":"Solving real-world optimization tasks using physics-informed neural computing","volume":"14","author":"Seo","year":"2024","journal-title":"Scientific Reports"},{"issue":"5","key":"10.1016\/j.neunet.2026.109698_bib0034","doi-asserted-by":"crossref","first-page":"2042","DOI":"10.4208\/cicp.OA-2020-0193","article-title":"On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs","volume":"28","author":"Shin","year":"2020","journal-title":"Communications in Computational Physics"},{"key":"10.1016\/j.neunet.2026.109698_bib0035","article-title":"Enhanced physics-informed neural networks with augmented lagrangian relaxation method (AL-PINNs)","author":"Son","year":"2023","journal-title":"arXiv"},{"key":"10.1016\/j.neunet.2026.109698_bib0036","series-title":"Proceedings of Cyber-Physical Systems and Internet of Things Week 2023","first-page":"170","author":"Tan","year":"2023"},{"key":"10.1016\/j.neunet.2026.109698_bib0037","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2021.113887","article-title":"Optimally weighted loss functions for solving PDEs with neural networks","volume":"405","author":"Van Der Meer","year":"2022","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"10.1016\/j.neunet.2026.109698_bib0038","unstructured":"Wang, S., Yu, X., & Perdikaris, P. (2020, July 28). When and why PINNs fail to train: A neural tangent kernel perspective. arXiv. https:\/\/doi.org\/10.48550\/arXiv.2007.14527."},{"key":"10.1016\/j.neunet.2026.109698_bib0039","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2024.113112","article-title":"A practical PINN framework for multi-scale problems with multi-magnitude loss terms","volume":"510","author":"Wang","year":"2024","journal-title":"Journal of Computational Physics"},{"issue":"3","key":"10.1016\/j.neunet.2026.109698_bib0040","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Physica D: Nonlinear Phenomena"},{"key":"10.1016\/j.neunet.2026.109698_bib0041","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":"Xiang","year":"2022","journal-title":"Neurocomputing"},{"key":"10.1016\/j.neunet.2026.109698_bib0042","article-title":"Analysis of attractor behavior and predictability in a coupled lorenz model","author":"ZHANG","year":"2023","journal-title":"Chinese Journal of Atmospheric Sciences"},{"issue":"9","key":"10.1016\/j.neunet.2026.109698_bib0043","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1175\/JTECH-D-20-0001.1","article-title":"An adaptive optimal interpolation based on analog forecasting: Application to SSH in the gulf of mexico","volume":"37","author":"Zhen","year":"2020","journal-title":"Journal of Atmospheric and Oceanic Technology"}],"container-title":["Neural Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893608026011536?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893608026011536?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,11,2]],"date-time":"2026-11-02T07:55:45Z","timestamp":1793606145000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0893608026011536"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2027,2]]},"references-count":41,"alternative-id":["S0893608026011536"],"URL":"https:\/\/doi.org\/10.1016\/j.neunet.2026.109698","relation":{},"ISSN":["0893-6080"],"issn-type":[{"value":"0893-6080","type":"print"}],"subject":[],"published":{"date-parts":[[2027,2]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Physics-informed hybrid neural networks with data constraints: A case study on lorenz63","name":"articletitle","label":"Article Title"},{"value":"Neural Networks","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.neunet.2026.109698","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"109698"}}