{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T07:52:52Z","timestamp":1771487572994,"version":"3.50.1"},"reference-count":51,"publisher":"AIP Publishing","issue":"7","funder":[{"DOI":"10.13039\/100020518","name":"Centre for Mechanical and Aerospace Science and Technologies, University of Beira Interior","doi-asserted-by":"publisher","award":["UIDB\/00151\/2020"],"award-info":[{"award-number":["UIDB\/00151\/2020"]}],"id":[{"id":"10.13039\/100020518","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100020518","name":"Centre for Mechanical and Aerospace Science and Technologies, University of Beira Interior","doi-asserted-by":"publisher","award":["UIDP\/00151\/2020"],"award-info":[{"award-number":["UIDP\/00151\/2020"]}],"id":[{"id":"10.13039\/100020518","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Project GreenAuto: Green Innovation for the Automotive Industry","award":["C644867037-00000013"],"award-info":[{"award-number":["C644867037-00000013"]}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2025,7,1]]},"abstract":"<jats:p>Turbulence remains a complex aspect of fluid mechanics, where accurately reconstructing high-resolution turbulent flow fields is particularly challenging due to frequently sparse and incomplete experimental data. This study leverages Physics-Informed Neural Networks (PINNs) to reconstruct turbulent flow fields from non-uniform, discrete supervisory data, improving adaptability across varying flow conditions and reducing reliance on direct Reynolds stress measurements\u2014often unavailable in practice. Our novel approach enables indirect learning of Reynolds stress tensors from collected velocity and pressure data, streamlining the computational process bypassing complex turbulence model equations. Specifically, the framework excels in navigating complex dynamics within critical engineering structures, such as T-junctions and sudden expansions, typical in water distribution networks. Extensive testing across data scenarios\u2014with sparsity levels of 100%, 10%, 5%, and 2%\u2014demonstrates the model's robustness, maintaining high accuracy in flow prediction and turbulence characterization. The model notably attained L2 errors of 0.0413 and 0.0434 for pressure and velocity reconstruction in critical areas of the 5% dataset. For a sparsity of 2%, the L2 errors were 0.0837 and 0.0769. These results highlight the potential of our PINN framework to complement traditional turbulence modeling approaches by enabling real-time flow monitoring and predictive analytics within digital twin systems, particularly in scenarios where reduced computational cost is essential and full-resolution Reynolds-Averaged Navier\u2013Stokes or Direct Numerical Simulations are impractical.<\/jats:p>","DOI":"10.1063\/5.0269730","type":"journal-article","created":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T15:51:57Z","timestamp":1757346717000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["A sparse and decoupled supervision framework for multi-field reconstruction in turbulent flows using physics-informed neural networks"],"prefix":"10.1063","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9326-865X","authenticated-orcid":false,"given":"David","family":"Gomes","sequence":"first","affiliation":[{"name":"Department of Electromechanical Engineering, University of Beira Interior 1 , Rua Marqu\u00eas de D'\u00c1vila e Bolama, 6201-001 Covilh\u00e3,","place":["Portugal"]},{"name":"C-MAST-Center for Mechanical and Aerospace Science and Technologies 2 , Rua Marqu\u00eas de D'\u00c1vila e Bolama, 6201-001 Covilh\u00e3,","place":["Portugal"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9540-8900","authenticated-orcid":false,"given":"Ant\u00f3nio","family":"Esp\u00edrito Santo","sequence":"additional","affiliation":[{"name":"Department of Electromechanical Engineering, University of Beira Interior 1 , Rua Marqu\u00eas de D'\u00c1vila e Bolama, 6201-001 Covilh\u00e3,","place":["Portugal"]},{"name":"IT-Institute of Telecommunications 3 , Covilh\u00e3,","place":["Portugal"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7019-3766","authenticated-orcid":false,"given":"Jos\u00e9 C.","family":"P\u00e1scoa","sequence":"additional","affiliation":[{"name":"Department of Electromechanical Engineering, University of Beira Interior 1 , Rua Marqu\u00eas de D'\u00c1vila e Bolama, 6201-001 Covilh\u00e3,","place":["Portugal"]},{"name":"C-MAST-Center for Mechanical and Aerospace Science and Technologies 2 , Rua Marqu\u00eas de D'\u00c1vila e Bolama, 6201-001 Covilh\u00e3,","place":["Portugal"]}]}],"member":"317","published-online":{"date-parts":[[2025,7,3]]},"reference":[{"key":"2025090811513785200_c1","doi-asserted-by":"publisher","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","year":"2019","journal-title":"J. Comput. Phys."},{"key":"2025090811513785200_c2","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"2025090811513785200_c3","doi-asserted-by":"publisher","first-page":"073603","DOI":"10.1063\/5.0054312","article-title":"Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels","volume":"33","year":"2021","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c4","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/j.taml.2020.01.039","article-title":"Physics-informed deep learning for incompressible laminar flows","volume":"10","year":"2020","journal-title":"Theor. Appl. Mech. Lett."},{"key":"2025090811513785200_c5","doi-asserted-by":"publisher","first-page":"113119","DOI":"10.1063\/5.0232534","article-title":"A deep learning framework for solving the prediction and reconstruction problem of bingham fluid flow field","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c6","doi-asserted-by":"publisher","first-page":"85187","DOI":"10.1063\/5.0218611","article-title":"Reconstruction of the turbulent flow field with sparse measurements using physics-informed neural network","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c7","doi-asserted-by":"publisher","first-page":"35107","DOI":"10.1063\/5.0190138","article-title":"Data-driven discovery of turbulent flow equations using physics-informed neural networks","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c8","doi-asserted-by":"publisher","first-page":"109232","DOI":"10.1016\/j.ijheatfluidflow.2023.109232","article-title":"Studying turbulent flows with physics-informed neural networks and sparse data","volume":"104","year":"2023","journal-title":"Int. J. Heat Fluid Flow"},{"key":"2025090811513785200_c9","doi-asserted-by":"publisher","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","year":"2022","journal-title":"J. Comput. Phys."},{"key":"2025090811513785200_c10","doi-asserted-by":"publisher","first-page":"105724","DOI":"10.1016\/j.engappai.2022.105724","article-title":"Reconstruction of hydrofoil cavitation flow based on the chain-style physics-informed neural network","volume":"119","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"2025090811513785200_c11","doi-asserted-by":"publisher","first-page":"2238849","DOI":"10.1080\/19942060.2023.2238849","article-title":"Reconstruction of 3D flow field around a building model in wind tunnel: A novel physics-informed neural network framework adopting dynamic prioritization self-adaptive loss balance strategy","volume":"17","year":"2023","journal-title":"Eng. Appl. Comput. Fluid Mech."},{"key":"2025090811513785200_c12","doi-asserted-by":"publisher","first-page":"091006","DOI":"10.1115\/1.4065165","article-title":"Unsupervised denoising and super-resolution of vascular flow data by physics-informed machine learning","volume":"146","year":"2024","journal-title":"J. Biomech. Eng."},{"key":"2025090811513785200_c13","article-title":"Theoretical framework and use of CNN reconstruction with optimal sparse sensor placement in a flow field","year":"2025"},{"key":"2025090811513785200_c14","article-title":"Artificial neural network softsensor for water distribution networks","year":"2024"},{"key":"2025090811513785200_c15","doi-asserted-by":"publisher","first-page":"7590","DOI":"10.3390\/s22197590","article-title":"An interoperable digital twin with the IEEE 1451 Standards","volume":"22","year":"2022","journal-title":"Sensors"},{"key":"2025090811513785200_c16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/IECON51785.2023.10312113","article-title":"Enhancing IEC 61499 with an IEEE 1451 TIM Function Block","year":"2023"},{"key":"2025090811513785200_c17","doi-asserted-by":"publisher","first-page":"103947","DOI":"10.1016\/j.ijengsci.2023.103947","article-title":"On Modal decomposition as surrogate for charge-conservative EHD modelling of taylor cone jets","volume":"193","year":"2023","journal-title":"Int. J. Eng. Sci."},{"key":"2025090811513785200_c18","doi-asserted-by":"publisher","first-page":"073606","DOI":"10.1063\/5.0211680","article-title":"Flow field reconstruction from sparse sensor measurements with physics-informed neural networks","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/S10409-022-22302-X\/METRICS","article-title":"A practical approach to flow field reconstruction with sparse or incomplete data through physics informed neural network","volume":"39","year":"2023","journal-title":"Acta Mech. Sin."},{"key":"2025090811513785200_c20","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1108\/HFF-05-2023-0239","article-title":"Time-averaged flow field reconstruction based on a multifidelity model using physics-informed neural network (PINN) and nonlinear information fusion","volume":"34","year":"2024","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"2025090811513785200_c21","doi-asserted-by":"publisher","first-page":"123625","DOI":"10.1063\/5.0243548","article-title":"Multiple scale method integrated physics-informed neural networks for reconstructing transient natural convection","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c22","doi-asserted-by":"publisher","first-page":"27124","DOI":"10.1063\/5.0136886","article-title":"Physics-informed neural networks for gravity currents reconstruction from limited data","volume":"35","year":"2023","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c23","doi-asserted-by":"publisher","first-page":"117104","DOI":"10.1063\/5.0231684","article-title":"Flow reconstruction with uncertainty quantification from noisy measurements based on Bayesian physics-informed neural networks","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c24","doi-asserted-by":"publisher","first-page":"085144","DOI":"10.1063\/5.0218499","article-title":"Trade-off between reconstruction accuracy and physical validity in modeling turbomachinery PIV data by physics-informed convolutional neural networks","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c25","doi-asserted-by":"publisher","first-page":"423","DOI":"10.3390\/W13040423","article-title":"Deep learning method based on physics informed neural network with resnet block for solving fluid flow problems","volume":"13","year":"2021","journal-title":"Water"},{"key":"2025090811513785200_c26","doi-asserted-by":"publisher","first-page":"4337","DOI":"10.1007\/s40808-024-02025-z","article-title":"Numerical modeling of turbulent flow interactions with vegetation in channels with fixed beds","volume":"10","year":"2024","journal-title":"Model. Earth Syst. Environ."},{"key":"2025090811513785200_c27","doi-asserted-by":"publisher","first-page":"109645","DOI":"10.1016\/j.geomorph.2025.109645","article-title":"Flow dynamics and bed morphology in a narrow channels: A comparative study of experimental and numerical approaches to velocity distribution","volume":"474","year":"2025","journal-title":"Geomorphology"},{"key":"2025090811513785200_c28","doi-asserted-by":"publisher","first-page":"123616","DOI":"10.1063\/5.0241201","article-title":"Coherence mode and floquet analysis on flow past a rectangular cylinder with small angle of attack","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c29","doi-asserted-by":"publisher","first-page":"119729","DOI":"10.1016\/j.enconman.2025.119729","article-title":"Wake dynamics of a wind turbine under real-time varying inflow turbulence: A coherence mode perspective","volume":"332","year":"2025","journal-title":"Energy Convers. Manage."},{"key":"2025090811513785200_c30","unstructured":"S.\n            Ghosh\n          , A.Chakraborty, G. O.Brikis, and B.Dey, \u201cRANS-PINN based simulation surrogates for predicting turbulent flows,\u201d arXiv:2306.06034 (2023)."},{"key":"2025090811513785200_c31","doi-asserted-by":"publisher","first-page":"105266","DOI":"10.1016\/j.compfluid.2021.105266","article-title":"Machine learning for vortex induced vibration in turbulent flow","volume":"235","year":"2022","journal-title":"Comput. Fluids"},{"key":"2025090811513785200_c32","doi-asserted-by":"publisher","first-page":"034605","DOI":"10.1103\/PhysRevFluids.9.034605","article-title":"Turbulence model augmented physics-informed neural networks for mean-flow reconstruction","volume":"9","year":"2024","journal-title":"Phys. Rev. Fluids"},{"key":"2025090811513785200_c33","doi-asserted-by":"publisher","first-page":"43","DOI":"10.3390\/fluids8020043","article-title":"Turbulence modeling for physics-informed neural networks: Comparison of different RANS models for the backward-facing step flow","volume":"8","year":"2023","journal-title":"Fluids"},{"key":"2025090811513785200_c34","doi-asserted-by":"publisher","first-page":"65141","DOI":"10.1063\/5.0155087\/18021017\/065141_1_5.0155087.PDF","article-title":"Spatiotemporal parallel physics-informed neural networks: A framework to solve inverse problems in fluid mechanics","volume":"35","year":"2023","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c35","doi-asserted-by":"publisher","first-page":"279","DOI":"10.3390\/fluids9120279","article-title":"Optimising physics-informed neural network solvers for turbulence modelling: A study on solver constraints against a data-driven approach","volume":"9","year":"2024","journal-title":"Fluids"},{"key":"2025090811513785200_c36","doi-asserted-by":"publisher","first-page":"055130","DOI":"10.1063\/5.0090050","article-title":"Physics-informed data based neural networks for two-dimensional turbulence","volume":"34","year":"2022","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c37","volume-title":"An Introduction to Computational Fluid Dynamics: The Finite Volume Method","year":"2007","edition":"2"},{"key":"2025090811513785200_c38","article-title":"Von karman institute for fluid dynamics","volume-title":"Computational Fluid Dynamics: An Introduction","year":"2009"},{"key":"2025090811513785200_c39","volume-title":"Fundamentals of Fluid Mechanics","year":"2004"},{"key":"2025090811513785200_c40","volume-title":"Introduction to Fluid Mechanics","year":"2004"},{"key":"2025090811513785200_c41","volume-title":"Turbulent Flows","year":"2000"},{"key":"2025090811513785200_c42","doi-asserted-by":"publisher","first-page":"109951","DOI":"10.1016\/j.jcp.2020.109951","article-title":"NSFnets (Navier-Stokes Flow Nets): Physics-informed neural networks for the incompressible Navier-Stokes equations","volume":"426","year":"2021","journal-title":"J. Comput. Phys."},{"key":"2025090811513785200_c43","doi-asserted-by":"publisher","first-page":"1","DOI":"10.48550\/arXiv.1502.05767","article-title":"Automatic differentiation in machine learning: A survey","volume":"18","year":"2015","journal-title":"J. Mach. Learn. Res."},{"key":"2025090811513785200_c44","doi-asserted-by":"publisher","first-page":"113837","DOI":"10.1016\/j.jcp.2025.113837","article-title":"On the preprocessing of physics-informed neural networks: How to better utilize data in fluid mechanics","volume":"528","year":"2025","journal-title":"J. Comput. Phys."},{"key":"2025090811513785200_c45","doi-asserted-by":"publisher","first-page":"126826","DOI":"10.1016\/j.neucom.2023.126826","article-title":"Hyper-parameter tuning of physics-informed neural networks: Application to helmholtz problems","volume":"561","year":"2023","journal-title":"Neurocomputing"},{"key":"2025090811513785200_c46","article-title":"Adam: A method for stochastic optimization","year":"2014"},{"key":"2025090811513785200_c47","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/BF01589116\/METRICS","article-title":"On the limited memory BFGS method for large scale optimization","volume-title":"Math. Program.","year":"1989"},{"key":"2025090811513785200_c48","doi-asserted-by":"publisher","first-page":"52102","DOI":"10.1063\/5.0205454","article-title":"Data-driven surrogate modelling of multistage Taylor cone-jet dynamics","volume":"36","year":"2024","journal-title":"Phys. Fluids"},{"key":"2025090811513785200_c49","volume-title":"Deep Learning","year":"2016"},{"key":"2025090811513785200_c50","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1137\/19M1274067","article-title":"DeepXDE: A deep learning library for solving differential equations","volume":"63","year":"2021","journal-title":"SIAM Rev."},{"key":"2025090811513785200_c51","unstructured":"D. Gomes , A.Esp\u00edrito Santo, and J.P\u00e1scoa (2025). \u201cMulti-field reconstruction with physics-informed neural networks (MFR_PINN),\u201d GitHub.https:\/\/github.com\/david-e-gomes\/MFR_PINN.git."}],"container-title":["Physics of Fluids"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/pof\/article-pdf\/doi\/10.1063\/5.0269730\/20579402\/075113_1_5.0269730.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/pof\/article-pdf\/doi\/10.1063\/5.0269730\/20579402\/075113_1_5.0269730.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T15:52:00Z","timestamp":1757346720000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/pof\/article\/37\/7\/075113\/3351672\/A-sparse-and-decoupled-supervision-framework-for"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,1]]},"references-count":51,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2025,7,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0269730","relation":{},"ISSN":["1070-6631","1089-7666"],"issn-type":[{"value":"1070-6631","type":"print"},{"value":"1089-7666","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,7]]},"published":{"date-parts":[[2025,7,1]]},"article-number":"075113"}}