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Observer-based neural networks for flow estimation and control

Published online by Cambridge University Press:  27 April 2026

Tarcísio Costa Déda Open the ORCID record for Tarcísio Costa Déda [Opens in a new window] ,
Brener Ramos ,
William Wolf Open the ORCID record for William Wolf [Opens in a new window]  and
Scott T.M. Dawson Open the ORCID record for Scott T.M. Dawson [Opens in a new window]
Tarcísio Costa Déda*
Affiliation:
School of Mechanical Engineering, University of Campinas, R. Mendeleyev 200, Campinas, São Paulo, Brazil
Brener Ramos
Affiliation:
School of Mechanical Engineering, University of Campinas, R. Mendeleyev 200, Campinas, São Paulo, Brazil
William Wolf
Affiliation:
School of Mechanical Engineering, University of Campinas, R. Mendeleyev 200, Campinas, São Paulo, Brazil
Scott T.M. Dawson
Affiliation:
Mechanical, Materials, and Aerospace Engineering Department, Illinois Institute of Technology, 10 W 35th St, Chicago, IL, USA
*
Corresponding author: Tarcísio Costa Déda, tarcisio.deda@gmail.com
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Abstract

Neural network observers (NNOs) are proposed for online estimation of fluid flows, addressing a key challenge in flow control: obtaining flow states online from a limited set of sparse and noisy sensor data. For this task, we propose a generalisation of the classical Luenberger observer. In the present framework, the estimation loop is composed of subsystems modelled as neural networks (NNs). By combining flow information from selected probes and a neural network surrogate model (NNSM) of the flow system, we train NNOs capable of fusing information to provide the best estimation of the states, that can in turn be fed back to a neural network controller (NNC). The NNO capabilities are demonstrated for three nonlinear dynamical systems. First, a variation of the Kuramoto–Sivashinsky (KS) equation with control inputs is studied, where variables are sparsely probed. We show that the NNO is able to track states even when probes are contaminated with random noise or with sensors at insufficient sample rates to match the control time step. Then, a confined cylinder flow is investigated, where velocity signals along the cylinder wake are estimated by using a small set of wall pressure sensors. In both the KS and cylinder problems, we show that the estimated states can be used to enable closed-loop control, taking advantage of stabilising NNCs. Finally, we present a legacy dataset of a turbulent boundary layer experiment, where convolutional NNs are employed to implement the models required for the estimation loop. We show that, by combining low-resolution noise-corrupted sensor data with an imperfect NNSM, it is possible to produce more accurate and robust estimates. Our approach presents better robustness to noise when compared with direct reconstructions via super-resolution NNs and predictions from graph NNs and Fourier neural operators.

JFM classification

Flow Control: Flow Control Flow Control: Instability control Mathematical Foundations: Machine learning

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JFM Papers
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Journal of Fluid Mechanics , Volume 1034 , 10 May 2026 , A15
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© The Author(s), 2026. Published by Cambridge University Press

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References

Afri, C., Andrieu, V., Bako, L. & Dufour, P. 2016 State and parameter estimation: a nonlinear Luenberger observer approach. IEEE Trans. Autom. Control 62 (2), 973–980.10.1109/TAC.2016.2566804CrossRefGoogle Scholar
Amaral, F.R., Cavalieri, A.V.G., Martini, E., Jordan, P. & Towne, A. 2021 Resolvent-based estimation of turbulent channel flow using wall measurements. J. Fluid Mech. 927, A17.10.1017/jfm.2021.764CrossRefGoogle Scholar
Audiffred, D.B.S., Cavalieri, A.V.G., Brito, P.P.C. & Martini, E. 2023 Experimental control of Tollmien–Schlichting waves using the Wiener–Hopf formalism. Phys. Rev. Fluids 8 (7), 073902.10.1103/PhysRevFluids.8.073902CrossRefGoogle Scholar
Bieker, K., Peitz, S., Brunton, S.L., Kutz, J.N. & Dellnitz, M. 2020 Deep model predictive flow control with limited sensor data and online learning. Theor. Comput. Fluid Dyn. 34 (4), 1–15.10.1007/s00162-020-00520-4CrossRefGoogle Scholar
Brunton, S.L. & Noack, B.R. 2015 Closed-loop turbulence control: progress and challenges. Appl. Mech. Rev. 67 (5), 050801.10.1115/1.4031175CrossRefGoogle Scholar
Brunton, S.L., Noack, B.R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508.10.1146/annurev-fluid-010719-060214CrossRefGoogle Scholar
Brunton, S.L., Proctor, J.L. & Kutz, J.N. 2016 Sparse identification of nonlinear dynamics with control (SINDYc). IFAC-PapersOnLine 49 (18), 710–715.10.1016/j.ifacol.2016.10.249CrossRefGoogle Scholar
Castellanos, R., Cornejo Maceda, G.Y., De La Fuente, I., Noack, B.R., Ianiro, A. & Discetti, S. 2022 Machine-learning flow control with few sensor feedback and measurement noise. Phys. Fluids 34 (4), 047118.10.1063/5.0087208CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75–110.10.1017/S0022112094000431CrossRefGoogle Scholar
Déda, T.C. & Wolf, W.R. 2022 Extremum seeking control applied to airfoil trailing-edge noise suppression. AIAA J. 60, 823–843.10.2514/1.J060634CrossRefGoogle Scholar
Déda, T., Wolf, W.R. & Dawson, S.T.M. 2023 Backpropagation of neural network dynamical models applied to flow control. Theor. Comput. Fluid Dyn. 37 (1), 1–25.10.1007/s00162-023-00641-6CrossRefGoogle Scholar
Déda, T.C., Wolf, W.R. & Dawson, S.T.M. 2024 Neural networks in feedback for flow analysis and control. Phys. Rev. Fluids 9 (6), 063904.10.1103/PhysRevFluids.9.063904CrossRefGoogle Scholar
Fabbiane, N., Semeraro, O., Bagheri, S. & Henningson, D.S. 2014 Adaptive and model-based control theory applied to convectively unstable flows. Appl. Mech. Rev. 66 (6), 060801.10.1115/1.4027483CrossRefGoogle Scholar
Fan, D., Yang, L., Wang, Z., Triantafyllou, M.S. & Karniadakis, E. 2020 Reinforcement learning for bluff body active flow control in experiments and simulations. Proc. Natl Acad. Sci. 117 (42), 26091–26098.10.1073/pnas.2004939117CrossRefGoogle ScholarPubMed
Friedland, B. 2012 Control System Design: An Introduction to State-Space Methods. Courier Corporation.Google Scholar
Fukami, K., Fukagata, K. & Taira, K. 2019 Super-resolution reconstruction of turbulent flows with machine learning. J. Fluid Mech. 870, 106–120.10.1017/jfm.2019.238CrossRefGoogle Scholar
Fukami, K., Fukagata, K. & Taira, K. 2023 Super-resolution analysis via machine learning: a survey for fluid flows. arXiv preprint arXiv:2301.10937.10.1007/s00162-023-00663-0CrossRefGoogle Scholar
Fukami, K., Iwatani, Y., Maejima, S., Asada, H. & Kawai, S. 2025 Compact representation of transonic airfoil buffet flows with observable-augmented machine learning. J. Fluid Mech. 1021, A39.10.1017/jfm.2025.10741CrossRefGoogle Scholar
Fukami, K. & Taira, K. 2025 Observable-augmented manifold learning for multi-source turbulent flow data. J. Fluid Mech. 1010, R4.10.1017/jfm.2025.383CrossRefGoogle Scholar
Gad-el Ha, M. 2001 Flow control: the future. J. Aircraft 38, 402–418.10.2514/2.2796CrossRefGoogle Scholar
Graff, J., Medina, A. & Lagor, F. 2023 Information-based sensor placement for data-driven estimation of unsteady flows. arXiv preprint arXiv:2303.12260.10.2514/1.J063015CrossRefGoogle Scholar
Guastoni, L., Rabault, J., Schlatter, P., Azizpour, H. & Vinuesa, R. 2023 Deep reinforcement learning for turbulent drag reduction in channel flows. Eur. Phys. J. E 46 (4), 27.10.1140/epje/s10189-023-00285-8CrossRefGoogle ScholarPubMed
Kossaifi, J., et al. 2025 A library for learning neural operators. arXiv preprint arXiv:2412.10354.Google Scholar
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. & Anandkumar, A. 2021 Fourier neural operator for parametric partial differential equations. arXiv:2010.08895.Google Scholar
Li, J. & Zhang, M. 2022 Reinforcement-learning-based control of confined cylinder wakes with stability analyses. J. Fluid Mech. 932, A44.10.1017/jfm.2021.1045CrossRefGoogle Scholar
Luenberger, D.G. 2007 Observing the state of a linear system. IEEE Trans. Milit. Electron. 8 (2), 74–80.10.1109/TME.1964.4323124CrossRefGoogle Scholar
Manohar, K., Brunton, B.W., Kutz, J.N. & Brunton, S.L. 2018 Data-driven sparse sensor placement for reconstruction: demonstrating the benefits of exploiting known patterns. IEEE Control Syst. Mag. 38 (3), 63–86.Google Scholar
Martini, E., Cavalieri, A.V.G., Jordan, P., Towne, A. & Lesshafft, L. 2020 Resolvent-based optimal estimation of transitional and turbulent flows. J. Fluid Mech. 900, A2.10.1017/jfm.2020.435CrossRefGoogle Scholar
Miotto, R., Wolf, W. & Zigunov, F. 2025 Pressure field reconstruction with SIREN: a mesh-free approach for image velocimetry in complex noisy environments. Exp. Fluids 66, 151.10.1007/s00348-025-04074-1CrossRefGoogle Scholar
Morimoto, M., Fukami, K., Zhang, K. & Fukagata, K. 2022 Generalization techniques of neural networks for fluid flow estimation. Neural Comput. Appl. 34 (5), 3647–3669.10.1007/s00521-021-06633-zCrossRefGoogle Scholar
Morton, J., Witherden, F.D., Jameson, A. & Kochenderfer, M.J. 2018 Deep dynamical modeling and control of unsteady fluid flows. arXiv preprint arXiv:1805.07472.Google Scholar
Mousavi, H. & Eldredge, J.D. 2025 Low-order flow reconstruction and uncertainty quantification in disturbed aerodynamics using sparse pressure measurements. J. Fluid Mech. 1013, A41.10.1017/jfm.2025.10253CrossRefGoogle Scholar
Nair, N.J. & Goza, A. 2020 Leveraging reduced-order models for state estimation using deep learning. J. Fluid Mech. 897, R1.10.1017/jfm.2020.409CrossRefGoogle Scholar
Nakamura, T. & Fukagata, K. 2022 Robust training approach of neural networks for fluid flow state estimations. Intl J. Heat Fluid Flow 96, 108997.10.1016/j.ijheatfluidflow.2022.108997CrossRefGoogle Scholar
Page, J. 2024 Super-resolution with dynamics in the loss. arXiv preprint arXiv:2410.20884.Google Scholar
Paris, R., Beneddine, S. & Dandois, J. 2021 Robust flow control and optimal sensor placement using deep reinforcement learning. J. Fluid Mech. 913, A25.CrossRefGoogle Scholar
Park, J. & Choi, H. 2020 Machine-learning-based feedback control for drag reduction in a turbulent channel flow. J. Fluid Mech. 904, A24.10.1017/jfm.2020.690CrossRefGoogle Scholar
Payne, N.E., Zhang, Y. & Cattafesta, L.N. 2024 Co-flow jet design optimization. In AIAA Aviation Forum and Ascend 2024, p. 3631.Google Scholar
Pfaff, T., Fortunato, M., Sanchez-Gonzalez, A. & Battaglia, P.W. 2021 Learning mesh-based simulation with graph networks. arXiv:2010.03409.Google Scholar
Rabault, J., Kuchta, M., Jensen, A., Réglade, U. & Cerardi, N. 2019 Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control. J. Fluid Mech. 865, 281–302.10.1017/jfm.2019.62CrossRefGoogle Scholar
Raibaudo, C., Zhong, P., Noack, B.R. & Martinuzzi, R.J. 2020 Machine learning strategies applied to the control of a fluidic pinball. Phys. Fluids 32 (1), 015108.10.1063/1.5127202CrossRefGoogle Scholar
Ramos, L.D., Di Meglio, F., Morgenthaler, V., da Silva, L.F. & Bernard, P. 2020 Numerical design of Luenberger observers for nonlinear systems. In 2020 59th IEEE Conference on Decision and Control (CDC), pp. 5435–5442. IEEE.10.1109/CDC42340.2020.9304163CrossRefGoogle Scholar
Ramos, B.L.O., Wolf, W.R., Yeh, C.-A. & Taira, K. 2019 Active flow control for drag reduction of a plunging airfoil under deep dynamic stall. Phys. Rev. Fluids 4, 074603.10.1103/PhysRevFluids.4.074603CrossRefGoogle Scholar
Ren, F., Rabault, J. & Tang, H. 2021 Applying deep reinforcement learning to active flow control in weakly turbulent conditions. Phys. Fluids 33 (3), 037121.10.1063/5.0037371CrossRefGoogle Scholar
Sashittal, P. & Bodony, D.J. 2021 Data-driven sensor placement for fluid flows. Theor. Comput. Fluid Dyn. 35 (5), 709–729.10.1007/s00162-021-00584-wCrossRefGoogle Scholar
Sonoda, T., Liu, Z., Itoh, T. & Hasegawa, Y. 2023 Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow. J. Fluid Mech. 960, A30.10.1017/jfm.2023.147CrossRefGoogle Scholar
de Souza, L.F., Wolf, W.R., Safari, M. & Yeh, C.-A. 2025 Control of deep dynamic stall by duty-cycle actuation informed by stability analysis. AIAA J. 63 (11), 4958–4969.10.2514/1.J064980CrossRefGoogle Scholar
Taira, K. & Nair, A.G. 2022 Network-based analysis of fluid flows: progress and outlook. Prog. Aerosp. Sci. 131, 100823.10.1016/j.paerosci.2022.100823CrossRefGoogle Scholar
Towne, A., Lozano-Durán, A. & Yang, X. 2020 Resolvent-based estimation of space–time flow statistics. J. Fluid Mech. 883, A17.10.1017/jfm.2019.854CrossRefGoogle Scholar
Varela, P., Suárez, P., vila, A.-Á., Francisco, M., Arnau, R., Jean, F., Bernat, G.-C., Miguel, L., Lehmkuhl, O. & Vinuesa, R. 2022 Deep reinforcement learning for flow control exploits different physics for increasing Reynolds number regimes. Actuators 11 (12), 359.10.3390/act11120359CrossRefGoogle Scholar
Visbal, M.R. & Benton, S.I. 2018 Exploration of high-frequency control of dynamic stall using large-eddy simulations. AIAA J. 56 (8), 2974–2991.10.2514/1.J056720CrossRefGoogle Scholar
Willcox, K. 2006 Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput. Fluids 35 (2), 208–226.10.1016/j.compfluid.2004.11.006CrossRefGoogle Scholar
Williams, J., Zahn, O. & Kutz, J.N. 2022 Data-driven sensor placement with shallow decoder networks. arXiv preprint arXiv:2202.05330.Google Scholar
Yadaiah, N. & Sowmya, G. 2006 Neural network based state estimation of dynamical systems. In The 2006 IEEE International Joint Conference on Neural Network Proceedings, pp. 1042–1049. IEEE.10.1109/IJCNN.2006.246803CrossRefGoogle Scholar
Yeh, C.-A., Meena, M.G. & Taira, K. 2021 Network broadcast analysis and control of turbulent flows. J. Fluid Mech. 910, A15.10.1017/jfm.2020.965CrossRefGoogle Scholar
Zeitz, M. 1987 The extended Luenberger observer for nonlinear systems. Syst. Control Lett. 9 (2), 149–156.10.1016/0167-6911(87)90021-1CrossRefGoogle Scholar
Zhong, Y., Fukami, K., An, B. & Taira, K. 2023 Sparse sensor reconstruction of vortex-impinged airfoil wake with machine learning. Theor. Comput. Fluid Dyn. 37 (2), 269–287.10.1007/s00162-023-00657-yCrossRefGoogle Scholar
Zigunov, F. 2020 Detailed Flow Field Study of an Upswept Cylinder Wake and Experimental Optimization Using Active Flow Control. The Florida State University.Google Scholar
Zigunov, F. & Charonko, J.J. 2024 One-shot omnidirectional pressure integration through matrix inversion. Meas. Sci. Technol. 35 (12), 125301.10.1088/1361-6501/ad7486CrossRefGoogle Scholar
Zigunov, F., Sellappan, P. & Alvi, F. 2022a A bluff body flow control experiment with distributed actuation and genetic algorithm-based optimization. Exp. Fluids 63 (1), 23.10.1007/s00348-021-03356-8CrossRefGoogle Scholar
Zigunov, F., Sellappan, P., Alvi, F., Ozawa, Y., Saito, Y., Nonomura, T. & Asai, K. 2022b Time-resolved particle image velocimetry and pressure sensitive paint measurements of afterbody flow dynamics. Phys. Rev. Fluids 7 (2), 024701.10.1103/PhysRevFluids.7.024701CrossRefGoogle Scholar
Zigunov, F., Song, M.J., Sellappan, P. & Alvi, F.S. 2023 Multiaxis shock vectoring control of overexpanded supersonic jet using a genetic algorithm. J. Propul. Power 39 (2), 249–257.10.2514/1.B38846CrossRefGoogle Scholar
Supplementary material: File

Déda et al. supplementary movie 1

Time response of KS cases.
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Déda et al. supplementary movie 2

Closed-loop control response of the cylinder flow with different sensor configurations.
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Déda et al. supplementary movie 3

Boundary layer estimation via different methods.
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Déda et al. supplementary movie 4

Boundary layer estimation with noisy or lacking boundary conditions.
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Supplementary material: File

Déda et al. supplementary movie 5

Boundary layer estimation via graph neural networks, Fourier neural operators and neural network observers.
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File 39 MB