Reconstruction of unsteady fluid-structure interaction flows from sparse measurements taken at the interface can facilitate data-driven modeling or real-time control; however, current methods often lead to nonoptimal sparse measurement placement and hinder insights into underlying physics and mechanisms. This paper proposes an embedded methodology whereby a concrete autoencoder is used to simultaneously optimize the sparse measurement placement and the flow reconstruction process. The concrete autoencoder uses a concrete selector layer as an encoder such that a low-dimensional state is indicated by concrete sparse measurements rather than abstract latent variables. The decoder can then be set up with any reasonable neural network depending on the problem of the fluid-structural interaction problem to be solved. The proposed method is applied to the sparse sensing and reconstruction of unsteady pressure distributions for an airfoil and a swept wing. In addition, the data-driven aerodynamic models based on sparse measurements are coupled with the structural models to efficiently predict the problem of transonic aeroelasticity. The numerical results demonstrate that the proposed method outperforms the conventional sparse sensing based on proper orthogonal decomposition.

[1] Chen D., Kaiser F., Hu J., Rival D. E., Fukami K. and Taira K., "Sparse Pressure-Based Machine Learning Approach for Aerodynamic Loads Estimation During Gust Encounters," AIAA Journal, Vol. 62, No. 1, 2024, pp. 275-290. https://doi.org/10.2514/1.J063263 LinkGoogle Scholar

[2] Mousavi H. and Eldredge J. D., "Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements," Journal of Fluid Mechanics, Vol. 1013, 2025, p. A41.https://doi.org/10.48550/arXiv.2501.03406 Google Scholar

[3] Lumley J., Stochastic Tools in Turbulence, Academic Press, New York, 1970, p. 57. Google Scholar

[4] Schmid P. J., "Dynamic Mode Decomposition of Numerical and Experimental Data," Journal of Fluid Mechanics, Vol. 656, 2010, pp. 5-28. https://doi.org/10.1017/S0022112010001217 CrossrefGoogle Scholar

[5] Sirovich L. and Everson R., "Karhunen-Loève Procedure for Gappy Data," Journal of the Optical Society of America A, Vol. 12, No. 8, 1995, pp. 1657-1664. https://doi.org/10.1364/JOSAA.12.001657 Google Scholar

[6] Bui-Thanh T., Damodaran M. and Willcox K., "Aerodynamic Data Reconstruction and Inverse Design Using Proper Orthogonal Decomposition," AIAA Journal, Vol. 42, No. 8, 2004, pp. 1505-1516. https://doi.org/10.2514/1.2159 LinkGoogle Scholar

[7] Venturi D. and Karniadakis G. E., "Gappy Data and Reconstruction Procedures for Flow Past a Cylinder," Journal of Fluid Mechanics, Vol. 519, 2004, pp. 315-336. https://doi.org/10.1017/S0022112004001338 CrossrefGoogle Scholar

[8] Podvin B., Fraigneau Y., Lusseyran F. and Gougat P., "A Reconstruction Method for the Flow Past an Open Cavity," Journal of Fluids Engineering, Vol. 128, No. 3, 2006, pp. 531-540. https://doi.org/10.1115/1.2175159 CrossrefGoogle Scholar

[9] Willcox K., "Unsteady Flow Sensing and Estimation via the Gappy Proper Orthogonal Decomposition," Computers and Fluids, Vol. 35, No. 2, 2006, pp. 208-226. https://doi.org/10.1016/j.compfluid.2004.11.006 CrossrefGoogle Scholar

[10] Zhao X., Deng Z. C. and Zhang W. W., "Sparse Reconstruction of Surface Pressure Coefficient Based on Compressed Sensing," Experiments in Fluids, Vol. 63, No. 10, 2022. https://doi.org/10.1007/s00348-022-03505-7 Google Scholar

[11] Manohar K., Brunton B. W., Kutz J. N. and Brunton S. L., "Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns," IEEE Control Systems, Vol. 38, No. 3, 2018, pp. 63-86. https://doi.org/10.1109/MCS.2018.2810460 CrossrefGoogle Scholar

[12] Donoho D. L., "Compressed Sensing," IEEE Transactions on Information Theory, Vol. 52, No. 4, 2006, pp. 1289-1306. https://doi.org/10.1109/TIT.2006.871582 CrossrefGoogle Scholar

[13] Tropp J. A., "Greed Is Good: Algorithmic Results for Sparse Approximation," IEEE Transactions on Information Theory, Vol. 50, No. 10, 2004, pp. 2231-2242. https://doi.org/10.1109/TIT.2004.834793 Google Scholar

[14] Joshi S. and Boyd S., "Sensor Selection via Convex Optimization," IEEE Transactions on Signal Processing, Vol. 57, No. 2, 2009, pp. 451-462. https://doi.org/10.1109/TSP.2008.2007095 CrossrefGoogle Scholar

[15] Chaturantabut S. and Sorensen D. C., "Nonlinear Model Reduction via Discrete Empirical Interpolation," SIAM Journal on Scientific Computing, Vol. 32, No. 5, 2010, pp. 2737-2764. https://doi.org/10.1137/090766498 CrossrefGoogle Scholar

[16] Drmac Z. and Gugercin S., "A New Selection Operator for the Discrete Empirical Interpolation Method---Improved a Priori Error Bound and Extensions," SIAM Journal on Scientific Computing, Vol. 38, No. 2, 2016, pp. A631-A648. https://doi.org/10.1137/15M1019271 Google Scholar

[17] Callaham J. L., Maeda K. and Brunton S. L., "Robust Flow Reconstruction from Limited Measurements via Sparse Representation," Physical Review Fluids, Vol. 4, No. 10, 2019, Paper 103907. https://doi.org/10.1103/PhysRevFluids.4.103907 CrossrefGoogle Scholar

[18] Dubois P., Gomez T., Planckaert L. and Perret L., "Machine Learning for Fluid Flow Reconstruction from Limited Measurements," Journal of Computational Physics, Vol. 448, 2022, Paper 110733. https://doi.org/10.1016/j.jcp.2021.110733 CrossrefGoogle Scholar

[19] Nair N. J. and Goza A., "Leveraging Reduced-Order Models for State Estimation Using Deep Learning," Journal of Fluid Mechanics, Vol. 897, 2020, Paper R1. https://doi.org/10.1017/jfm.2020.409 Google Scholar

[20] Zhang X., Ji T., Xie F., Zheng H. and Zheng Y., "Unsteady Flow Prediction from Sparse Measurements by Compressed Sensing Reduced Order Modeling," Computer Methods in Applied Mechanics and Engineering, Vol. 393, 2022, Paper 114800. https://doi.org/10.1016/j.cma.2022.114800 CrossrefGoogle Scholar

[21] Luo Z. H., Wang L. Y., Xu J., Chen M., Yuan J. P. and Tan A. C. C., "Flow Reconstruction from Sparse Sensors Based on Reduced-Order Autoencoder State Estimation," Physics of Fluids, Vol. 35, No. 7, 2023, Paper 075127. https://doi.org/10.1063/5.0155039 Google Scholar

[22] Erichson N. B., Mathelin L., Yao Z., Brunton S. L., Mahoney M. W. and Kutz J. N., "Shallow Neural Networks for Fluid Flow Reconstruction with Limited Sensors," Proceedings of the Royal Society A, Vol. 476, No. 2238, 2020. https://doi.org/10.1098/rspa.2020.0097 Google Scholar

[23] Adrian R., "On the Role of Conditional Averages in Turbulence Theory," Proceedings of the Symposium on Turbulence in Liquids, 1975, pp. 323-332, https://scholarsmine.mst.edu/sotil/34. Google Scholar

[24] Kalman R. E., "A New Approach to Linear Filtering and Prediction Problems," Journal of Basic Engineering, Vol. 82, No. 1, 1960, pp. 35-45. https://doi.org/10.1115/1.3662552 CrossrefGoogle Scholar

[25] Loiseau J. C., Noack B. R. and Brunton S. L., "Sparse Reduced-Order Modelling: Sensor-Based Dynamics to Full-State Estimation," Journal of Fluid Mechanics, Vol. 844, 2018, pp. 459-490. https://doi.org/10.1017/jfm.2018.147 CrossrefGoogle Scholar

[26] Graff J., Medina A. and Lagor F. D., "Information-Based Sensor Placement for Data-Driven Estimation of Unsteady Flows," AIAA Journal, Vol. 61, No. 11, 2023, p. 4864. https://doi.org/10.2514/1.J063015 LinkGoogle Scholar

[27] Murata T., Fukami K. and Fukagata K., "Nonlinear Mode Decomposition with Convolutional Neural Networks for Fluid Dynamics," Journal of Fluid Mechanics, Vol. 882, 2020, Paper A13. https://doi.org/10.1017/jfm.2019.822 CrossrefGoogle Scholar

[28] Abid A., Balin M. F. and Zou J., "Concrete Autoencoders: Differentiable Feature Selection and Reconstruction," Proceedings of the 36th International Conference on Machine Learning, Vol. 97, 2019, pp. 444-453. https://doi.org/10.48550/arXiv.1901.09346 Google Scholar

[29] Fukami K., Murata T., Zhang K., Fukagata K., Fukami K., Murata T., Zhang K. and Fukagata K., "Sparse Identification of Nonlinear Dynamics with Low-Dimensionalized Flow Representations," Journal of Fluid Mechanics, Vol. 926, 2021, Paper A10. https://doi.org/10.1017/jfm.2021.697 CrossrefGoogle Scholar

[30] Maddison C., Mnih A. and Teh Y., "The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables," International Conference on Learning Representations, 2017, pp. 1-20. https://doi.org/10.48550/arXiv.1611.00712 Google Scholar

[31] Gumbel E. J., "Statistical Theory of Extreme Values and Some Practical Applications: A Series of Lectures," US Government Printing Office, 1954. Google Scholar

[32] Proctor J. L., Brunton S. L. and Kutz J. N., "Dynamic Mode Decomposition with Control," SIAM Journal on Applied Dynamical Systems, Vol. 15, No. 1, 2016, pp. 142-161. https://doi.org/10.1137/15M1013857 CrossrefGoogle Scholar

[33] Yao X., Huang R. and Hu H., "Data-Driven Modeling of Transonic Unsteady Flows and Efficient Analysis of Fluid-Structure Stability," Journal of Fluids and Structures, Vol. 111, 2022, Paper 103549. https://doi.org/10.1016/j.jfluidstructs.2022.103549 CrossrefGoogle Scholar

[34] Paduart J., Lauwers L., Swevers J., Smolders K., Schoukens J. and Pintelon R., "Identification of Nonlinear Systems Using Polynomial Nonlinear State Space Models," Automatica, Vol. 46, No. 4, 2010, pp. 647-656. https://doi.org/10.1016/j.automatica.2010.01.001 Google Scholar

[35] Decuyper J., De Troyer T., Runacres M. C., Tiels K. and Schoukens J., "Nonlinear State-Space Modelling of the Kinematics of an Oscillating Circular Cylinder in a Fluid Flow," Mechanical Systems and Signal Processing, Vol. 98, 2018, pp. 209-230. https://doi.org/10.1016/j.ymssp.2017.04.048 CrossrefGoogle Scholar

[36] Fonzi N., Brunton S. L. and Fasel U., "Data-Driven Modeling for Transonic Aeroelastic Analysis," Journal of Aircraft, Vol. 61, No. 2, 2024, pp. 625-637. https://doi.org/10.2514/1.C037409 LinkGoogle Scholar

[37] Landon R. H. and Davis S. S., "Compendium of Unsteady Aerodynamic Measurements," Advisory Group for Aerospace Research and Development, AGARD-R-702, Aug. 1982, https://www.abbottaerospace.com/downloads/agard-r-702. Google Scholar

[38] Yao X., Huang R., Hu H. and Liu H., "Transonic Aerodynamic-Structural Coupling Characteristics Predicted by Nonlinear Data-Driven Modeling Approach," AIAA Journal, Vol. 62, No. 3, 2024, pp. 1159-1178. https://doi.org/10.2514/1.J063360 LinkGoogle Scholar

[39] Yates E. C., "AGARD Standard Aeroelastic Configurations for Dynamic Response. Candidate Configuration I.-Wing 445.6," NASA-TM-100492, NASA Langley Research Center, Hampton, VA, 1987, pp. v-73. Google Scholar

[40] Hüllermeier E. and Waegeman W., "Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods," Machine Learning, Vol. 110, No. 3, 2021, pp. 457-506. https://doi.org/10.1007/s10994-021-05946-3 CrossrefGoogle Scholar

[41] Yu C. and Rosendo A., "Risk-Aware Model-Based Control," Frontiers in Robotics and AI Volume, Vol. 8, March 2021, pp. 1-14. https://doi.org/10.3389/frobt.2021.617839 Google Scholar