Symbolic Deep Learning for Structural System Identification
Publication: Journal of Structural Engineering
Volume 148, Issue 9
Abstract
Closed-form model expression is commonly required for parametric data assimilation (e.g., model updating, damage quantification, and so on). However, epistemic bias due to fixing the model class is a challenging issue for structural identification. Furthermore, it is sometimes hard to derive explicit expressions for structural mechanisms such as damping and nonlinear restoring forces. Although existing model class selection methods are beneficial to reduce the model uncertainty, the primary issue lies in their limitation to a small number of predefined model choices. We propose a symbolic deep learning framework that alleviates the constraint of fixed model classes and lets the data more flexibly determine the model type and discover the symbolic invariance of the structural system. A design principle for symbolic neural networks has been developed to leverage domain knowledge and translate data to flexibly symbolic equations of motion with a good predictive capacity for new data. A two-stage model selection strategy is proposed to conduct adaptive pruning on network and equation levels by balancing the model sparsity and the goodness of fit. The proposed method’s expressive strengths and weaknesses have been analyzed in several numerical case studies, including systems with nonlinear damping, restoring force, and chaotic behavior. Results from an experimental case study revealed the potential of the proposed method for flexibly interpreting hidden mechanisms for real-world applications. Finally, we discuss necessary improvements to transfer this computational method for practical applications.
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Data Availability Statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The work is supported in part by the Beijing Outstanding Young Scientist Program (No. BJJWZYJH012019100020098) as well as the Intelligent Social Governance Platform, Major Innovation & Planning Interdisciplinary Platform for the “Double-First Class” Initiative, Renmin University of China. We also wish to acknowledge the support in part by the Engineering for Civil Infrastructure program at National Science Foundation under Grant No. CMMI-2013067, the research award from MathWorks, the Tier 1 Seed Grant Program at Northeastern University, and the Thornton Tomasetti Student Innovation Fellowship. We also appreciate the significant contributions from reviewers, editors, and the journal staff.
References
Abazarsa, F., F. Nateghi, S. F. Ghahari, and E. Taciroglu. 2013. “Blind modal identification of non-classically damped systems from free or ambient vibration records.” Earthquake Spectra 29 (4): 1137–1157. https://doi.org/10.1193/031712EQS093M.
Al-Hababi, T., M. Cao, B. Saleh, N. F. Alkayem, and H. Xu. 2020. “A critical review of nonlinear damping identification in structural dynamics: Methods, applications, and challenges.” Sensors 20 (24): 7303. https://doi.org/10.3390/s20247303.
Allemang, R. J., D. L. Brown, and W. Fladung. 1994. “Modal parameter estimation: A unified matrix polynomial approach.” In Vol. 2251 of Proc., 12th Int. Modal Analysis Conf., 501. Bethel, CT: Society for Experimental Mechanics.
Avci, O., O. Abdeljaber, S. Kiranyaz, M. Hussein, M. Gabbouj, and D. J. Inman. 2021. “A review of vibration-based damage detection in civil structures: From traditional methods to machine learning and deep learning applications.” Mech. Syst. Sig. Process. 147 (Jan): 107077. https://doi.org/10.1016/j.ymssp.2020.107077.
Behmanesh, I., and B. Moaveni. 2016. “Accounting for environmental variability, modeling errors, and parameter estimation uncertainties in structural identification.” J. Sound Vib. 374 (Jul): 92–110. https://doi.org/10.1016/j.jsv.2016.03.022.
Behmanesh, I., B. Moaveni, G. Lombaert, and C. Papadimitriou. 2015. “Hierarchical Bayesian model updating for structural identification.” Mech. Syst. Sig. Process. 64–65 (Dec): 360–376. https://doi.org/10.1016/j.ymssp.2015.03.026.
Brincker, R., L. Zhang, and P. Andersen. 2000. “Modal identification from ambient responses using frequency domain decomposition.” In Proc., 18th Int. Modal Analysis Conf. (IMAC). Bethel, CT: Society for Experimental Mechanics.
Brualdi, R. A. 1977. Introductory combinatorics. New York: Pearson Education India.
Brunton, S. L., B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz. 2017. “Chaos as an intermittently forced linear system.” Nat. Commun. 8 (1): 1–9. https://doi.org/10.1038/s41467-017-00030-8.
Carden, E. P., and J. M. Brownjohn. 2008. “ARMA modelled time-series classification for structural health monitoring of civil infrastructure.” Mech. Syst. Sig. Process. 22 (2): 295–314. https://doi.org/10.1016/j.ymssp.2007.07.003.
Chen, Z., Y. Liu, and H. Sun. 2021. “Physics-informed learning of governing equations from scarce data.” Nat. Commun. 12 (1): 6136. https://doi.org/10.1038/s41467-021-26434-1.
Cunha, Á., and E. Caetano. 2006. “Experimental modal analysis of civil engineering structures.” Sound Vib. 40 (6): 12–20.
Deraemaeker, A., E. Reynders, G. De Roeck, and J. Kullaa. 2008. “Vibration-based structural health monitoring using output-only measurements under changing environment.” Mech. Syst. Sig. Process. 22 (1): 34–56. https://doi.org/10.1016/j.ymssp.2007.07.004.
Dong, C., X. Ye, and T. Jin. 2018. “Identification of structural dynamic characteristics based on machine vision technology.” Measurement 126 (Oct): 405–416. https://doi.org/10.1016/j.measurement.2017.09.043.
Feng, D., and M. Q. Feng. 2018. “Computer vision for SHM of civil infrastructure: From dynamic response measurement to damage detection: A review.” Eng. Struct. 156 (Feb): 105–117. https://doi.org/10.1016/j.engstruct.2017.11.018.
Feng, D., H. Sun, and M. Q. Feng. 2015. “Simultaneous identification of bridge structural parameters and vehicle loads.” Comput. Struct. 157 (Sep): 76–88. https://doi.org/10.1016/j.compstruc.2015.05.017.
Fine, T. L. 2006. Feedforward neural network methodology. New York: Springer.
Ghahari, S., F. Abazarsa, M. Ghannad, M. Celebi, and E. Taciroglu. 2014. “Blind modal identification of structures from spatially sparse seismic response signals.” Struct. Control Health Monit. 21 (5): 649–674. https://doi.org/10.1002/stc.1593.
Goulet, J.-A., and I. F. Smith. 2013. “Structural identification with systematic errors and unknown uncertainty dependencies.” Comput. Struct. 128 (Nov): 251–258. https://doi.org/10.1016/j.compstruc.2013.07.009.
Gul, M., and F. N. Catbas. 2008. “Ambient vibration data analysis for structural identification and global condition assessment.” J. Eng. Mech. 134 (8): 650–662. https://doi.org/10.1061/(ASCE)0733-9399(2008)134:8(650).
Han, Q., Q. Ma, J. Xu, and M. Liu. 2021. “Structural health monitoring research under varying temperature condition: A review.” J. Civ. Struct. Health Monit. 11 (1): 149–173. https://doi.org/10.1007/s13349-020-00444-x.
Higham, D. J., and N. J. Higham. 2016. MATLAB guide. Philadelphia, PA: Society for Industrial and Applied Mathematics.
Hornik, K. 1991. “Approximation capabilities of multilayer feedforward networks.” Neural Netw. 4 (2): 251–257. https://doi.org/10.1016/0893-6080(91)90009-T.
Huang, Y., C. Shao, B. Wu, J. L. Beck, and H. Li. 2019. “State-of-the-art review on Bayesian inference in structural system identification and damage assessment.” Adv. Struct. Eng. 22 (6): 1329–1351. https://doi.org/10.1177/1369433218811540.
Hung, S.-L., C.-S. Huang, C. Wen, and Y. Hsu. 2003. “Nonparametric identification of a building structure from experimental data using wavelet neural network.” Comput.-Aided Civ. Infrastruct. Eng. 18 (5): 356–368. https://doi.org/10.1111/1467-8667.t01-1-00313.
Ismail, M., F. Ikhouane, and J. Rodellar. 2009. “The hysteresis Bouc-Wen model, a survey.” Arch. Comput. Methods Eng. 16 (2): 161–188. https://doi.org/10.1007/s11831-009-9031-8.
Khuc, T., and F. N. Catbas. 2018. “Structural identification using computer vision–based bridge health monitoring.” J. Struct. Eng. 144 (2): 04017202. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001925.
Kijewski, T., and A. Kareem. 2003. “Wavelet transforms for system identification in civil engineering.” Comput.-Aided Civ. Infrastruct. Eng. 18 (5): 339–355. https://doi.org/10.1111/1467-8667.t01-1-00312.
Kim, S., P. Y. Lu, S. Mukherjee, M. Gilbert, L. Jing, V. čeperić, and M. Soljačić. 2020. “Integration of neural network-based symbolic regression in deep learning for scientific discovery.” IEEE Trans. Neural Networks Learn. Syst. 32 (9): 4166–4177. https://doi.org/10.1109/TNNLS.2020.3017010.
Kingma, D. P., and J. Ba. 2014. “Adam: A method for stochastic optimization.” Preprint, submitted December 22, 2014. http://arxiv.org/abs/1412.6980.
Krishnan Nair, K., and A. S. Kiremidjian. 2007. “Time series based structural damage detection algorithm using Gaussian mixtures modeling.” J. Dyn. Syst. Meas. Control 129 (3): 285–293. https://doi.org/10.1115/1.2718241.
Lagaros, N. D., and M. Papadrakakis. 2012. “Neural network based prediction schemes of the non-linear seismic response of 3D buildings.” Adv. Eng. Software 44 (1): 92–115. https://doi.org/10.1016/j.advengsoft.2011.05.033.
Lai, Z., and S. Nagarajaiah. 2019. “Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior.” Mech. Syst. Sig. Process. 117 (Feb): 813–842. https://doi.org/10.1016/j.ymssp.2018.08.033.
Lee Rodgers, J., and W. A. Nicewander. 1988. “Thirteen ways to look at the correlation coefficient.” Am. Stat. 42 (1): 59–66. https://doi.org/10.1080/00031305.1988.10475524.
Long, Z., Y. Lu, and B. Dong. 2019. “PDE-net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network.” J. Comput. Phys. 399 (Dec): 108925. https://doi.org/10.1016/j.jcp.2019.108925.
Lu, Z., H. Pu, F. Wang, Z. Hu, and L. Wang. 2017. “The expressive power of neural networks: A view from the width.” In Proc., 31st Int. Conf. on Neural Information Processing Systems, 6232–6240. La Jolla, CA: Neural Information Processing Systems Foundation.
Malekjafarian, A., and E. J. OBrien. 2014. “Identification of bridge mode shapes using short time frequency domain decomposition of the responses measured in a passing vehicle.” Eng. Struct. 81 (Dec): 386–397. https://doi.org/10.1016/j.engstruct.2014.10.007.
Martius, G., and C. H. Lampert. 2016. “Extrapolation and learning equations.” Preprints, submitted October 10, 2016. http://arxiv.org/abs/1610.02995.
Mikolov, T., M. Karafiát, L. Burget, J. Cernockỳ, and S. Khudanpur. 2010. “Recurrent neural network based language model.” In Vol. 2 of Interspeech, 1045–1048. Red Hook, NY: Curran Associates.
Muto, M., and J. L. Beck. 2008. “Bayesian updating and model class selection for hysteretic structural models using stochastic simulation.” J. Vib. Control 14 (1–2): 7–34. https://doi.org/10.1177/1077546307079400.
Newmark, N. M. 1959. “A method of computation for structural dynamics.” J. Eng. Mech. Div. 85 (3): 67–94. https://doi.org/10.1061/JMCEA3.0000098.
Pai, S. G., M. Sanayei, and I. F. Smith. 2021. “Model-class selection using clustering and classification for structural identification and prediction.” J. Comput. Civ. Eng. 35 (1): 04020051. https://doi.org/10.1061/(ASCE)CP.1943-5487.0000932.
Rudin, L. I., S. Osher, and E. Fatemi. 1992. “Nonlinear total variation based noise removal algorithms.” Physica D 60 (1–4): 259–268. https://doi.org/10.1016/0167-2789(92)90242-F.
Rudy, S. H., S. L. Brunton, J. L. Proctor, and J. N. Kutz. 2017. “Data-driven discovery of partial differential equations.” Sci. Adv. 3 (4): e1602614. https://doi.org/10.1126/sciadv.1602614.
Sahoo, S., C. Lampert, and G. Martius. 2018. “Learning equations for extrapolation and control.” Preprint, submitted June 19, 2018. http://arxiv.org/abs/1806.07259v1.
Schmidt, M., and H. Lipson. 2009. “Distilling free-form natural laws from experimental data.” Science 324 (5923): 81–85. https://doi.org/10.1126/science.1165893.
Singhose, W., D. Kim, and M. Kenison. 2008. “Input shaping control of double-pendulum bridge crane oscillations.” J. Dyn. Syst. Meas. Control 130 (30): 034504. https://doi.org/10.1115/1.2907363.
Spencer, B. F., Jr., V. Hoskere, and Y. Narazaki. 2019. “Advances in computer vision-based civil infrastructure inspection and monitoring.” Engineering 5 (2): 199–222. https://doi.org/10.1016/j.eng.2018.11.030.
Tropp, J. A., and A. C. Gilbert. 2007. “Signal recovery from random measurements via orthogonal matching pursuit.” IEEE Trans. Inf. Theory 53 (12): 4655–4666. https://doi.org/10.1109/TIT.2007.909108.
Wu, R.-T., and M. R. Jahanshahi. 2019. “Deep convolutional neural network for structural dynamic response estimation and system identification.” J. Eng. Mech. 145 (1): 04018125. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001556.
Xia, Y., B. Chen, S. Weng, Y.-Q. Ni, and Y.-L. Xu. 2012. “Temperature effect on vibration properties of civil structures: A literature review and case studies.” J. Civ. Struct. Health Monit. 2 (1): 29–46. https://doi.org/10.1007/s13349-011-0015-7.
Yang, Y., Y. Li, and K. C. Chang. 2014. “Constructing the mode shapes of a bridge from a passing vehicle: A theoretical study.” Smart Struct. Syst. 13 (5): 797–819. https://doi.org/10.12989/sss.2014.13.5.797.
Yang, Y., and J. P. Yang. 2018. “State-of-the-art review on modal identification and damage detection of bridges by moving test vehicles.” Int. J. Struct. Stab. Dyn. 18 (2): 1850025. https://doi.org/10.1142/S0219455418500256.
Yarnold, M. T., F. L. Moon, and A. Emin Aktan. 2015. “Temperature-based structural identification of long-span bridges.” J. Struct. Eng. 141 (11): 04015027. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001270.
Ye, X., T. Jin, and C. Yun. 2019. “A review on deep learning-based structural health monitoring of civil infrastructures.” Smart Struct. Syst. 24 (5): 567–586. https://doi.org/10.12989/sss.2019.24.5.567.
Yuen, K.-V. 2010a. Bayesian methods for structural dynamics and civil engineering. Hoboken, NJ: Wiley.
Yuen, K.-V. 2010b. “Recent developments of Bayesian model class selection and applications in civil engineering.” Struct. Saf. 32 (5): 338–346. https://doi.org/10.1016/j.strusafe.2010.03.011.
Yuen, K.-V., and H.-Q. Mu. 2015. “Real-time system identification: An algorithm for simultaneous model class selection and parametric identification.” Comput.-Aided Civ. Infrastruct. Eng. 30 (10): 785–801. https://doi.org/10.1111/mice.12146.
Zhang, R., Z. Chen, S. Chen, J. Zheng, O. Büyüköztürk, and H. Sun. 2019. “Deep long short-term memory networks for nonlinear structural seismic response prediction.” Comput. Struct. 220 (Aug): 55–68. https://doi.org/10.1016/j.compstruc.2019.05.006.
Zhang, R., Y. Liu, and H. Sun. 2020a. “Physics-guided convolutional neural network (phyCNN) for data-driven seismic response modeling.” Eng. Struct. 215 (Jul): 110704. https://doi.org/10.1016/j.engstruct.2020.110704.
Zhang, R., Y. Liu, and H. Sun. 2020b. “Physics-informed multi-LSTM networks for metamodeling of nonlinear structures.” Comput. Methods Appl. Mech. Eng. 369 (Sep): 113226. https://doi.org/10.1016/j.cma.2020.113226.
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Journal of Structural Engineering
Volume 148 • Issue 9 • September 2022
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© 2022 American Society of Civil Engineers.
History
Received: Oct 15, 2021
Accepted: Mar 22, 2022
Published online: Jun 24, 2022
Published in print: Sep 1, 2022
Discussion open until: Nov 24, 2022
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