Abstract
This research adopts emerging machine learning techniques to tackle the soil–structure interaction analysis problems of laterally loaded piles through physics-informed neural networks (PINNs), which employs prior physical information in the form of partial differential equations during the model training, eliminating the tremendous data requirement in the traditional data-driven machine learning methods. The formulations to describe the problem are discussed, and the corresponding governing equations are derived. A PINN framework, including neural networks architecture and loss functions, is developed for the machine learning-based solution and elaborated with details. The corresponding model training process is presented, based on which the surrogate model construction and back analysis implementation are introduced to demonstrate the effectiveness and flexibility of the proposed method. This method has been demonstrated for its accuracy via several examples with benchmark solutions from the existing well-developed methods. Finally, a case study of the uncertainty evaluation of a laterally loaded pile is conducted to illustrate its high computational efficiency and advantages in potential engineering applications.
Similar content being viewed by others
Data availability
All data generated or used during the study are available from the corresponding author by request.
References
AASHTO (2010) AASHTO LRFD bridge design specifications. American Association of State Highway and Transportation Officials, Washington
Abadi M,Barham P, Chen J, Chen Z, Davis A, Dean J, et al (2016) TensorFlow: a system for large-scale machine learning. In: 12th USENIX symposium on operating systems design and implementation (OSDI 16)
Abiodun OI, Jantan A, Omolara AE, Dada KV, Mohamed NA, Arshad H (2018) State-of-the-art in artificial neural network applications: a survey. Heliyon 4(11):e00938
Allotey N, El Naggar MH (2008) Generalized dynamic Winkler model for nonlinear soil–structure interaction analysis. Can Geotech J 45(4):560–573
API (2000) Recommended practice for planning, designing, and constructing fixed offshore platforms-working stress design (RP 2A-WSD). American Petroleum Institute,
Apostol TM (1991) Calculus, vol 1. Wiley, Hoboken
Bai X-D, Cheng W-C, Li G (2021) A comparative study of different machine learning algorithms in predicting EPB shield behaviour: a case study at the Xi’an metro, China. Acta Geotech 16:4061–4080
Basir S, Senocak I (2022) Physics and equality constrained artificial neural networks: application to forward and inverse problems with multi-fidelity data fusion. J Comput Phys 463:111301
Cai S, Mao Z, Wang Z, Yin M, Karniadakis GE (2021) Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mechanica Sinica 37(12):1727–1738
Chang FK, Markmiller JF, Yang J, Kim Y (2011) Structural health monitoring. In: Stephen B (ed) System health management: with aerospace application, 1st edn. Wiley, pp 419–428
Choi YS, Basu D, Salgado R, Prezzi M (2014) Response of laterally loaded rectangular and circular piles in soils with properties varying with depth. J Geotech Geoenviron Eng 140(4):04013049
Corliss G, Faure C, Griewank A, Hascoet L, Naumann U (2002) Automatic differentiation of algorithms: from simulation to optimization. Springer, Berlin
Dash SR, Bhattacharya S, Blakeborough A (2010) Bending–buckling interaction as a failure mechanism of piles in liquefiable soils. Soil Dyn Earthq Eng 30(1–2):32–39
Ellingwood B, Galambos TV, MacGregor JG, Cornell CA (1980) Development of a probability based load criterion for American National Standard A58, NBS Special Publication 577. National Bureau of Standards, Washington
Ellingwood BR, Tekie PB (1999) Wind load statistics for probability-based structural design. J Struct Eng 125(4):453–463
Fan H, Liang R (2013) Performance-based reliability analysis of laterally loaded drilled shafts. J Geotech Geoenviron Eng 139(12):2020–2027
Farrokh M, Dizaji MS, Joghataie A (2015) Modeling hysteretic deteriorating behavior using generalized Prandtl neural network. J Eng Mech 141(8):04015024
Gazetas G, Dobry R (1984) Horizontal response of piles in layered soils. J Geotech Eng 110(1):20–40
Gong W, Tang H, Juang CH, Wang L (2020) Optimization design of stabilizing piles in slopes considering spatial variability. Acta Geotech 15:3243–3259
Gupta B, Basu D (2018) Applicability of Timoshenko, Euler-Bernoulli and rigid beam theories in analysis of laterally loaded monopiles and piles. Géotechnique 68(9):772–785
Haghighat E, Raissi M, Moure A, Gomez H, Juanes R (2021) A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput Methods Appl Mech Eng 379:113741
He Z, Nguyen H, Vu TH, Zhou J, Asteris PG, Mammou A (2022) Novel integrated approaches for predicting the compressibility of clay using cascade forward neural networks optimized by swarm-and evolution-based algorithms. Acta Geotech 17:1257–1272
Hsiao C-H, Chen AY, Ge L, Yeh F-H (2022) Performance of artificial neural network and convolutional neural network on slope failure prediction using data from the random finite element method. Acta Geotech 17:5801–5811
Hu B, Gong Q, Zhang Y, Yin Y, Chen W (2022) Characterizing uncertainty in geotechnical design of energy piles based on Bayesian theorem. Acta Geotech 17(9):4191–4206
Huynh TQ, Nguyen TT, Nguyen H (2022) Base resistance of super-large and long piles in soft soil: performance of artificial neural network model and field implications. Acta Geotech 18:2755–2775
Joghataie A, Farrokh M (2008) Dynamic analysis of nonlinear frames by prandtl neural networks. J Eng Mech ASCE 134(11):961–969
Johnson R, Zhang T (2013) Accelerating stochastic gradient descent using predictive variance reduction. In: Advances in neural information processing systems 26
Jong S, Ong D, Oh E (2021) State-of-the-art review of geotechnical-driven artificial intelligence techniques in underground soil–structure interaction. Tunn Undergr Space Technol 113:103946
Kardani N, Bardhan A, Gupta S, Samui P, Nazem M, Zhang Y, Zhou A (2021) Predicting permeability of tight carbonates using a hybrid machine learning approach of modified equilibrium optimizer and extreme learning machine. Acta Geotech 17:1239–1255
Kavitha P, Beena K, Narayanan K (2016) A review on soil–structure interaction analysis of laterally loaded piles. Innov Infrastruct Solut 1(1):1–15
Kim Y, Jeong S, Won J (2009) Effect of lateral rigidity of offshore piles using proposed py curves in marine clay. Mar Georesour Geotechnol 27(1):53–77
Kingma DP, Ba J (2014) Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980
Lagaros ND, Papadrakakis M (2012) Neural network based prediction schemes of the non-linear seismic response of 3D buildings. Adv Eng Softw 44(1):92–115
Leung YF, Klar A, Soga K, Hoult N (2017) Superstructure–foundation interaction in multi-objective pile group optimization considering settlement response. Can Geotech J 54(10):1408–1420
Li Z, Kotronis P, Escoffier S, Tamagnini C (2016) A hypoplastic macroelement for single vertical piles in sand subject to three-dimensional loading conditions. Acta Geotech 11:373–390
Li Z, Ren A, Li J, Qiu Q, Yuan B, Draper J, Wang Y (2017) Structural design optimization for deep convolutional neural networks using stochastic computing. In: Design, automation and test in Europe conference and exhibition (DATE), 2017. IEEE
Lombardi D, Bhattacharya S (2016) Evaluation of seismic performance of pile-supported models in liquefiable soils. Earthq Eng Struct Dyn 45(6):1019–1038
Matlock H (1970) Correlation for design of laterally loaded piles in soft clay. In: Offshore technology conference. OnePetro
Matlock H, Reese LC (1960) Generalized solutions for laterally loaded piles. J Soil Mech Found Div 86(5):63–92
McClelland B, Focht J (1956) Soil modulus for laterally loaded piles. J Soil Mech Found Div 82(4):1081-1–1081-22
Mishra S, Molinaro R (2022) Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. IMA J Numer Anal 42(2):981–1022
Mohan D, Shrivastava S (1971) Nonlinear behavior of single vertical pile under lateral loads. In: Offshore technology conference. OTC
Montavon G, Orr G, Müller K-R (2012) Neural networks: tricks of the trade. Springer, Berlin
Nadeem M, Chakraborty T, Matsagar V (2015) Nonlinear buckling analysis of slender piles with geometric imperfections. J Geotech Geoenviron Eng 141(1):06014014
Nguyen T, Ly D-K, Huynh TQ, Nguyen TT (2023) Soft computing for determining base resistance of super-long piles in soft soil: a coupled SPBO-XGBoost approach. Comput Geotech 162:105707
Nguyen T, Ly K-D, Nguyen-Thoi T, Nguyen B-P, Doan N-P (2022) Prediction of axial load bearing capacity of PHC nodular pile using Bayesian regularization artificial neural network. Soils Found 62(5):101203
Nguyen V, Dackermann U, Li J, Alamdari MM, Mustapha S, Runcie P, Ye L (2015) Damage identification of a concrete arch beam based on frequency response functions and artificial neural networks. Electron J Struct Eng 14(1):75–84
Niaki SA, Haghighat E, Campbell T, Poursartip A, Vaziri R (2021) Physics-informed neural network for modelling the thermochemical curing process of composite-tool systems during manufacture. Comput Methods Appl Mech Eng 384:113959
Ong DEL, Barla M, Cheng JW-C, Choo CS, Sun M, Peerun MI (2022) Complex soil–pipe interaction: challenges in geological characterization and construction. In: Sustainable pipe jacking technology in the urban environment: recent advances and innovations. Springer, pp 43–101
Ouyang W, Liu S-W, Wan J, Yang Y (2021) Euler-Bernoulli pile element for nonlinear buckling analysis of single piles in slope. Int J Geomech 21(9):04021170
Ouyang W, Wan J, Liu S-W, Li X (2021) Line-finite-element implementation for driven steel H-piles in layered sands considering post-driving residual stresses. Adv Struct Eng 24(7):1384–1398
Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G et al (2019) Pytorch: an imperative style, high-performance deep learning library. In: Advances in neural information processing systems 32
Pham TA, Ly H-B, Tran VQ, Giap LV, Vu H-LT, Duong H-AT (2020) Prediction of pile axial bearing capacity using artificial neural network and random forest. Appl Sci 10(5):1871
Raissi M, Perdikaris P, Karniadakis GE (2017) Physics informed deep learning (part I): data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561.
Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707
Randolph MF (1981) The response of flexible piles to lateral loading. Geotechnique 31(2):247–259
Reese LC, H Matlock (1956) Non-dimensional solutions for laterally loaded piles with soil modulus assumed proportional to depth. In: Proceedings of the 8th Texas conference SMFE, The Univ. of Texas
Reese LC, Matlock H (1956) Non-dimensional solutions for laterally loaded piles with soil modulus assumed proportional to depth. In: Proceedings of the 8th Texas conference on soil mechanics and foundation engineering, The Univ. of Texas, pp 633–649
Shahin MA (2014) Load–settlement modeling of axially loaded drilled shafts using CPT-based recurrent neural networks. Int J Geomech 14(6):06014012
Shahin MA (2014) Load–settlement modeling of axially loaded steel driven piles using CPT-based recurrent neural networks. Soils Found 54(3):515–522
Sharma S, Sharma S, Athaiya A (2017) Activation functions in neural networks. Towards Data Sci 6(12):310–316
Shen W, Teh C (2004) Analysis of laterally loaded piles in soil with stiffness increasing with depth. J Geotech Geoenviron Eng 130(8):878–882
Tang C, Phoon K-K (2018) Evaluation of model uncertainties in reliability-based design of steel H-piles in axial compression. Can Geotech J 55(11):1513–1532
Vega-Posada CA, Gallant AP, Areiza-Hurtado M (2020) Simple approach for analysis of beam-column elements on homogeneous and non-homogeneous elastic soil. Eng Struct 221:111110
Wang H, Lehane B, Bransby M, Wang L, Hong Y (2022) Field and numerical study of the lateral response of rigid piles in sand. Acta Geotech 17(12):5573–5584
Wang S, Teng Y, Perdikaris P (2021) Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM J Sci Comput 43(5):A3055–A3081
Wang T, Altabey WA, Noori M, Ghiasi R (2020) A deep learning based approach for response prediction of beam-like structures. Struct Durab Health Monit 14(4):315
Wu D, Broms BB, Choa V (1998) Design of laterally loaded piles in cohesive soils using py curves. Soils Found 38(2):17–26
Xu G, Wu H-Z, Shi Y-Q (2016) Structural design of convolutional neural networks for steganalysis. IEEE Signal Process Lett 23(5):708–712
Yan C, Vescovini R, Dozio L (2022) A framework based on physics-informed neural networks and extreme learning for the analysis of composite structures. Comput Struct 265:106761
Zhang N, Zhang N, Zheng Q, Xu Y-S (2022) Real-time prediction of shield moving trajectory during tunnelling using GRU deep neural network. Acta Geotech 17(4):1167–1182
Zhang R, Liu Y, Sun H (2020) Physics-informed multi-LSTM networks for metamodeling of nonlinear structures. Comput Methods Appl Mech Eng 369:113226
Zhang W, Li H, Tang L, Gu X, Wang L, Wang L (2022) Displacement prediction of Jiuxianping landslide using gated recurrent unit (GRU) networks. Acta Geotech 17(4):1367–1382
Zhang X, Tang L, Ling X, Chan A (2020) Critical buckling load of pile in liquefied soil. Soil Dyn Earthq Eng 135:106197
Zhou J, Dai Y, Huang S, Armaghani DJ, Qiu Y (2022) Proposing several hybrid SSA—machine learning techniques for estimating rock cuttability by conical pick with relieved cutting modes. Acta Geotech 18(3):1431–1446
Zio E (2013) Monte Carlo simulation: the method. Springer, Berlin
Zobeiry N, Humfeld KD (2021) A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications. Eng Appl Artif Intell 101:104232
Acknowledgements
The work described in this paper was partially supported by Grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU/21E/15203121), and a Grant from the National Natural Science Foundation of China (No. 52008410). This work is also partially supported by a grant (BBTH) from Hong Kong Branch of National Rail Transit Electrification and Automation Engineering Technology Research Center.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ouyang, W., Li, G., Chen, L. et al. Machine learning-based soil–structure interaction analysis of laterally loaded piles through physics-informed neural networks. Acta Geotech. (2024). https://doi.org/10.1007/s11440-023-02179-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11440-023-02179-7