Neural networks in stochastic mechanics. (English) Zbl 1043.74053
From the summary: We present the state-of-the-art of the application of neural networks in stochastic mechanics. The use of these artificial intelligence numerical devices is almost exclusively carried out in combination with Monte Carlo simulation for calculating the probability distributions of response variables, specific failure probabilities or statistical quantities. To that purpose the neural networks are trained with a few samples obtained by conventional Monte Carlo techniques, and used henceforth to obtain the responses for the rest of samples. The advantage of this approach over standard Monte Carlo techniques lies in fast computation of output samples which is characteristic of neural networks in comparison to lengthy calculation required by finite element solvers. The paper considers this combined method as applied to three categories of stochastic mechanics problems, namely those modelled with random variables, random fields, and random processes.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74H50 | Random vibrations in dynamical problems in solid mechanics |
| 74K99 | Thin bodies, structures |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
Keywords:
Monte Carlo methodSoftware:
itsmrReferences:
| [1] | Atkinson, K. E., The Numerical Solution of Integral Equations of the Second Kind (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0899.65077 |
| [2] | Ayyub, B.; McCuen, R. H.; Sundararajan, C., Simulation-based reliability methods, Probabilistic Structural Mechanics Handbook (1995), New York: Chapman and Hall, New York |
| [3] | Bender, A. M.; Orszag, S. A., Advanced Mathematical Methods for Scientific and Engineers (1978), New York: McGraw-Hill, New York · Zbl 0417.34001 |
| [4] | Binh, T. T.; Korn, U.; Reusch, B., Scalar Optimization with Linear and Nonlinear Constraints using Erolution Strategies, Lecture Notes in Computer Science (1997), Berlin: Springer Verlag, Berlin |
| [5] | Box, G. E.P.; Draper, N. R., Empirical Model Building and Response Surfaces (1987), New York: John Wiley and Sons, New York · Zbl 0614.62104 |
| [6] | Brockwell, P. J.; Davies, R. A., Introduction to Time Series and Forecasting (1996), New York: Springer Verlag, New York · Zbl 0868.62067 |
| [7] | Cabral, S. V.S.; Katafygiotis, L. S.; Corotis, R. B.; Schuëller, G. I.; Shinozuka, M., Neural network based response surface method and adaptive importance sampling for reliability analysis of large structural systems, Structural Safety and Reliability-ICOSSAR 01, 46-46 (2001), Lisse, The Netherlands: A.A. Balkema Publishers, Lisse, The Netherlands |
| [8] | Chapman, O. J.; Crossland, A. D.; Sundararajan, C., Neural networks in probabilistic structural mechanics, Probabilistic Structural Mechanics Handbook (1995), New York: Chapman and Hall, New York |
| [9] | Chen, H.; Qi, G.; Yang, J.; Amini, F., Neural network for dynamic model identification, Journal of Engineering Mechanics, 121, 1377-1381 (1995) · doi:10.1061/(ASCE)0733-9399(1995)121:12(1377) |
| [10] | Cichocki, A.; Unbehauen, R., Neural Networks for Optimization and Signal Processing (1993), Chichester: John Wiley and Sons, Chichester · Zbl 0824.68101 |
| [11] | Cressie, N. A.C., Statistics for Spatial Data (1993), New York: John Wiley and Sons, New York |
| [12] | Ditlevsen, O.; Madsen, H. O., Structural Reliability Methods (1999), Chichester: John Wiley and Sons, Chichester |
| [13] | Faravelli, L., Response-surface approach for reliability analysis, Journal of Engineering Mechanics, 115, 2763-2781 (1989) · doi:10.1061/(ASCE)0733-9399(1989)115:12(2763) |
| [14] | Fenton, G. A.; Vanmarcke, E. H., Simulation of random fields via local average subdivision, Journal of Engineering Mechanics, 116, 1733-1749 (1990) · doi:10.1061/(ASCE)0733-9399(1990)116:8(1733) |
| [15] | Ghanem, R. G.; Spanos, P. D., Stochastic Finite Elements: A Spectral Approach (1991), New York: Springer Verlag, New York · Zbl 0722.73080 |
| [16] | Ghanem, R. G.; Spanos, P. D., Spectral techniques for stochastic finite elements, Archives of Computational Methods in Engineering, 4, 63-100 (1997) |
| [17] | Hartman, E. J.; Keeler, J. D.; Kowalski, J. M., Layered neural networks with Gaussian hidden units as universal approximations, Neural Computation, 2, 210-215 (1990) · doi:10.1162/neco.1990.2.2.210 |
| [18] | Hernández, D. B., Lectures on Probability and Second Order Random Fields (1995), Singapore: World Scientific, Singapore · Zbl 0838.60034 |
| [19] | Hornik, K. M.; Stinchcommbe, M.; White, H., Multilayer feedforward networks are universal approximators, Neural Networks, 2, 359-366 (1989) · Zbl 1383.92015 · doi:10.1016/0893-6080(89)90020-8 |
| [20] | Hurtado, J.E. (2001a), “Stochastic finite element analysis using point estimates”Proceedings of the European Conference on Computational Mechanics ECCM-01 (CD-ROM), Cracow, Poland. · Zbl 1017.74072 |
| [21] | Hurtado, J.E. (2001b), “Analysis of one-dimensional stochastic finite elements using neural networks,”Probabilistic Engineering Mechanics (to appear). |
| [22] | Hurtado, J. E.; Barbat, A. H., Monte Carlo techniques in computational stochastic mechanics, Archives of Computational Methods in Engineering, 5, 3-30 (1998) |
| [23] | Hurtado, J.E. and Alvarez, D.A. (2000), “Reliability assessment of structural systems using neural networks,” InProceedings of the European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS-2000 (CD-ROM), ECCOMAS, Barcelona. |
| [24] | Hurtado, J.E. and Alvarez, D.A. (2001), “Neural network-based reliability analysis: A comparative study”,Computational Methods in Applied Mechanics and Engineering (to appear). · Zbl 1016.74044 |
| [25] | Hurtado, J. E.; Alvarez, D. A.; Barbat, A. H.; Schuëller, G. I.; Spanos, P. D., Monte Carlo analysis of structural systems using neural networks, Monte Carlo Simulation, 265-271 (2001), Lisse, The Netherlands: A.A. Balkema, Publishers, Lisse, The Netherlands |
| [26] | Hurtado, J. E.; Barbat, A. H.; Corotis, R. B.; Schuëller, G. I.; Shinozuka, M., Analysis of stochastic finite elements via neural classifiers, Structural Safety and Reliability-ICOSSAR 01, 132-132 (2001), Lisse, The Netherlands: A.A. Balkema Publishers, Lisse, The Netherlands |
| [27] | Hurtado, J. E.; Londoño, J. M.; Meza, M. A., On the applicability of neural networks for soil dynamics amplification analysis, Soil Dynamics and Earthquake Engineering, 21, 619-631 (2001) · doi:10.1016/S0267-7261(01)00037-9 |
| [28] | Idriss, I. M.; Sun, J., User’s Manual for SHAKE-91 (1992), Richmond: University of California, Richmond |
| [29] | Ishibuchi, H.; Tanaka, H., Fuzzy regression analysis using neural networks, Fuzzy Sets and Systems, 50, 257-265 (1987) · doi:10.1016/0165-0114(92)90224-R |
| [30] | Jang, J. S.R.; Sun, C. T.; Mizutani, E., Neuro-Fuzzy and Soft Computing (1997), Upper Saddle River: Prentice Hall, Upper Saddle River |
| [31] | Kleiber, M.; Hien, T. D., The Stochastic Finite Element Method (1992), Chichester: John Wiley and Sons, Chichester · Zbl 0902.73004 |
| [32] | Lin, C. T.; Lee, C. S.G., Neural Fuzzy Systems (1996), Upper Saddle River: Prentice Hall, Upper Saddle River |
| [33] | Luo, F. L.; Unbehauen, R., Applied Neural Networks for Signal Processing (1998), Cambridge: Cambridge University Press, Cambridge |
| [34] | Mantoglu, A., Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method, Mathematical Geology, 19, 129-149 (1987) |
| [35] | McCulloch, W. S.; Pitts, W., A logical calculus of ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 5, 115-133 (1943) · Zbl 0063.03860 · doi:10.1007/BF02478259 |
| [36] | Melchers, R. E., Structural Reliability: Analysis and Prediction (1999), Chichester: John Wiley and Sons, Chichester |
| [37] | Noble, B., Methods based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (1958), New York: Chelsea Publishing Company, New York · Zbl 0082.32101 |
| [38] | Papadrakakis, M., Papadopoulos, V., Lagaros, N.D. (1996), “Structural reliability analysis of elastic-plastic structures using neural networks and Monte Carlo simulation”,Computer Methods in Applied Mechanics and Engineering, 145-163. · Zbl 0893.73079 |
| [39] | Pollock, D. S.G., A Handbook of Time-Series analysis, Signal Processing and Dynamics (1999), San Diego: Academic Press, San Diego · Zbl 0953.62090 |
| [40] | Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical Recipes in FORTRAN (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0778.65002 |
| [41] | Resnikoff, H. L.; Wells, R. O., Wavelet Analysis (1998), New York: Springer Verlag, New York · Zbl 0922.42020 |
| [42] | Ripley, B. D., Pattern Recognition and Neural Networks (1996), Cambridge: Cambridge University Press, Cambridge · Zbl 0853.62046 |
| [43] | Rubinstein, R. Y., Monte Carlo Optimization, Simulation and Sensitivity of Queuing Networks (1992), Malabar, FL: Krieger Publishing Company, Malabar, FL |
| [44] | Saraiva, J. M.F.; Ebecken, N. F.F., Application of neural networks in structural reliability analysis, Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 14, 167-180 (1998) |
| [45] | Sasaki, T.; Corotis, R. B.; Schuëller, G. I.; Shinozuka, M., A neural network-based response surface approach for computing failure probabilities, Structural Safety and Reliability-ICOSSAR 01, 257-257 (2001), Lisse, The Netherlands: A.A. Balkema Publishers, Lisse, The Netherlands |
| [46] | Schuëller, G. I.; Stix, R., A critical appraisa of methods to determine failure probabilities, Structural Safety, 4, 293-309 (1987) · doi:10.1016/0167-4730(87)90004-X |
| [47] | Shinozuka, M.; Sato, Y., Simulation of nonstationary random processes, Journal of the Engineering Mechanics Division ASCE, 93, 11-40 (1967) |
| [48] | Shinozuka, M.; Nomoto, T., Response Variability due to Spatial Randomness of Material Properties (1980), New York: Dept. of Civil Engineering, Columbia University, New York |
| [49] | Shinozuka, M.; Schuëller, G. I.; Shinozuka, M., Stochastic fields and their digital simulation, Stochastic Methods in Structural Dynamics (1987), Dordrecht: Martinus Nijhoff Publishers, Dordrecht · Zbl 0653.73037 |
| [50] | Soong, T. T.; Grigoriu, M., Random Vibration of Mechanical and Structural Systems (1993), Upper Saddler River: Prentice Hall, Upper Saddler River · Zbl 0788.73005 |
| [51] | Szidarovszky, F.; Bahill, A. T., Linear System Theory (1992), Boca Ratón: CRC Press, Boca Ratón · Zbl 0770.93001 |
| [52] | Tajimi, H., A statistical method of determining the maximum response of a building structure during an earthquake, Proc. of the Second World Conference on Earthquake Engineering, 2, 781-797 (1960) |
| [53] | Theodoridis, S.; Koutroumbas, K., Pattern Recognition (1999), London: Academic Press, London |
| [54] | Wan, E.A. (1993),Finite Impulse Response Neural Networks With Applications in Time Series Prediction, Ph.D. Thesis, Stanford University. |
| [55] | Wacszczyszyn, Z., Neural Networks in the Analysis and Design of Structures (1999), Wien: Springer Verlag, Wien |
| [56] | Yagawa, G.; Okuda, H., Neural networks in computational mechanics, Archives of Computational Methods in Engineering, 3, 435-512 (1996) |
| [57] | Youla, D. C., The solution of a homogeneous Wiener-Hopf Integral equation occurring in the expansion of second-order random functions, IRE Transactions on Information Theory, September, 187-193 (1957) · doi:10.1109/TIT.1957.1057414 |
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