Computer-Aided Civil and Infrastructure Engineering 25 (2010) 322–333 Recurrent Neural Networks for Uncertain Time-Dependent Structural Behavior W. Graf, ∗ S. Freitag, M. Kaliske & J.-U. Sickert Institute for Structural Analysis, Technische Universit¨ at Dresden, 01062 Dresden, Germany Abstract: In this article, an approach is introduced which permits the numerical prediction of future struc- tural responses in dependency of uncertain load pro- cesses and environmental influences. The approach is based on recurrent neural networks trained by time- dependent measurement results. Thereby, the uncertainty of the measurement results is modeled as fuzzy processes which are considered within the recurrent neural network approach. An efficient solution for network training and prediction is developed utilizing α-cuts and interval arith- metic. The capability of the approach is demonstrated by means of the prediction of the long-term structural be- havior of a reinforced concrete plate strengthened by a textile reinforced concrete layer. 1 INTRODUCTION The artificial neural network concept is adapted from the structure and the functionality of the human brain. It is a powerful tool to capture and learn functional de- pendencies in data. An overview of neural network ap- plications in civil engineering is given in Adeli (2001). Nowadays, neural networks are used in multiple engi- neering applications, for example, parameter identifi- cation (Nov ´ ak and Lehk´ y, 2006), response surface ap- proximation (Liebscher et al., 2007), lifetime prediction based on accelerated test data (Freitag et al., 2009), time series prediction (M ¨ oller and Reuter, 2007), system identification (Jiang and Adeli, 2005), damage detection (Jiang and Adeli, 2007), earthquake prediction (Adeli and Panakkat, 2009), etc. The widely used type in en- gineering applications is the multilayer perceptron net- work with feed forward architecture (see, e.g., Haykin, 1999). To consider time-dependent effects of structural behavior advanced network architectures have to be ap- ∗ To whom correspondence should be addressed. E-mail: wolfgang. graf@tu-dresden.de. plied. For example, the rate-dependency of materials is considered in Jung and Ghaboussi (2006) by means of additional input neurons in a feed forward network structure to approximate time-dependent constitutive behavior. Moreover, recurrent neural networks have been developed for temporal signal processing. Besides wavelet neural networks (see Jiang and Adeli, 2005; 2007), fully and partially recurrent neural networks (see Zell, 1996), are suitable for mapping structural pro- cesses, obtained by experiments or numerical analy- ses, onto time-dependent structural responses (see, e.g., Puscasu et al., 2009; Panakkat and Adeli, 2009). This provides a rational basis for the prediction of time- dependent effects of the structural behavior addressed here. If a structural process is observed experimentally with the help of measurement devices, it is not pos- sible to assign precise values to the observed events. That means, data uncertainty occurs which may result from scale-dependent effects, varying boundary con- ditions which are not considered, inaccuracies in the measurements, and incomplete sets of observations. Therefore, measured results are more or less charac- terized by data uncertainty that originates in impreci- sion. According to their nature, the imprecision results from epistemic uncertainty, which is regarded here. An adequate description of epistemic uncertain data suc- ceeds according to M ¨ oller and Beer (2008) by means of nonprobabilistic uncertainty models. A very popu- lar nonprobabilistic uncertainty model is the interval, which is well-applicable to describe uncertainty math- ematically if only a value range between bounds is known. A generalization and enhancement of the inter- val model by means of a gradual assignment is a fuzzy set. Thereby, the elements of an interval are assessed or weighted by membership values. In this contribu- tion, the imprecision is modeled by means of fuzzy sets. However, intervals are also regarded, as they represent in view of the numerical treatment a special case of C  2010 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/j.1467-8667.2009.00645.x