## 6 Conclusions  
This research presents a computational study of different surrogate strategies to account for uncertainty propagation in nonlinear MDOF structural systems. A nine-DOF structure is selected for the numerical study for generating the training and validation samples. MIMO-NARX modeling is established to capture the nonlinear dynamic behavior in MOR-reduced coordinates using the results from dynamic time history simulations. The NARX coefficients are surrogated using meta-models including Kriging, PCE, and APC with uncertain variables of ground motion and structural mass. The MCS results show good fitness of responses between MIMO-NARX models and the traditional dynamic simulation using the integration algorithm. It is worth noting that the absolute velocity responses are used in NARX training for improving the nonlinear capture effectiveness. The accuracy of the four reduced coordinates decreases with the increase of frequency due to difficulty in capturing high-frequency nonlinearity. Good performances are observed for all three meta-models in absolute velocity response prediction when compared with MCS. Generally, a slight decrease in accuracy is observed in displacement response prediction obtained by integrating the velocity responses due to the drift errors caused by error accumulation at the end of time history analysis. The accuracy of both inter-story velocity and displacement responses show different degrees of decrease compared with absolute velocity and displacement responses. The integral and accumulation drift errors do not necessarily affect the meta-model predicted peak displacement responses, which are more concerned with structural safety and serviceability. From a statistical point of view, all three meta-models surrogated MIMO-NARX models have nearly similar accuracy for most of the three hundred MCS validation cases except for some outliers. PCE- and APC- are shown to be superior to Kriging-NARX in terms of the counts of good fitness according to some statistical information. Considering the advantages of APC over PCE in the scope of application, APC-NARX is best suitable to replace the original model in a large number of MCS. It is illustrated that the MIMO-NARX modeling with surrogated coefficients by meta-models provides an effective and efficient tool for uncertainty quantification of nonlinear structures in earthquake engineering. This study takes a nine-DOF structure as a proof of concept to verify the effectiveness of the proposed method. The nonlinear performance of the Bouc-Wen model in the MDOF  
Figure 20: Relationship between the mean ductility ratio and the mean fitness (300 samples). (**a**) Outputs in reduced coordinates; (**b**) Absolute velocity responses; (**c**) Absolute displacement responses; (**d**) Inter-story velocity responses; (**e**) Inter-story displacement responses. system changes with different stochastic seismic excitations and structural properties. The choice of the basic functions in MIMO-NARX, which depends on different nonlinear types of structural models, remains to be studied in the future. It should be noticed that the accuracy of the MIMO-NARX model is affected and limited by the intensity of nonlinearity. Although the accuracy of the three meta-model-surrogated MIMO-NARX models is significantly decreased with the intensity of structural ductility increase, it still can be demonstrated that the Kriging/PCE/APC-NARX strategy is basically accurate for most of the MCS validation cases within a certain nonlinearity range. Future research may focus on finding more suitable NARX basis functions to eliminate the adverse effect of high nonlinearity on accuracy. Conceptualization, C.C. ; methodology, C.C., X.G. and M.C. ; software, X.G. and M.C. ; validation, M.C. ; formal analysis, M.C. ; investigation, X.G. and M.C. ; resources, X.G. and M.C. ; data curation, M.C. ; writing--original draft preparation, M.C. ; writing--review and editing, C.C. ; visualization, M.C. ; supervision, T.G. ; project administration, T.G. ; funding acquisition, T.G. and W.X.