A New Approach for Interval Dynamic Analysis of Train-Bridge System Based on Bayesian Optimization

A train-bridge system (TBS) is inevitably subjected to parameter uncertainty, which leads to variability in its dynamic responses. In practice, it is difficult to characterize parameter uncertainty using precise probability density functions due to lack of sufficient statistical information. In such situations, uncertain parameters are usually modeled as uncertain-but-bounded parameters; this is also known as interval uncertainty. This paper aims to determine the dynamic response bounds of a TBS subjected to interval uncertainty. In mathematics, estimation of dynamic response bounds can be pursued in the context of optimization, that is, the minimization or maximization of an objective function. The solver in this context shares common features of a black-box function, such as high computational cost and no closed-form solution. In view of this, the present study proposes an efficient Bayesian optimization approach for estimating the dynamic response bounds of a TBS. Specifically, a Bayesian modeling approach employing a Gaussian process prior is proposed to replace the current expensive-to-run original model solver, along with an acquisition function that trades off exploration and exploitation of the search space. By doing so, the optimization of a complex, intractable black-box function is converted to the maximization of a computationally efficient acquisition function that has a closed-form expression and is differentiable. Two test functions are provided in order to demonstrate the applicability of the proposed Bayesian optimization methodology for finding the global minimum. It is demonstrated that the Bayesian optimization methodology is efficient and effective in solving the optimization problem with a limited number of function evaluations. Next, the proposed Bayesian optimization approach is utilized for interval dynamic analysis (IDA) of the TBS. The computational accuracy and efficiency of the proposed method is compared with a direct Monte Carlo simulation (MCS) estimator, which is used as a reference solution because of its generality, robustness, and easy implementation. The comparison results show that the proposed Bayesian optimization method is feasible and reliable for IDA of the TBS in terms of both computational accuracy and efficiency. Last, the influence of the interval change ratios of the system parameters on dynamic responses is investigated. The results reveal that an increase in the parameter uncertainty level results in a higher uncertainty bound on the dynamic responses.