Surrogate-aided Bayesian calibration with adaptive learning strategies

Bayesian inference or calibration for engineering systems requires sampling from the posterior distribution of the system model parameters. For applications with complex numerical models, the computational burden for this sampling, requiring a large number of calls to that model, can be prohibitive. For alleviating this burden, this work investigates an adaptive surrogate model implementation for approximating the numerical model predictions within the process of the Bayesian inference. Gaussian Process (GP) regression is adopted as surrogate modelling technique, while the formulation is embedded within a sequential Monte Carlo (MC) approach, using a series of intermediate auxiliary densities to efficiently derive samples from the posterior target density. The adaptive training of the GP is established by iteratively adding training points to inform the target posterior density approximation. At each iteration, a GP model is constructed using the current set of training points to approximate the system response and, consequently, the posterior distribution. This approximation is then used to facilitate sampling from the posterior. The iterative procedure terminates once the posterior density approximations from successive iterations exhibit sufficient similarity. Convergence is assessed using a combination of criteria, including the stability of the most probable parameter estimates and the similarity of the approximated posterior densities. If convergence is not established, the surrogate model is refined through an adaptive design of computer experiments (DoE) that uses the weighted integrated mean squared error as acquisition function. Through proper selection of the weights, exploitation and exploration strategies are promoted to improve, respectively, efficiency and robustness for the GP approximation. This weight selection combines information from both the target posterior density and the intermediate, auxiliary densities. Strategies are examined for improving computational efficiency for the DoE optimization and for the sequential Monte Carlo sampling once the surrogate model has been updated. Three illustrative examples are examined, considering both static and dynamic Bayesian calibration problems. Results demonstrate the computational efficiency offered through the adaptive GP formulation and provide insights for the best adaptation strategies and for the appropriate selection of response quantities to be adopted as outputs within the GP implementation.