Solving Bayesian Inverse Problems Using Gaussian Process Regression with Goal-Oriented Active Learning

Sequential design focuses on designing computer experiments with limited computational budgets. It aims to create efficient surrogate models to replace complex computer codes. Some sequential design strategies can be understood within the Stepwise Uncertainty Reduction (SUR) framework. In the SUR framework, each new design point is chosen by minimizing the expectation of a metric of uncertainty with respect to the yet unknown new data point. These methods offer an accessible framework for sequential experiment design, including almost sure convergence for common uncertainty functionals. This article introduces two goal-oriented strategies tailored for surrogate models used within a Bayesian inverse problem. The Constraint Set Query (CSQ) strategy is adapted from Maximum Mean Squared Error (MMSE) designs, where the search space is constrained in a ball for the Mahalanobis distance around the maximum a posteriori. The second, known as the IP-SUR (Inverse Problem SUR) strategy, uses a posterior-weighted integrated mean squared prediction error as the uncertainty metric and is derived from SUR methods. It is tractable for Gaussian process surrogates and comes with a theoretical guarantee for the almost sure convergence of the uncertainty functional. In various test cases, both strategies are shown to outperform standard goal-oriented designs for fine-tuning surrogate models.