Several inverse and optimization problems involve the use of simulation models. Examples of such problems include simulation-based optimization, simulation-based (parameter) inference, identifying quasi-optimal regions in computer-aided design, etc. The simulation models represent complex real-world phenomena and therefore are often computationally expensive to evaluate. Consequently, it is desirable to minimise the number of simulations involved in solving the inverse and optimization problems. Examples of such approaches include methods that build a computationally cheaper approximation, or a surrogate model (often a machine learning model) as a replacement of the simulation model such as Bayesian optimization. Such an approach is increasingly being used in different inverse problems. The project will involve exploration of statistical sampling methods that create training sets for surrogate models in a data-efficient manner (selecting simulation locations such that the information gain for the surrogate model is maximised), along with bespoke formulations of machine learning models tailored for inverse problems.