Active learning for multi-objective optimization of processes and energy systems

In superstructure optimization of processes and energy systems, the design space is defined as the combination of unit considerations, process conditions and model parameters that might be subjected to uncertainty. Most of the time, decision makers are not looking for a single best solution, but rather are interested in analyzing a set of Pareto-optimal superstructure designs. The generation of Pareto-optimal solutions is computationally expensive, especially if nonlinear process evaluations or simulation is required. In our approach, we address the question of how to efficiently generate Paretooptimal sets of solutions by applying machine learning concepts. Using the criteria of Pareto-optimality to evaluate the performance of a set of design space variables and the corresponding solution, we train our algorithm on predicting if a solution is belonging to the Pareto frontier. Following the approach presented by Zuluaga et al., (2016) and applied to the design of materials by Jablonka et al., (2021), an adaptive learning concept is used to systematically identify the next best function evaluation to improve the confidence of the Pareto-frontier definition. Gaussian process surrogate models provide a prediction of the mean and the standard deviation of the relevant objectives. Design points with high probability to of being in the Pareto-optimal domain are evaluated by the original model, increasing the confidence with which the Pareto front is predicted. Simultaneously, the design space is continuously reduced by discarding the design points for which the probability of being in the set of Pareto-optimal solutions is low. The procedure is stopped when all points are labeled as Pareto-optimal or discarded. The algorithm is applied to the design of a utility superstructure for an industrial energy system. Our algorithm is compared and benchmarked with quasi random sampling of the design space.