Data-Driven Prior Construction in Hilbert Spaces for Bayesian Optimization

We propose a variant of Bayesian optimisation in which probability distributions are constructed using uncertainty quantification (UQ) techniques. These distributions serve as prior knowledge of the search space. They are introduced into the acquisition function to provide an enhanced guide to selecting enrichment points. Different versions of the proposed method are analysed, depending on the type of distribution used (normal, lognormal, etc.), and compared with traditional Bayesian optimisation using test functions. The results demonstrate competitive performance with selective improvements depending on problem structure, achieving faster convergence in specific cases. As an application, we consider a structural shape optimisation problem. The initial geometry is that of an L-shaped plate, and the aim is to minimize the volume subject to a horizontal displacement constraint expressed as a penalty. In this case, our approach acts as an initial step to quickly identify a promising region, while efficiently training the underlying surrogate model in said regions. Next, a gradient-based optimisation process adjusts the final design, harnessing the trained surrogate, leading to a volume reduction of more than 30% while satisfying the displacement constraint and incurring no new functional evaluations of the true and expensive objective.