  Black-box optimization is a powerful approach for discovering global optima
in noisy and expensive black-box functions, a problem widely encountered in
real-world scenarios. Recently, there has been a growing interest in leveraging
domain knowledge to enhance the efficacy of machine learning methods. Partial
Differential Equations (PDEs) often provide an effective means for elucidating
the fundamental principles governing the black-box functions. In this paper, we
propose PINN-BO, a black-box optimization algorithm employing Physics-Informed
Neural Networks that integrates the knowledge from Partial Differential
Equations (PDEs) to improve the sample efficiency of the optimization. We
analyze the theoretical behavior of our algorithm in terms of regret bound
using advances in NTK theory and prove that the use of the PDE alongside the
black-box function evaluations, PINN-BO leads to a tighter regret bound. We
perform several experiments on a variety of optimization tasks and show that
our algorithm is more sample-efficient compared to existing methods.
