Bayesian optimization (BO) is a powerful tool for optimizing non-convex black-box functions that are expensive and/or time-consuming to evaluate and subject to random noise in their observations. Many important real-world science and engineering problems belong to this class such as optimizing over high-fidelity computer simulations, tuning hyperparameters in machine learning algorithms, and efficient material and drug discovery. Traditionally, BO has been deployed as a purely black-box optimizer. However, this black-box approach can lead to significant performance losses, especially in high-dimensional, intricately constrained design spaces, such as those that appear in materials and molecular optimization. In most real-world applications, however, only a portion of the model is unknown, suggesting that we might (substantially) improve performance by “peeking inside the black box.” In this talk, I will present an overview of advanced “physics-informed” Bayesian optimization (PIBO) methods recently developed by the Paulson Lab that selectively exploit problem structure to achieve state-of-the-art performance. Specifically, I will focus on a PIBO approach called MolDAIS, designed for molecular property optimization under small budgets. MolDAIS is built upon a conjecture that the properties of interest depend on a small subset of key molecular descriptors that can be actively learned from data using a particular type of sparsity-inducing probabilistic surrogate model. I will illustrate the effectiveness of MolDAIS on several molecule design benchmark problems as well as a real-world application to discovery of high-performance, low-cost organic electrode materials. By working closely with experimental collaborators, we used a variant of MolDAIS to find candidate materials with specific energy and cycling stability that match or surpass current state-of-the-art organic electrodes in aqueous zinc-ion batteries, while being synthesizable at a fraction of the cost.