Microbiomes interact dynamically with their environment to perform exploitable functions such as production of valuable metabolites and degradation of toxic metabolites for a wide range of applications in human health, agriculture, and environmental cleanup. Developing computational models to predict the key bacterial species and environmental factors to build and optimize such functions are crucial to accelerate microbial community engineering. However, there is an unknown web of interactions that determine the highly complex and dynamic behavior of these systems, which precludes the development of models based on known mechanisms. By contrast, entirely data-driven machine learning models can produce physically unrealistic predictions and often require significant amounts of experimental data to learn system behavior. We develop a physically-constrained recurrent neural network that preserves model flexibility but is constrained to produce physically consistent predictions and show that it can outperform existing machine learning methods in the prediction of certain experimentally measured species abundance and metabolite concentrations. Further, we present a closed-loop, Bayesian experimental design algorithm to guide data collection by selecting experimental conditions that simultaneously maximize information gain and target microbial community functions. Using a bioreactor case study, we demonstrate how the proposed framework can be used to efficiently navigate a large design space to identify optimal operating conditions. The proposed methodology offers a flexible machine learning approach specifically tailored to optimize microbiome target functions through the sequential design of informative experiments that seek to explore and exploit community functions.