Bayesian optimal experimental design (BOED)—including active learning, Bayesian optimization, and sensor placement—provides a probabilistic framework to maximize the expected information gain (EIG) or mutual information (MI) for uncertain parameters or quantities of interest with limited experimental data. However, evaluating the EIG remains prohibitive for largescale complex models due to the need to compute double integrals with respect to both the parameter and data distributions. In this work, we develop a fast and scalable computational framework to solve Bayesian optimal experimental design (OED) problems governed by partial differential equations (PDEs) with application to optimal sensor placement by maximizing the EIG. We (1) exploit the low-rank structure of the Jacobian of the parameter-to-observable map to extract the intrinsic low-dimensional data-informed subspace, and (2) employ a series of approximations of the EIG that reduce the number of PDE solves while retaining a high correlation with the true EIG. This allows us to propose an efficient offline–online decomposition for the optimization problem, using a new swapping greedy algorithm for both OED problems and goal-oriented linear OED problems. The offline stage dominates the cost and entails precomputing all components requiring PDE solusion. The online stage optimizes sensor placement and does not require any PDE solves. We provide a detailed error analysis with an upper bound for the approximation error in evaluating the EIG for OED and goal-oriented OED linear cases. Finally, we evaluate the EIG with a derivative-informed projected neural network (DIPNet) surrogate for parameter-to-observable maps. With this surrogate, no further PDE solves are required to solve the optimization problem. We provided an analysis of the error propagated from the DIPNet approximation to the approximation of the normalization constant and the EIG under suitable assumptions. We demonstrate the efficiency and scalability of the proposed methods for both linear inverse problems, in which one seeks to infer the initial condition for an advection–diffusion equation, and nonlinear inverse problems, in which one seeks to infer coefficients for a Poisson problem, an acoustic Helmholtz problem and an advection–diffusion–reaction problem. This dissertation is based on the following articles: A fast and scalable computational framework for large-scale and high-dimensional Bayesian optimal experimental design by Keyi Wu, Peng Chen, and Omar Ghattas [88]; An efficient method for goal-oriented linear Bayesian optimal experimental design: Application to optimal sensor placement by Keyi Wu, Peng Chen, and Omar Ghattas [89]; and Derivative-informed projected neural network for large-scale Bayesian optimal experimental design by Keyi Wu, Thomas O’Leary-Roseberry, Peng Chen, and Omar Ghattas [90]. This material is based upon work partially funded by DOE ASCR DE-SC0019303 and DESC0021239, DOD MURI FA9550-21-1-0084, and NSF DMS-2012453