We consider optimal experimental design (OED) for nonlinear Bayesian inverse problems governed by large-scale partial differential equations (PDEs). For the optimality criteria of Bayesian OED, we consider both expected information gain and summary statistics including the trace and determinant of the information matrix that involves the evaluation of the parameter-to-observable (PtO) map and its derivatives. However, it is prohibitive to compute and optimize these criteria when the PDEs are very expensive to solve, the parameters to estimate are high-dimensional, and the optimization problem is combinatorial, high-dimensional, and non-convex. To address these challenges, we develop an accurate, scalable, and efficient computational framework to accelerate the solution of Bayesian OED. In particular, the framework is developed based on derivative-informed neural operator (DINO) surrogates with proper dimension reduction techniques and a modified swapping greedy algorithm. We demonstrate the high accuracy of the DINO surrogates in the computation of the PtO map and the optimality criteria compared to high-fidelity finite element approximations. We also show that the proposed method is scalable with increasing parameter dimensions. Moreover, we demonstrate that it achieves high efficiency with over 1000X speedup compared to a high-fidelity Bayesian OED solution for a three-dimensional PDE example with tens of thousands of parameters, including both online evaluation and offline construction costs of the surrogates.

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