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.

翻译：我们针对由大规模偏微分方程控制的高维非线性贝叶斯反问题，研究了最优实验设计问题。在贝叶斯最优实验设计的最优性准则方面，我们同时考虑了期望信息增益以及涉及参数-可观测映射及其导数计算的汇总统计量（包括信息矩阵的迹与行列式）。然而，当偏微分方程求解代价极高、待估参数维度较大且优化问题具有组合性、高维性与非凸性时，上述准则的计算与优化将变得难以实现。为克服这些挑战，我们构建了一个精确、可扩展且高效的计算框架，用于加速贝叶斯最优实验设计的求解。该框架基于导数信息驱动的神经算子代理模型，结合适当的降维技术与改进型交换贪婪算法。与高保真有限元近似相比，我们验证了该神经算子代理模型在计算参数-可观测映射及最优性准则方面具有高精度，同时证明了所提方法在参数维度增加时具有良好的可扩展性。此外，针对一个含数万参数的三维偏微分方程实例（包含代理模型的在线评估与离线构建成本），该方法实现了超过1000倍的高保真贝叶斯最优实验设计解算加速比。