We develop a new computational framework to solve sequential Bayesian optimal experimental design (SBOED) problems constrained by large-scale partial differential equations with infinite-dimensional random parameters. We propose an adaptive terminal formulation of the optimality criteria for SBOED to achieve adaptive global optimality. We also establish an equivalent optimization formulation to achieve computational simplicity enabled by Laplace and low-rank approximations of the posterior. To accelerate the solution of the SBOED problem, we develop a derivative-informed latent attention neural operator (LANO), a new neural network surrogate model that leverages (1) derivative-informed dimension reduction for latent encoding, (2) an attention mechanism to capture the dynamics in the latent space, (3) an efficient training in the latent space augmented by projected Jacobian, which collectively leads to an efficient, accurate, and scalable surrogate in computing not only the parameter-to-observable (PtO) maps but also their Jacobians. We further develop the formulation for the computation of the MAP points, the eigenpairs, and the sampling from posterior by LANO in the reduced spaces and use these computations to solve the SBOED problem. We demonstrate the superior accuracy of LANO compared to two other neural architectures and the high accuracy of LANO compared to the finite element method (FEM) for the computation of MAP points and eigenvalues in solving the SBOED problem with application to the experimental design of the time to take MRI images in monitoring tumor growth. We show that the proposed computational framework achieves an amortized $180\times$ speedup.

翻译：本文提出了一种新的计算框架，用于求解受大规模偏微分方程约束且具有无限维随机参数的序列贝叶斯最优实验设计问题。我们针对SBOED提出了一种自适应终端的优化准则表述，以实现自适应的全局最优性。同时，通过后验分布的拉普拉斯近似与低秩近似，我们建立了一种等效的优化表述以实现计算简化。为加速SBOED问题的求解，我们开发了一种导数信息潜在注意力神经算子——这是一种新型神经网络代理模型，其优势在于：（1）利用导数信息降维技术进行潜在编码；（2）通过注意力机制捕捉潜在空间中的动态特性；（3）结合投影雅可比矩阵在潜在空间中进行高效训练。这些特性共同使得该代理模型在计算参数到观测量的映射及其雅可比矩阵时，兼具高效性、精确性与可扩展性。我们进一步构建了在降维空间中计算最大后验点、特征对及后验采样的LANO实现方案，并利用这些计算求解SBOED问题。通过MRI监测肿瘤生长成像时间实验设计的应用案例，我们验证了LANO相较于其他两种神经架构的优越精度，以及在求解SBOED问题时计算最大后验点和特征值方面相较于有限元方法的高精度。实验表明，所提出的计算框架实现了平均180倍的加速比。