基于导数信息潜在注意力神经算子的序列无限维贝叶斯最优实验设计
Sequential infinite-dimensional Bayesian optimal experimental design with derivative-informed latent attention neural operator
AI总结：
本文提出导数信息潜在注意力神经算子LANO，结合Laplace与低秩近似，高效求解无限维随机参数约束下的序列贝叶斯最优实验设计问题，并在MRI肿瘤监测实验中实现180倍加速。
摘要：
我们开发了一个新的计算框架，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（SBOED）问题。我们提出了SBOED最优性准则的自适应终端公式，以实现自适应全局最优性。我们还建立了一个等价的优化公式，借助后验的Laplace近似和低秩近似实现计算上的简化。为了加速SBOED问题的求解，我们开发了导数信息潜在注意力神经算子（LANO），这是一种新的神经网络代理模型，它利用：（1）用于潜在编码的导数信息降维；（2）用于捕捉潜在空间中动态的注意力机制；（3）在由投影Jacobian增强的潜在空间中进行高效训练，这些共同构成一个高效、准确且可扩展的代理模型，不仅能够计算参数到观测量（PtO）映射，还能计算其Jacobian。我们进一步建立了在降维空间中通过LANO计算最大后验概率（MAP）点、特征对以及从后验中采样的公式，并利用这些计算来求解SBOED问题。我们展示了LANO相较于另外两种神经架构的优越精度，以及在求解SBOED问题时LANO相较于有限元方法（FEM）在计算MAP点和特征值方面的高精度，并将其应用于监测肿瘤生长时MRI成像时间的实验设计。我们表明，所提出的计算框架实现了摊销180倍的加速。
英文摘要：
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.
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