We present LazyDINO, a transport map variational inference method for fast, scalable, and efficiently amortized solutions of high-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable (PtO) maps. Our method consists of an offline phase in which we construct a derivative-informed neural surrogate of the PtO map using joint samples of the PtO map and its Jacobian. During the online phase, when given observational data, we seek rapid posterior approximation using surrogate-driven training of a lazy map [Brennan et al., NeurIPS, (2020)], i.e., a structure-exploiting transport map with low-dimensional nonlinearity. The trained lazy map then produces approximate posterior samples or density evaluations. Our surrogate construction is optimized for amortized Bayesian inversion using lazy map variational inference. We show that (i) the derivative-based reduced basis architecture [O'Leary-Roseberry et al., Comput. Methods Appl. Mech. Eng., 388 (2022)] minimizes the upper bound on the expected error in surrogate posterior approximation, and (ii) the derivative-informed training formulation [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] minimizes the expected error due to surrogate-driven transport map optimization. Our numerical results demonstrate that LazyDINO is highly efficient in cost amortization for Bayesian inversion. We observe one to two orders of magnitude reduction of offline cost for accurate posterior approximation, compared to simulation-based amortized inference via conditional transport and conventional surrogate-driven transport. In particular, LazyDINO outperforms Laplace approximation consistently using fewer than 1000 offline samples, while other amortized inference methods struggle and sometimes fail at 16,000 offline samples.

翻译：本文提出LazyDINO，一种基于传输映射的变分推断方法，用于解决具有昂贵参数-观测（PtO）映射的高维非线性贝叶斯反演问题，该方法具有快速、可扩展和高效分摊的特性。我们的方法包含离线与在线两个阶段：在离线阶段，我们利用PtO映射及其雅可比矩阵的联合样本构建PtO映射的导数信息神经代理模型；在线阶段，当给定观测数据时，我们通过代理驱动的惰性映射训练[Brennan等人，NeurIPS，(2020)]实现快速后验近似，该惰性映射是一种具有低维非线性特征的结构利用型传输映射。训练后的惰性映射可生成近似后验样本或密度估计值。我们的代理构建专为基于惰性映射变分推断的分摊式贝叶斯反演而优化。我们证明：（i）基于导数的降基架构[O'Leary-Roseberry等人，Comput. Methods Appl. Mech. Eng., 388 (2022)]能够最小化代理后验近似期望误差的上界；（ii）导数信息训练框架[O'Leary-Roseberry等人，J. Comput. Phys., 496 (2024)]可最小化由代理驱动传输映射优化引起的期望误差。数值结果表明，LazyDINO在贝叶斯反演的成本分摊方面具有显著优势。与基于仿真的条件传输分摊推断及传统代理驱动传输方法相比，为达到精确后验近似所需的离线计算成本降低了一到两个数量级。特别值得注意的是，在离线样本数少于1000的情况下，LazyDINO始终优于拉普拉斯近似方法，而其他分摊推断方法即便使用16000个离线样本仍表现不佳甚至失效。