LazyDINO：通过结构利用与代理驱动的测度传输实现快速、可扩展且高效摊销的贝叶斯反演
LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport
AI总结：
LazyDINO 利用导数感知神经代理与惰性映射变分推断，在昂贵 PtO 映射的高维贝叶斯反演中实现高效摊销，以极低离线样本成本超越拉普拉斯近似等现有方法。
摘要：
我们提出 LazyDINO，一种传输映射变分推断方法，用于对具有昂贵参数到观测量（PtO）映射的高维非线性贝叶斯反演问题进行快速、可扩展且高效摊销的求解。我们的方法包含一个离线阶段，在该阶段中，我们利用 PtO 映射及其雅可比矩阵的联合样本，构建 PtO 映射的导数感知神经代理模型。在在线阶段，当给定观测数据时，我们利用代理驱动训练一个惰性映射 [Brennan et al., NeurIPS, (2020)]，即一种具有低维非线性的结构利用传输映射，来寻求快速的后验近似。训练后的惰性映射随后产生近似的后验样本或密度评估。我们的代理构建针对使用惰性映射变分推断的摊销贝叶斯反演进行了优化。我们证明：(i) 基于导数的降维基架构 [O'Leary-Roseberry et al., Comput. Methods Appl. Mech. Eng., 388 (2022)] 最小化了代理后验近似中期望误差的上界，并且 (ii) 导数感知训练公式 [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] 最小化了由代理驱动的传输映射优化引起的期望误差。我们的数值结果表明，LazyDINO 在贝叶斯反演的成本摊销方面具有很高的效率。与通过条件传输的基于仿真的摊销推断以及传统的代理驱动传输相比，我们观察到在实现精确后验近似时，离线成本降低了一到两个数量级。特别是，LazyDINO 在使用少于 1000 个离线样本的情况下始终优于拉普拉斯近似，而其他摊销推断方法在 16,000 个离线样本时仍然表现挣扎，有时甚至失败。
英文摘要：
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.
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