变分目标用于反问题中的摊销贝叶斯推断：后验条件化的作用
Variational objectives for amortized Bayesian inference in inverse problems: The role of posterior conditioning
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中文总结 AI 辅助
本研究比较三种变分目标（VAE-KL、VAE-JS、VAE-JSWA）在反问题摊销贝叶斯推断中的表现，发现后验条件化显著影响目标选择，病态问题中JSWA更优，并建议采用几何自适应变分推断。
中文摘要 AI 辅助
变分自编码器（VAEs）为反问题中的摊销贝叶斯推断提供了一种高效方法，但后验精度可能强烈依赖于变分正则化的选择，特别是当反问题包含弱可识别参数方向时。本研究考察了三种目标：反向Kullback-Leibler公式（VAE-KL）、非对称Jensen-Shannon公式（VAE-JS）以及Jensen-Shannon-Wasserstein公式（VAE-JSWA），后者用平方2-Wasserstein距离替换反向Kullback-Leibler正则化器，同时保留前向Kullback-Leibler后验监督。使用全协方差高斯编码器和预训练的基于物理的代理模型进行摊销后验推断。在广义Fisher基中发展了局部线性-高斯分析，以刻画三种目标的方差相关梯度。这些公式首先在具有已知后验解的线性-高斯基准上进行评估，随后在非线性基于物理的反问题（包括由线性ODE控制的反问题和两个PDE约束问题）上进行测试。在良条件基准中，VAE-KL比其他公式略优，所有三种方法产生相当的后验近似，而在强病条件基准中，VAE-JSWA提供显著更低的后验误差。非线性基于物理的问题表现出类似的依赖于条件数的趋势，随着后验病条件程度的增加，基于JS的公式提供更大益处。这些结果表明后验条件化是选择变分目标的重要因素，并激励了几何自适应的变分推断用于贝叶斯反问题。
英文摘要
Variational autoencoders (VAEs) offer an efficient approach to amortized Bayesian inference for inverse problems, but posterior accuracy can depend strongly on the choice of variational regularization, particularly when the inverse problem contains weakly identified parameter directions. This study investigates three objectives: a reverse Kullback--Leibler formulation (VAE-KL), an asymmetric Jensen--Shannon formulation (VAE-JS), and a Jensen--Shannon--Wasserstein formulation (VAE-JSWA), which replaces the reverse Kullback--Leibler regularizer with the squared 2-Wasserstein distance while retaining forward-Kullback--Leibler posterior supervision. A full-covariance Gaussian encoder and a pre-trained physics-based surrogate are used for amortized posterior inference. A local linear--Gaussian analysis in the generalized Fisher basis is developed to characterize the variance-dependent gradients of the three objectives. The formulations are first evaluated using linear--Gaussian benchmarks with known posterior solutions and subsequently tested on nonlinear physics-based inverse problems, including an inverse problem governed by a linear ODE and two PDE-constrained problems. VAE-KL performs slightly better than the other formulations in the well-conditioned benchmark, where all three approaches yield comparable posterior approximations, whereas VAE-JSWA provides substantially lower posterior errors in the strongly ill-conditioned benchmark. The nonlinear physics-based problems exhibit a similar conditioning-dependent trend, with JS-based formulations providing greater benefit as posterior ill-conditioning increases. These results indicate that posterior conditioning is an important factor in selecting variational objectives and motivate geometry-adaptive variational inference for Bayesian inverse problems.
发表机构
- Indian Institute of Technology Palakkad（帕拉卡德印度理工学院）
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