Inverse problems of partial differential equations are ubiquitous across various scientific disciplines and can be formulated as statistical inference problems using Bayes' theorem. To address large-scale problems, it is crucial to develop discretization-invariant algorithms, which can be achieved by formulating methods directly in infinite-dimensional space. We propose a novel normalizing flow based infinite-dimensional variational inference method (NF-iVI) to extract posterior information efficiently. Specifically, by introducing well-defined transformations, the prior in Bayes' formula is transformed into post-transformed measures that approximate the true posterior. To circumvent the issue of mutually singular probability measures, we formulate general conditions for the employed transformations. As guiding principles, these conditions yield four concrete transformations. Additionally, to minimize computational demands, we have developed a conditional normalizing flow variant, termed CNF-iVI, which is adept at processing measurement data of varying dimensions while requiring minimal computational resources. We apply the proposed algorithms to two typical inverse problems governed by a simple smooth equation and the steady-state Darcy flow equation. Numerical results confirm our theoretical findings, illustrate the efficiency of our algorithms, and verify the discretization-invariant property.

翻译：偏微分方程反问题广泛存在于各科学领域，并可通过贝叶斯定理表述为统计推断问题。为处理大规模问题，发展离散化无关算法至关重要，这可通过在无限维空间中直接构建方法实现。本文提出一种基于归一化流的无限维变分推断新方法（NF-iVI），用于高效提取后验信息。具体而言，通过引入良定义的变换，将贝叶斯公式中的先验分布转化为逼近真实后验的变换后测度。为规避概率测度相互奇异的问题，我们建立了所用变换的通用条件。这些条件作为指导原则，导出了四种具体变换。此外，为降低计算需求，我们开发了条件归一化流变体（CNF-iVI），该变体能够处理不同维度的测量数据，同时仅需极少计算资源。我们将所提算法应用于两个典型反问题：受简单光滑方程控制的系统及稳态达西流动方程。数值结果验证了理论结论，展示了算法的高效性，并证实了离散化无关特性。