发表机构
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对无穷维贝叶斯推理问题，提出基于无穷维连续归一化流的先验模型，结合训练与采样算法，在三类逆问题上验证了算法有效性。
AI 中文摘要
本文针对偏微分方程的逆问题开展无穷维贝叶斯推理研究，其模型参数属于无穷维希尔伯特空间。为有效融合先验信息，我们提出一种新颖的基于连续归一化流的无穷维模型。具体而言，通过在无穷维空间中引入定义良好的神经常微分方程，可将简单的参考测度转换为编码先验信息的更复杂测度。我们建立了相应的理论框架，以确保所提出的无穷维贝叶斯先验的适定性。同时，针对两种不同的数据设置提供了该先验的训练方法，并为所得贝叶斯后验提供了两种采样算法。将所提框架应用于三类典型逆问题：简单光滑逆问题、逆散射问题及逆热传导问题。数值实验验证了理论分析，并证明了所提算法的有效性。
英文摘要
This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.
Comments41 pages
