Inverse problems, particularly those governed by Partial Differential Equations (PDEs), are prevalent in various scientific and engineering applications, and uncertainty quantification (UQ) of solutions to these problems is essential for informed decision-making. This second part of a two-paper series builds upon the foundation set by the first part, which introduced CUQIpy, a Python software package for computational UQ in inverse problems using a Bayesian framework. In this paper, we extend CUQIpy's capabilities to solve PDE-based Bayesian inverse problems through a general framework that allows the integration of PDEs in CUQIpy, whether expressed natively or using third-party libraries such as FEniCS. CUQIpy offers concise syntax that closely matches mathematical expressions, streamlining the modeling process and enhancing the user experience. The versatility and applicability of CUQIpy to PDE-based Bayesian inverse problems are demonstrated on examples covering parabolic, elliptic and hyperbolic PDEs. This includes problems involving the heat and Poisson equations and application case studies in electrical impedance tomography and photo-acoustic tomography, showcasing the software's efficiency, consistency, and intuitive interface. This comprehensive approach to UQ in PDE-based inverse problems provides accessibility for non-experts and advanced features for experts.

翻译：反问题，特别是由偏微分方程（PDE）控制的反问题，在科学与工程应用中普遍存在，对这些问题的解进行不确定性量化（UQ）对于决策至关重要。作为两篇系列论文的第二部分，本文在第一部分奠定的基础上展开。第一部分介绍了CUQIpy，这是一个基于贝叶斯框架、用于反问题计算不确定性量化的Python软件包。本文通过一个通用框架扩展了CUQIpy的能力，使其能够求解基于PDE的贝叶斯反问题，该框架允许在CUQIpy中集成PDE，无论是原生表达还是使用第三方库（如FEniCS）。CUQIpy提供了简洁的语法，与数学表达式高度匹配，从而简化了建模过程并提升了用户体验。本文通过涵盖抛物型、椭圆型和双曲型PDE的示例，展示了CUQIpy在基于PDE的贝叶斯反问题中的通用性和适用性。这些示例包括涉及热方程和泊松方程的问题，以及电阻抗断层成像和光声断层成像的应用案例研究，充分体现了该软件的高效性、一致性和直观界面。这种针对基于PDE的反问题的不确定性量化的综合性方法，既为非专业人士提供了易用性，又为专家提供了高级功能。