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Inverse problems, particularly those governed by Partial DifferentialEquations (PDEs), are prevalent in various scientific and engineeringapplications, and uncertainty quantification (UQ) of solutions to theseproblems is essential for informed decision-making. This second part of atwo-paper series builds upon the foundation set by the first part, whichintroduced CUQIpy, a Python software package for computational UQ in inverseproblems using a Bayesian framework. In this paper, we extend CUQIpy'scapabilities to solve PDE-based Bayesian inverse problems through a generalframework that allows the integration of PDEs in CUQIpy, whether expressednatively or using third-party libraries such as FEniCS. CUQIpy offers concisesyntax that closely matches mathematical expressions, streamlining the modelingprocess and enhancing the user experience. The versatility and applicability ofCUQIpy to PDE-based Bayesian inverse problems are demonstrated on examplescovering parabolic, elliptic and hyperbolic PDEs. This includes problemsinvolving the heat and Poisson equations and application case studies inelectrical impedance tomography (EIT) and photo-acoustic tomography (PAT),showcasing the software's efficiency, consistency, and intuitive interface.This comprehensive approach to UQ in PDE-based inverse problems providesaccessibility for non-experts and advanced features for experts.

CUQIpy -- Part II: computational uncertainty quantification for PDE-based inverse problems in Python