  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.
