This project aims to develop innovative numerical methods for reconstructing cardiac electrical potentials from highly sparse torso potential measurements ('inverse problem of electrocardiog-raphy'). A central innovation lies in replacing traditional handcrafted regularizers with learned regularizers in the corresponding variational formulation, seamlessly integrated within a finite element framework on unstructured grids. The reliability of the proposed method will be complemented by proving consistency with a limiting functional, deriving a posteriori error estimates, and quantifying measurement uncertainty through a multilevel version of MCMC.