Abstract: We study the inverse problem of unique recovery of a complex-valued scalar function V : M × C → C, defined over a smooth compact Riemannian manifold (M, g) with smooth boundary, given the Dirichlet to Neumann map, in a suitable sense, for the elliptic semi-linear equation −∆gu + V (x, u) = 0. We show that uniqueness can be proved for a large class of non-linearities. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex geometric optic solutions for the linearized operator and the resulting non-linear interactions. These interactions result in the study of a weighted transform along geodesics, that we call the Jacobi weighted ray transform. The talk is based on a recent joint work with Lauri Oksanen.