  We consider the inverse problem of recovering a connection on a complex
vector bundle over a compact smooth Riemannian manifold with boundary from a
Dirichlet-to-Neumann (DN) map at a high fixed frequency. We construct Gaussian
beams using the language of jet bundles and show that their value at the
boundary can be recovered from those DN maps. This allows us to show
injectivity up to gauge on manifolds whose non-abelian X-ray transform is
injective. We also study DN maps with a cubic nonlinearity and show how to
recover the broken non-abelian X-ray transform from them. This transform maps a
connection to its parallel transport along broken geodesics with endpoints on
the boundary. We show that the broken non-abelian X-ray transform is always
injective up to gauge equivalence, regardless of the manifold's geometry.
