The pluripotential theory on a normal projective variety is closely related to the pluripotential theory on its Berkovich analytification. In particular, thanks to the celebrated work of Berman--Boucksom--Jonsson, we know that the space of finite energy non-Archimedean metrics associated with an ample line bundle is isomorphic to a subspace of the space of geodesic rays in the space Archimedean potentials. In this talk, I will explain an extension of this point of view to transcendental big classes. We show that Boucksom--Jonsson's envelope conjecture holds in our transcendental theory and our theory coincides with Boucksom--Jonsson's on smooth projective varieties. Part of the talk is based on a joint work with Tamás Darvas and Kewei Zhang.