We first generalize the theory of Poisson-Lie groups and Lie bialgebras in the infinite-dimensional setting. The classical Drinfeld correspondence establishes the one-to-one correspondence between Poisson structures on a one-connected finite-dimensional Lie group and Lie bialgebra structures on its Lie algebra. We will extend this result in the infinite-dimensional setting for a class of regular Lie groups modeled on Fréchet or Silva locally convex topological vector spaces. Our framework includes important examples such as smooth loop groups of finite-dimensional Lie groups, analytic loop groups of finite-dimensional Lie groups, and diffeomorphism groups of some manifolds.