De Rham's theorem for Orlicz cohomology

We prove that the de Rham $L^\phi$-cohomology of a Riemannian manifold $M$ admitting a convenient triangulation $X$ is isomorphic to the simplicial $\ell^\phi$-cohomology of $X$ under some assumptions on the Young function $\phi$. This result implies the quasi-isometry invariance of the first cohomology.