The gluon fusion and vector-boson-fusion Higgs boson production modes are measured using proton--proton collisions in the $H\rightarrow W W^* \rightarrow e\nu \mu\nu$ decay channel. The Large Hadron Collider delivered proton-proton collisions at a centre-of-mass energy of 13 TeV between 2015 and 2018 which were recorded by the ATLAS detector, corresponding to an integrated luminosity of 139 fb$^{-1}$. The products of the total gluon fusion and vector-boson-fusion cross sections times the $H\rightarrow W W^*$ branching fraction are respectively measured to be 12.4\pm1.5~\mathrm{pb} and 0.79\;^{+0.19}_{-0.16}~\mathrm{pb}, in agreement with Standard Model predictions. Higgs boson production is further characterised through measurements of Simplified Template Cross Sections in a total of 11 categories.

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