This thesis presents a search for the Higgs boson decaying into invisible particles, such as dark matter particles, using proton-proton collision data. Although the invisible decay channel is very small in the Standard Model of particle physics (SM), it is highly motivated by some of new physics models including Higgs portal dark matter models. In addition, the present experimental measurements of Higgs boson decays allow non-SM decays as large as O(30\%). The data for this thesis were collected with the ATLAS detector at the LHC in 2015-2016 at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 36.1 fb$^{-1}$. The search uses the production of a Higgs boson via the vector boson fusion mode, which has the second largest cross section among Higgs boson production processes and drives the sensitivity for this search thus far. The data are found to be in agreement with the background contributions from SM processes. The first analysis places an observed (expected) upper limit of 0.37 (0.28), at 95\% confidence level, on the branching fraction of the Higgs boson decay to invisible particles, assuming a Higgs boson mass of 125 GeV. It is also interpreted in the context of Higgs portal dark matter models and compared to direct dark matter searches. After the publication of this result, a second analysis was performed with significant improvements, resulting in an observed (expected) upper limit of 0.20 (0.20), at 95\% confidence level, on the branching fraction of the Higgs boson decay to invisible particles.