This paper presents a measurement of $ZZ$ production with the ATLAS detector at the Large Hadron Collider. The measurement is carried out in the final state with two charged leptons and two neutrinos, using data collected during 2015 and 2016 in $pp$ collisions at $s=13$ TeV, corresponding to an integrated luminosity of 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$. The integrated cross-sections in the total and fiducial phase spaces are measured with an uncertainty of 7% and compared with Standard Model predictions, and differential measurements in the fiducial phase space are reported. No significant deviations from the Standard Model predictions are observed, and stringent constraints are placed on anomalous couplings corresponding to neutral triple gauge-boson interactions.