  A search for non-resonant Higgs boson pair production, as predicted by the
Standard Model, is presented, where one of the Higgs bosons decays via the
$H\rightarrow bb$ channel and the other via one of the $H \rightarrow
WW^*/ZZ^*/\tau\tau$ channels. The analysis selection requires events to have at
least two $b$-tagged jets and exactly two leptons (electrons or muons) with
opposite electric charge in the final state. Candidate events consistent with
Higgs boson pair production are selected using a multi-class neural network
discriminant. The analysis uses 139 fb$^{-1}$ of $pp$ collision data recorded
at a centre-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron
Collider. An observed (expected) upper limit of 1.2 ($0.9^{+0.4}_{-0.3}$) pb is
set on the non-resonant Higgs boson pair production cross-section at 95%
confidence level, which is equivalent to 40 ($29^{+14}_{-9}$) times the value
predicted in the Standard Model.
