  While the interactions between water molecules are dominated by strongly
directional hydrogen bonds (HBs), it was recently proposed that relatively
weak, isotropic van der Waals (vdW) forces are essential for understanding the
properties of liquid water and ice. This insight was derived from ab initio
computer simulations, which provide an unbiased description of water at the
atomic level and yield information on the underlying molecular forces. However,
the high computational cost of such simulations prevents the systematic
investigation of the influence of vdW forces on the thermodynamic anomalies of
water. Here we develop efficient ab initio-quality neural network potentials
and use them to demonstrate that vdW interactions are crucial for the formation
of water's density maximum and its negative volume of melting. Both phenomena
can be explained by the flexibility of the HB network, which is the result of a
delicate balance of weak vdW forces, causing e.g. a pronounced expansion of the
second solvation shell upon cooling that induces the density maximum.
