Whereas 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.water structure | van der Waals interactions | neural network potentials | ab initio liquid water | density-functional theory W ater is an exceptional liquid exhibiting several anomalies, of which the density maximum at 4°C is the most prominent (1). Together with the negative volume of melting, it is responsible for the fact that water freezes from the top down and ice floats on water. The unusual behavior of water can be directly related to its ability to form hydrogen bonds (HBs) which are of strongly directional nature and determine the microscopic structure of water (2, 3). To investigate the anomalies of water at the molecular level, atomistic computer simulations have become an essential tool. Important contributions have been made by simulations using simple empirical water models (3-8).Simulations based on ab initio molecular dynamics (AIMD) (9-11) allow determination of the properties of water with high predictive power and enable a detailed analysis of their underlying microscopic mechanisms. In contrast to empirical water models (5), which depend on experimental data resulting in a limited transferability, in AIMD the atomic forces that govern the molecular dynamics are obtained directly from quantum mechanics. Although this approach is in principle exact [in combination with methods that account for the quantum nature of the nuclei (12, 13)], ab initio simulations of condensed matter systems are feasible only if approximate but efficient methods such as density-functional theory (DFT) are used. Even then, however, simulations are restricted to short times and small systems. AIMD simulations have been used to a limited extent to investigate the phase behavior of water, for instance by estimating melting temperatures (14, 15) and vapor-liquid coexistence curves (16, 17). However, many fundamental thermodynamic properties of water have not been evaluated to date. To circumvent the limitations of on-the-fly AIMD, various efficient water potentia...