A classical density functional from machine learning and a convolutional neural network

We use machine learning methods to approximate a classical density functional. The functional 'learns' by comparing the density profile it generates with that of simulations. As a study case, we choose the model problem of a Lennard-Jones fluid in one dimension where there is no exact solution available. After separating the excess free energy functional into a repulsive and an attractive part, machine learning finds a functional for the attractive part in weighted-density form. The predictions of density profile at a hard wall shows good agreement when subject to thermodynamic conditions beyond those in the training set. This also holds for the equation of state if this is evaluated near the training temperature. We discuss the applicability to problems in higher dimensions.