Metadensity Functional Theory for Classical Fluids: Extracting the Pair Potential

The excess free energy functional of classical density functional theory depends upon the type of fluid model, specifically on the choice of (pair) potential. This functional is unknown in general and is approximated reliably only in special cases. We present a machine learning scheme for training a neural network that acts as a generic metadensity functional for truncated but otherwise arbitrary pair potentials. Automatic differentiation and neural functional calculus then yield, for one-dimensional fluids, accurate predictions for inhomogeneous states and immediate access to the pair distribution function. The approach provides a means of addressing a fundamental problem in the physics of liquids and for soft matter design: "How do we best invert structural data to obtain the pair potential?"