The bridge function as a functional of the radial distribution function: Operator learning and application

Properties of classical molecular systems can be calculated with integral equation theories based on the Ornstein-Zernike (OZ) equation and a complementing closure relation. One such closure relation is the hyper netted chain (HNC) approximation, which neglects the so-called bridge function. We present a new way to use machine learning to train a deep operator network to predict the bridge function, based on the radial distribution function as input. Bridge functions for the Lennard-Jones fluid are calculated from Monte Carlo simulations in a wide range of densities and temperatures. These results are used to train the deep operator network. This network is employed to improve the HNC closure by the prediction for the bridge function, and the resulting set of equations is solved iteratively. For assessment, we compare the radial distribution function and the pressure, calculated by the viral expression, with Monte Carlo results and standard HNC. We demonstrate that incorporating the neural network based bridge function in the closure relation leads to substantially improved predictions. Universality of our method is demonstrated by comparing results for the hard sphere fluid, calculated with our model trained on the Lennard-Jones fluid, with exact hard sphere results, showing overall good agreement.