Development of Operator Networks in Classical Density Functional Theory

Classical density functional theory (cDFT) offers a powerful statistical-mechanical approach for predicting the structure and thermodynamic properties of inhomogeneous fluids. The importance local properties and values in cDFT calculations, including external potential V_ext(x), density profile ρ(x) and the direct correlation function c₁[x,ρ(x)] are all functions defined in the physical space. Mapping between these functions can be expressed as an operator, and learned by operator learning methods.

In this work, we investigate the performance of several operator networks, including fully-connect DNN, some variances of Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) on their performance in learning the mapping of ρ(x) to c₁, training with data of the analytical solution of 1D hard rod fluid. DNN, CNN and its multi-scale form (MSCNN) with discrete kernel or trainable Gaussian function derivative kernel (G-CNN) are used as the branch net of some DeepONet implements. MSE loss and model prediction of excess free energy are used to evaluate the models, random cross-validation (CV) and leave-one-group CV are applied to show intrapolation and extrapolation ability of models, with some examples of operator-based predictios.

Results shows that FNO provides the best prediction. Squared Relu performs best among activation functions.

Quasi-local CNN-DeepONet is then applied to 3D CO₂ with a 1D potential, showing good correlation and ability to predict density for complex real fluids.