Neural Operators for Forward and Inverse Potential-Density Mappings in Classical Density Functional Theory

TL;DR

This paper evaluates neural operator architectures for modeling the complex functional relationships in classical density functional theory, demonstrating their accuracy in predicting free energy and density profiles in inhomogeneous fluids.

Contribution

It compares various neural operator models, identifying the most accurate architectures and activation functions for functional mappings in classical density functional theory.

Findings

FNO with squared ReLU achieves highest free energy prediction accuracy.

RMSCNN with Gaussian kernel performs best among DeepONet variants.

Neural operators effectively predict density profiles and free energy in inhomogeneous fluids.

Abstract

Neural operators are capable of capturing nonlinear mappings between infinite-dimensional functional spaces, offering a data-driven approach to modeling complex functional relationships in classical density functional theory (cDFT). In this work, we evaluate the performance of several neural operator architectures in learning the functional relationships between the one-body density profile $\rho(x)$, the one-body direct correlation function $c_1(x)$, and the external potential $V_{ext}(x)$ of inhomogeneous one-dimensional (1D) hard-rod fluids, using training data generated from analytical solutions of the underlying statistical-mechanical model. We compared their performance in terms of the Mean Squared Error (MSE) loss in establishing the functional relationships as well as in predicting the excess free energy across two test sets: (1) a group test set generated via random cross-validation (CV) to assess interpolation capability, and (2) a newly constructed dataset for leave-one-group CV to evaluate extrapolation performance. Our results show that FNO achieves the most accurate predictions of the excess free energy, with the squared ReLU activation function outperforming other activation choices. Among the DeepONet variants, the Residual Multiscale Convolutional Neural Network (RMSCNN) combined with a trainable Gaussian derivative kernel (GK-RMSCNN-DeepONet) demonstrates the best performance. Additionally, we applied the trained models to solve for the density profiles at various external potentials and compared the results with those obtained from the direct mapping $V_{ext} \mapsto \rho$ with neural operators, as well as with Gaussian Process Regression (GPR) combined with Active Learning by Error Control (ALEC), which has shown strong performance in previous studies.