Neural operators for forward and inverse potential–density mappings in classical density functional theory

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. In this work, we evaluate the performance of several neural operator architectures in learning the functional relationships between the one-body density profile ρ(x), the one-body direct correlation function c1(x), and the external potential Vext(x) of inhomogeneous one-dimensional hard-rod fluids, using training data generated from analytical solutions of the underlying statistical-mechanical model. Several variants of the Deep Operator Network (DeepONet) and the Fourier Neural Operator (FNO) were considered, each incorporating different machine-learning architectures, activation functions, and training strategies. These operator learning methods are benchmarked against a fully connected dense neural network, which serves as a baseline. We compared their performance in terms of the mean squared error 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 Vext ↦ ρ with neural operators, as well as with Gaussian process regression combined with active learning by error control, which has shown strong performance in previous studies. While the direct mapping from Vext ↦ ρ suffers from high extrapolation error and proves inefficient for out-of-distribution predictions, the neural-operator mapping ρ ↦ c1 can effectively be used to solve the density profile via the Euler-Lagrange equation or be integrated with other surrogate methods. Moreover, neural operators offer additional flexibility through specialized operations, such as significance-based predictions on uneven grids (as in GK-CNN-DeepONet) and adaptive grid resolution adjustment (as in FNO), both of which can enhance prediction accuracy.