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

Area of Science:

• Computational Physics
• Statistical Mechanics
• Machine Learning

Background:

• Neural operators offer a data-driven approach to model complex functional relationships.
• Classical density functional theory (DFT) involves intricate mappings between physical quantities.

Purpose of the Study:

• Evaluate neural operator architectures for learning relationships in one-dimensional hard-rod fluids.
• Benchmark Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) against dense neural networks.

Main Methods:

• Trained DeepONet and FNO variants on data from analytical solutions of hard-rod fluids.
• Assessed interpolation and extrapolation capabilities using cross-validation.
• Compared mean squared error and excess free energy prediction accuracy.

Main Results:

• FNO demonstrated superior accuracy in predicting excess free energy, especially with squared ReLU activation.
• GK-RMSCNN-DeepONet performed best among DeepONet variants.
• The neural-operator mapping ρ ↦ c1 proved effective for solving density profiles, outperforming direct Vext ↦ ρ mapping for extrapolation.

Conclusions:

• Neural operators, particularly FNO, are highly effective for modeling functional relationships in DFT.
• The ρ ↦ c1 mapping offers a robust approach for density profile prediction, with advantages in extrapolation.
• Specialized neural operator features enhance prediction accuracy and flexibility.