Neural density functional theory in higher dimensions with convolutional layers.

Area of Science:

• Computational physics
• Statistical mechanics
• Machine learning

Background:

• Recent advancements have enabled machine learning (ML) applications in classical density functional theory (DFT) for systems with one-dimensional (1D) inhomogeneities.
• Extending these ML-based DFT methods to higher dimensions is crucial for tackling more complex physical systems.

Purpose of the Study:

• To propose and implement a novel machine learning model for two-dimensional (2D) density functional theory.
• To adapt ML models, similar to weighted density functionals, for application in 2D inhomogeneous systems.

Main Methods:

• The proposed model utilizes fast convolutional layers exclusively.
• It is applied to a system of hard disks in fully 2D inhomogeneous scenarios.
• Training involved a combination of smooth and steplike external potentials in the fluid phase.

Main Results:

• The machine learning model demonstrated highly satisfactory agreement with simulation data for pair correlation functions.
• The model performed well even for external potentials not included in the training set, indicating robustness.
• Test particle geometry analysis confirmed the model's predictive capabilities.

Conclusions:

• The developed ML-based DFT model is effective for 2D inhomogeneous systems.
• The methodology shows significant potential for direct application to three-dimensional (3D) problems.
• This work advances the integration of machine learning in theoretical condensed matter physics.