Analytical classical density functionals from an equation learning network.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Machine Learning

Background:

• Classical free energy functionals are crucial for understanding fluid behavior.
• Previous machine learning approaches were limited by restricted functional spaces.

Purpose of the Study:

• To investigate the feasibility of using modified equation learning networks to derive analytic free energy functionals.
• To expand the functional space explored by machine learning in fluid thermodynamics.

Main Methods:

• Utilizing a modified equation learning network based on Martius and Lampert's work.
• Constructing free energy densities as functions of weighted densities using flexible basis functions.
• Applying the method to one-dimensional hard rod and Lennard-Jones fluids.

Main Results:

• Achieved a good approximation of the hard rod free energy functional and its direct correlation function.
• Successfully learned the full excess free energy functional for the Lennard-Jones fluid.
• Learned the excess free energy functional related to interparticle attractions for the Lennard-Jones fluid.
• Demonstrated good agreement between learned functionals and simulated density profiles.

Conclusions:

• The modified equation learning network effectively expands the functional space for machine learning optimization.
• This machine learning approach provides accurate analytic forms for free energy functionals in model fluids.
• The method shows promise for predicting thermodynamic properties of complex systems.