Physics-constrained Bayesian inference of state functions in classical density-functional theory.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Machine Learning

Background:

• Characterizing free energy landscapes is a key inverse problem in classical statistical mechanics.
• Existing methods often rely on optimization or struggle with complex systems like those with excluded volume interactions.

Purpose of the Study:

• To develop a novel, data-driven approach for inferring free energy landscapes from experimental data.
• To create an efficient learning algorithm that automates the construction of approximate free-energy functionals.
• To provide uncertainty quantification and human interpretability in the inference process.

Main Methods:

• Utilizing non-parametric Bayesian inference combined with physically motivated constraints.
• Developing an efficient learning algorithm that propagates prior physical assumptions through the model.
• Employing experimental data to probabilistically weigh model predictions.

Main Results:

• Successfully inferred accurate analytic expressions for free-energy functionals from small data samples.
• Demonstrated high data efficiency and robustness, particularly for systems with excluded volume interactions.
• Validated the approach on one-dimensional fluids for canonical and grand-canonical ensembles.

Conclusions:

• The proposed Bayesian inference method offers a powerful and data-efficient way to solve the inverse problem in statistical mechanics.
• The algorithm provides interpretable results with full uncertainty quantification.
• The framework is extendable to higher dimensions and can incorporate attractive interactions using standard techniques.