[Paper Review] Free energy calculation of crystalline solids using normalizing flow

Rasool Ahmad, Wei Cai|arXiv (Cornell University)|Nov 1, 2021

Machine Learning in Materials Science45 references4 citations

TL;DR

This paper introduces a normalizing flow-based framework, Boltzmann Generator, to compute absolute free energies of crystalline solids using unsupervised deep generative modeling. By training a normalizing flow to learn the equilibrium distribution of atomic configurations, the method enables accurate free energy calculations via free energy perturbation, achieving results comparable to established methods for diamond-cubic silicon across temperatures and defect states.

ABSTRACT

Taking advantage of the advances in generative deep learning, particularly normalizing flow, a framework, called Boltzmann Generator, has recently been proposed for the purpose of generating equilibrium atomic configurations from the canonical ensemble and determining the associated free energy. In this work, we revisit Boltzmann Generator to motivate the construction of the loss function from the statistical mechanical point of view, and to cast the training of the neural networks in a purely unsupervised manner that requires no samples of the atomic configurations from the equilibrium ensemble. We further show that the normalizing flow framework furnishes a reference thermodynamic system, very close to the real thermodynamic system under consideration, that is suitable for the well-established free energy perturbation methods to determine accurate free energy of solids. We then apply the normalizing flow to two problems: temperature-dependent Gibbs free energy of perfect crystal and formation free energy of monovacancy defect in a model system of diamond cubic Si. The results obtained from the normalizing flow are shown to be in good agreement with that obtained from independent well-established free energy methods.

Motivation & Objective

• To develop an unsupervised deep learning framework for generating equilibrium atomic configurations of crystalline solids without requiring pre-sampled data from the canonical ensemble.
• To enable accurate computation of absolute free energy of solids using normalizing flows by constructing a reference thermodynamic system close to the real system.
• To validate the method’s accuracy by computing temperature-dependent Gibbs free energy of perfect diamond-cubic Si and formation free energy of monovacancy defects.
• To demonstrate that normalizing flow can serve as a reliable alternative to traditional free energy methods such as thermodynamic integration and free energy perturbation.

Proposed method

• Utilizes normalizing flow to model the probability distribution of atomic configurations in the canonical ensemble (NVT) via a bijective, invertible transformation from a simple base distribution (e.g., Gaussian).
• Employs RealNVP-based coupling layers with learnable scaling and translation functions (s and t) implemented as deep neural networks to parameterize the transformation.
• Computes the Jacobian determinant of the transformation efficiently using the product of exponentiated scaling functions, enabling exact likelihood computation and density estimation.
• Trains the normalizing flow in an unsupervised manner using only the system’s Hamiltonian, without requiring equilibrium samples, by minimizing a loss function derived from statistical mechanics.
• Constructs a reference system via the learned flow model that is close to the real system, enabling application of well-established free energy perturbation (FEP) methods.
• Applies the framework to compute free energy differences using FEP between the reference system (generated by the flow) and the target system, ensuring high accuracy.

Experimental results

Research questions

• RQ1 Can a normalizing flow model be trained in an unsupervised manner to generate equilibrium atomic configurations of crystalline solids without requiring pre-sampled data?
• RQ2 How accurately can the normalizing flow framework compute the absolute Gibbs free energy of a perfect crystal across varying temperatures?
• RQ3 Can the method accurately predict the formation free energy of a monovacancy defect in diamond-cubic silicon compared to established methods?
• RQ4 How effective is the normalizing flow-generated reference system for free energy perturbation calculations in solid-state systems?

Key findings

• The unsupervised training of the normalizing flow model successfully generates equilibrium atomic configurations of crystalline solids without requiring any sampled configurations from the canonical ensemble.
• The computed temperature-dependent Gibbs free energy of perfect diamond-cubic Si using the normalizing flow method shows excellent agreement with results from established thermodynamic integration and free energy perturbation methods.
• The formation free energy of a monovacancy defect in diamond-cubic Si, computed via free energy perturbation using the flow-generated reference system, matches well with reference values obtained through conventional simulations.
• The Jacobian determinant of the flow transformation is efficiently computed as the sum of the log-scaling functions, enabling exact likelihood evaluation and enabling density estimation for thermodynamic calculations.
• The method achieves high accuracy in free energy prediction by constructing a reference system that is very close to the real system in phase space, minimizing the variance in FEP calculations.
• The framework demonstrates that normalizing flows can serve as a powerful, scalable, and accurate alternative to traditional free energy computation techniques in atomistic simulations of solids.