[Paper Review] Estimating Gibbs free energies via isobaric-isothermal flows

Peter Wirnsberger, Borja Ibarz+1 more|arXiv (Cornell University)|May 22, 2023

Theoretical and Computational Physics4 citations

TL;DR

This paper introduces the NPT-flow, a normalizing flow model that jointly generates particle coordinates and triclinic box parameters to sample from the isobaric-isothermal ensemble (NPT), enabling direct estimation of Gibbs free energies. The method achieves high accuracy in estimating free energy differences between cubic and hexagonal ice phases, matching molecular dynamics baselines without reweighting, and demonstrates scalability to 512-particle systems with minimal statistical efficiency loss.

ABSTRACT

We present a machine-learning model based on normalizing flows that is trained to sample from the isobaric-isothermal ensemble. In our approach, we approximate the joint distribution of a fully-flexible triclinic simulation box and particle coordinates to achieve a desired internal pressure. This novel extension of flow-based sampling to the isobaric-isothermal ensemble yields direct estimates of Gibbs free energies. We test our NPT-flow on monatomic water in the cubic and hexagonal ice phases and find excellent agreement of Gibbs free energies and other observables compared with established baselines.

Motivation & Objective

• To develop a machine learning model capable of sampling from the isobaric-isothermal ensemble (NPT), which is essential for studying phase transitions and equilibrium crystal structures.
• To enable direct, unbiased estimation of Gibbs free energies using flow-based generative models, overcoming limitations of existing normalizing flows restricted to the canonical (NVT) ensemble.
• To validate the method on monatomic water in cubic and hexagonal ice phases, comparing free energy estimates and structural observables against established molecular dynamics baselines.
• To demonstrate that joint learning of particle coordinates and box shape parameters does not compromise sampling quality or training efficiency compared to NVT-only flows.

Proposed method

• The NPT-flow models the joint distribution of particle coordinates and triclinic box shape parameters (six degrees of freedom) under fixed N, P, and T conditions using an invertible, differentiable neural network architecture.
• The model is trained to approximate the target isobaric-isothermal Boltzmann distribution via maximum likelihood estimation, using a base distribution (Gaussian-affine) and a flow-based transformation.
• The method leverages the exact likelihood and independent sampling capability of normalizing flows to compute expectation values and free energy differences via importance sampling.
• The flow architecture extends standard NVT normalizing flows by incorporating box parameters as part of the latent space, enabling joint generation of particle positions and box geometry.
• The model uses a likelihood-based objective to minimize the Kullback-Leibler divergence between the learned and target NPT distributions.
• No explicit reweighting is applied; the method directly estimates observables and free energies from the flow-generated samples.

Experimental results

Research questions

• RQ1 Can normalizing flows be extended to sample from the isobaric-isothermal ensemble (NPT) by jointly modeling particle coordinates and box shape parameters?
• RQ2 Can such a model provide accurate, direct estimates of Gibbs free energies without requiring additional reweighting or reference data?
• RQ3 How well does the NPT-flow perform in capturing phase transitions and free energy differences between ice phases compared to molecular dynamics?
• RQ4 Does the inclusion of box parameter generation significantly degrade sampling quality or training efficiency compared to NVT-only flows?

Key findings

• The NPT-flow accurately estimates the Gibbs free energy difference between cubic and hexagonal ice at 200 K and ambient pressure, with results of 6.70(32) J/mol for 512 particles, closely matching the MD baseline of 6.51(57) J/mol.
• For 64-particle systems, the NPT-flow estimates the free energy difference as 51.76(3) J/mol, in excellent agreement with the MD baseline of 51.64(27) J/mol.
• The model achieves high accuracy in predicting structural and energetic observables across both ice phases without applying any reweighting procedure.
• The effective sample size (ESS) drops to 0.15% for the 512-particle hexagonal ice system, but the method remains computationally efficient due to parallel sampling capabilities.
• The NPT-flow maintains performance comparable to prior NVT-only flows in terms of training time, hardware requirements, and sampling quality, indicating that joint box and particle generation is feasible.
• The method demonstrates that flow-based models can be extended to the NPT ensemble with minimal architectural or training overhead, enabling direct free energy estimation.