[Paper Review] Machine learnt approximations to the bridge function yield improved closures for the Ornstein-Zernike equation

Rhys E. A. Goodall, Alpha A. Lee|arXiv (Cornell University)|Apr 20, 2020

Block Copolymer Self-Assembly4 citations

TL;DR

This paper proposes a machine learning approach to learn the bridge function in the Ornstein-Zernike equation, improving closure approximations for inverse problems in soft matter. By training a machine learning model on simulation data to predict the bridge function, the method achieves more accurate one-step inversion of pair distribution functions into interaction potentials than traditional HNC or PY closures across diverse systems.

ABSTRACT

A key challenge for soft materials design and coarse-graining simulations is determining interaction potentials between components that give rise to desired condensed-phase structures. In theory, the Ornstein-Zernike equation provides an elegant framework for solving this inverse problem. Pioneering work in liquid state theory derived analytical closures for the framework. However, these analytical closures are approximations, valid only for specific classes of interaction potentials. In this work, we combine the physics of liquid state theory with machine learning to infer a closure directly from simulation data. The resulting closure is more accurate than commonly used closures across a broad range of interaction potentials. We show for two examples of a prototypical inverse design problem, fitting a coarse-grained simulation potential, that our approach leads to improved one-step inversion.

Motivation & Objective

• To address the challenge of accurately determining effective interaction potentials from target pair distribution functions in soft matter systems.
• To overcome limitations of analytical closures like HNC and PY, which fail for complex or non-universal interaction potentials.
• To develop a data-driven closure for the Ornstein-Zernike equation by learning the bridge function directly from simulation data.
• To improve the accuracy of one-step inverse design in coarse-graining and structural inversion problems.
• To demonstrate that ML-informed closures outperform standard approximations in reconstructing target structures from g(r).

Proposed method

• The bridge function B(r) is modeled as a machine learning function of physical correlation functions, including h(r), c(r), and S(q), to embed physical constraints.
• A local correlation (LC) model is trained on simulation data to predict B(r) directly, using features derived from the direct and total correlation functions.
• The method uses a physics-informed feature set to ensure the ML model respects known thermodynamic and structural relationships.
• The learned closure is applied in an inverse framework to reconstruct interaction potentials from target g(r) using a non-iterative, one-step approach.
• A Savitzky-Golay filter is applied to smooth the estimated potential and its gradient to ensure numerically stable forces.
• The approach is validated on two prototypical inverse design problems: one with a two-species system and another with a Lennard-Jones potential.

Experimental results

Research questions

• RQ1 Can a machine learning model trained on simulation data accurately predict the bridge function in the Ornstein-Zernike equation?
• RQ2 Does a data-driven closure for the bridge function outperform traditional analytical closures like HNC and PY in inverse problems?
• RQ3 Can the learned closure enable more accurate one-step inversion of pair distribution functions into interaction potentials?
• RQ4 How does the performance of the ML-based closure vary across different interaction potential types and structural features?
• RQ5 To what extent does the ML closure reduce the Wasserstein distance between target and simulated particle densities?

Key findings

• The machine learning-based closure reduced the Wasserstein distance between target and simulated particle densities to 0.037 in the first example, compared to 0.243 for IBI and 0.044 for HNC.
• In the second example, the ML closure achieved a Wasserstein distance of 0.038, outperforming IBI (0.306) and HNC (0.053).
• The learned closure better captured the complex step-like structure in g(r) around 1.6σ, where HNC and IBI failed to reproduce fine features.
• The method produced more accurate estimates of the bridge function than HNC or PY across a broad range of interaction potentials.
• The approach enabled improved one-step inversion in coarse-graining problems, reducing structural error without iterative optimization.
• The results demonstrate that ML can effectively learn physically meaningful corrections to analytical closures in liquid state theory.