Physics-informed machine learning of the correlation functions in bulk fluids

Aug 2, 2026

The work leverages physics-informed machine learning to solve the Ornstein–Zernike (OZ) equation, the central relation for computing pair correlation functions in liquid integral equation theory. Physics-informed neural networks and physics-informed neural operator networks are developed to address forward and inverse OZ problems for diverse bulk fluids. The proposed framework improves accuracy and computational efficiency versus prior PINN approaches, and supports accurate long-range correlation at higher densities. Results show strong potential for thermodynamic state theory applications through both fixed-parameter and parameterized OZ formulations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Introduction

Integral equation theory and the OZ equation

Physics-informed machine learning for equation solving

Method overview and contributions

Methodology

OZ equation approach for simple fluids

arXiv :2309 .00767v1 [physics .comp-ph] 2 Sep 2023 Physics-informed machine learning of the correlation functions in bulk fluids Wenqian Chen, a) Peiyuan Gao, a) and Panos Stinisa) (*Electronic mail: [peiyuan.gao@pnnl.gov](peiyuan.gao@pnnl.gov)) The Ornstein-Zernike (OZ) equation is the fundamental equation for pair correlation function computations in the modern integral equation theory for liquids. In this work, machine learning models, notably physics-informed neural networks and physics-informed neural operator networks, are explored to solve the OZ equation. The physics-informed machine learning models demonstrate great accuracy and high efficiency in solving the forward and inverse OZ problems of various bulk fluids. The results highlight the significant potential of physics-informed machine learning for applications in thermodynamic state theory. I. INTRODUCTION Statistical mechanics has been successful as a framework for modeling complex systems from microscale to macroscale. 1,2 The integral equation theory (IET) approach for liquids is one of the most widely used approaches that is based on statistical mechanics.3–5 It provides a closed analytical relation between the molecular interaction potentialsand microscopic correlation functions of liquids and liquid mixtures. The prediction of macroscopic properties from the knowledge of the microscopic structure allows for a detailed description of a wide diversity of bulk fluids. The basic concept in the theory is the Ornstein–Zernike (OZ) equation.6 With the aid of statistical mechanics, along with some approximation methods such as hypernetted chain (HNC), PercusYevick (PY), and Verlet modified (VM)4,7–10 closure approximations, many properties of bulk fluids can be calculated. The OZ equation has practical importance as a foundation for approximations for computing the pair correlation function of molecules or ions, or of colloidal particles in solution. 11,12 Furthermore, based on the OZ equation, several approaches such as the molecular Ornstein–Zernike (MOZ) equation 13 and the three-dimensional reference interaction site model (3D-RISM) theory 14–16 have been developed. These theories offer a rigorous framework for calculating equilibrium solvation properties without the need for costly dynamic simulations, which can be crucial for biochemical process and drug design.17–19 The integral equation theory of molecular liquids has been an active area of academic research. In recent years, the development of machine learning (ML) techniques offers a suite of powerful tools in function approximation and equation solving.20 ML approaches have been successfully applied to problems such as solving the direct correlation function of materials21,22 or learning closures within the OZ framework23. Among ML techniques, physics-informed neural networks24 (PINNs) , are a novel type of neural network that exploit the known laws of physics by including them in the loss function. PINNs are designed to address systems governed by established laws of physics. Their versatility allows them to tackle a range of challenges, ranging a)Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory, Richland, WA, 99354, USA from elementary algebraic equations to sophisticated physical phenomena, notably fluid dynamics25–27 , heat transfer28,29 . However, PINNs can sometimes struggle with the complexities introduced by parameterized partial differential equations (PDEs) . Such PDEs can include parameters that are related to specific material properties, which can increase significantly the required complexity of a general-purpose neural network. The deep operator network30 (DeepOnet) is designed for operator learning, where the inputs are processed separately and then combined to produce the final output. Hence, by substituting the general-purpose neural network with DeepOnet within the PINN framework, the physics-informed deep operator network (PIDee

FAQs

What is the role of the Ornstein–Zernike (OZ) equation in the presented work?

The OZ equation serves as the fundamental relation for computing pair correlation functions within modern integral equation theory for liquids.

Which machine learning models are explored to solve the OZ equation?

The study investigates physics-informed neural networks (PINNs) and physics-informed neural operator networks, specifically using a physics-informed deep operator network framework (PIDeepOnet).

How does the proposed framework improve upon earlier PINN approaches?

By tailoring a PINN for fixed-parameter OZ equations and using PIDeepOnet for parameterized OZ equations, the work integrates architectural and training enhancements (e.g., Fourier feature embedding, modified feedforward networks, and self-adaptive weighting) to achieve higher accuracy and efficiency, addressing issues like slow training and long-range prediction accuracy at higher densities.

Aug 2, 2026

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