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  • Open Access

Bridge function as a functional of the radial distribution function: Operator learning and application

Martin Panholzer*

Michael Haring and Thomas Wallek†

Robert E. Zillich‡

  • Uni Software Plus GmbH, Linzer Strasse 6, 4320 Perg, Austria

  • *Contact author: panholzer@unisoftwareplus.com
  • †Contact author: thomas.wallek@tugraz.at
  • ‡Contact author: robert.zillich@jku.at
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Phys. Rev. E 112, 035311 – Published 15 September, 2025

DOI: https://doi.org/10.1103/4962-9rch

Abstract

Properties of classical molecular systems can be calculated with integral equation theories based on the Ornstein-Zernike equation and a complementary closure relation. One such closure relation is the hypernetted chain (HNC) approximation, which neglects the so-called bridge function. We present a way to use machine learning to train a deep operator network to predict the bridge function, based on the radial distribution function as input. Bridge functions for the Lennard-Jones fluid are calculated from Monte Carlo simulations in a wide range of densities and temperatures. These results are used to train the deep operator network. This network is employed to improve the HNC closure by the prediction for the bridge function, and the resulting set of equations is solved iteratively. For assessment, we compare the radial distribution function and the pressure, calculated by the viral expression, with Monte Carlo results and standard HNC. We demonstrate that incorporating the neural-networkbased bridge function in the closure relation leads to substantially improved predictions. The universality of our method is demonstrated by comparing results for the Mie fluid and the hard-sphere fluid, calculated with our model trained on the Lennard-Jones fluid, with exact Mie and hardsphere results, showing overall good agreement.

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Outline

Information

Phys. Rev. E 112, 035311– Published 15 September, 2025
  • Received 26 May 2025
  • Accepted 12 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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