- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Learned free-energy functionals from pair-correlation matching for dynamical density functional theory

Karnik Ram1,2,*, Jacobus Dijkman3,4, René van Roij5, Jan-Willem van de Meent4, Bernd Ensing3, Max Welling4, and Daniel Cremers1,2

  • 1School of Computation, Information and Technology, TU Munich, Munich, Germany
  • 2Munich Center for Machine Learning, Munich, Germany
  • 3Van't Hoff Institute for Molecular Sciences, University of Amsterdam, Amsterdam, the Netherlands
  • 4Informatics Institute, University of Amsterdam, Amsterdam, the Netherlands
  • 5Institute for Theoretical Physics, Utrecht University, Utrecht, the Netherlands

  • *Contact author: karnik.ram@tum.de
PDF

Phys. Rev. E 112, 045314 – Published 17 October, 2025

DOI: https://doi.org/10.1103/22fd-ykkb

Abstract

Classical density functional theory (cDFT) and dynamical density functional theory (DDFT) are modern statistical mechanical theories for modeling many-body colloidal systems at the one-body density level. The theories hinge on knowing the excess free-energy accurately, which is, however, not feasible for most practical applications. Dijkman et al. [Phys. Rev. Lett. 134, 056103 (2025)] recently showed how a neural excess free-energy functional for cDFT can be learned from bulk simulations via pair-correlation matching. In this article, we demonstrate how this same functional can be applied to DDFT, without any retraining, to simulate nonequilibrium overdamped dynamics of inhomogeneous densities. We evaluate this on a three-dimensional Lennard-Jones system with planar geometry under various complex external potentials, and we observe good agreement of the dynamical densities with those from expensive Brownian dynamic simulations, up to the limit of the adiabatic approximation. We further develop and apply an extension of DDFT based on gradient flows to a grand-canonical system modeled after breakthrough gas adsorption studies, finding similarly good agreement. Our results demonstrate a practical route for leveraging learned free-energy functionals in DDFT, paving the way for accurate and efficient modeling of many-body nonequilibrium systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (43)

  1. E. d. A. Soares, A. G. Barreto, and F. W. Tavares, Classical density functional theory reveals structural information of H2 and CH4 fluids adsorbed in MOF-5, Fluid Phase Equilib. 574 113887 (2023).
  2. Q. Wei, T. Becherer, S. Angioletti-Uberti, J. Dzubiella, C. Wischke, A. T. Neffe, A. Lendlein, M. Ballauff, and R. Haag, Protein interactions with polymer coatings and biomaterials, Angew. Chem. Int. Ed. 53, 8004 (2014).
  3. R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv. Phys. 28, 143 (1979).
  4. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids: With Applications to Soft Matter (Academic Press, 2013).
  5. Y. Rosenfeld, Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing, Phys. Rev. Lett. 63, 980 (1989).
  6. S. C. Lin and M. Oettel, A classical density functional from machine learning and a convolutional neural network, SciPost Phys. 6, 001 (2019).
  7. P. Cats, S. Kuipers, S. de Wind, R. van Damme, G. M. Coli, M. Dijkstra, and R. van Roij, Machine-learning free-energy functionals using density profiles from simulations, APL Mater. 9, 031109 (2021).
  8. F. Sammüller, S. Hermann, D. de las Heras, and M. Schmidt, Neural functional theory for inhomogeneous fluids: Fundamentals and applications, Proc. Natl. Acad. Sci. USA 120, e2312484120 (2023).
  9. A. Simon, J. Weimar, G. Martius, and M. Oettel, Machine learning of a density functional for anisotropic patchy particles, J. Chem. Theory Comput. 20, 1062 (2024).
  10. J. Dijkman, M. Dijkstra, R. van Roij, M. Welling, J.-W. van de Meent, and B. Ensing, Learning neural free-energy functionals with pair-correlation matching, Phys. Rev. Lett. 134, 056103 (2025).
  11. U. M. B. Marconi and P. Tarazona, Dynamic density functional theory of fluids, J. Chem. Phys. 110, 8032 (1999).
  12. A. J. Archer and R. Evans, Dynamical density functional theory and its application to spinodal decomposition, J. Chem. Phys. 121, 4246 (2004).
  13. M. te Vrugt, H. Löwen, and R. Wittkowski, Classical dynamical density functional theory: from fundamentals to applications, Adv. Phys. 69, 121 (2020).
  14. A. Diaw and M. Murillo, A dynamic density functional theory approach to diffusion in white dwarfs and neutron star envelopes, Astrophys. J. 829, 16 (2016).
  15. A. M. Menzel, A. Saha, C. Hoell, and H. Löwen, Dynamical density functional theory for microswimmers, J. Chem. Phys. 144, 024115 (2016).
  16. M. te Vrugt and R. Wittkowski, Perspective: New directions in dynamical density functional theory, J. Phys.: Condens. Matter 35, 041501 (2023).
  17. L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures (Springer Science & Business Media, 2008).
  18. K. Neklyudov, J. Nys, L. Thiede, J. Carrasquilla, Q. Liu, M. Welling, and A. Makhzani, Wasserstein quantum Monte Carlo: a novel approach for solving the quantum many-body Schrödinger equation, Adv. Neural Inf. Proc. Syst. 36, 07050 (2024).
  19. M. Raissi, P. Perdikaris, and G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  20. M. Schmidt and J. M. Brader, Power functional theory for Brownian dynamics, J. Chem. Phys. 138, 214101 (2013).
  21. M. Schmidt, Power functional theory for many-body dynamics, Rev. Mod. Phys. 94, 015007 (2022).
  22. S. M. Tschopp and J. M. Brader, First-principles superadiabatic theory for the dynamics of inhomogeneous fluids, J. Chem. Phys. 157, 234108 (2022).
  23. G. S. Heffelfinger and F. v. Swol, Diffusion in Lennard-Jones fluids using dual control volume grand canonical molecular dynamics simulation (DCV-GCMD), J. Chem. Phys. 100, 7548 (1994).
  24. A. Ozcan, C. Perego, M. Salvalaglio, M. Parrinello, and O. Yazaydin, Concentration gradient driven molecular dynamics: a new method for simulations of membrane permeation and separation, Chem. Sci. 8, 3858 (2017).
  25. U. Thiele, I. Vancea, A. J. Archer, M. J. Robbins, L. Frastia, A. Stannard, E. Pauliac-Vaujour, C. P. Martin, M. O. Blunt, and P. J. Moriarty, Modelling approaches to the dewetting of evaporating thin films of nanoparticle suspensions, J. Phys.: Condens. Matter 21, 264016 (2009).
  26. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math. 84, 375 (2000).
  27. R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal. 29, 1 (1998).
  28. L. Chizat, G. Peyré, B. Schmitzer, and F.-X. Vialard, An interpolating distance between optimal transport and Fisher–Rao metrics, Found. Comput. Math. 18, 1 (2018).
  29. See Supplemental Material at http://link.aps.org/supplemental/10.1103/22fd-ykkb for two movie files showing dynamical versions of Figs. 3 and 4 in the main paper.
  30. A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).
  31. K. Fazel, N. Karimitari, T. Shah, C. Sutton, and R. Sundararaman, Improving the reliability of machine learned potentials for modeling inhomogeneous liquids, J. Comput. Chem. 45, 1821 (2024).
  32. D. Zwicker, py-pde: A python package for solving partial differential equations, J. Open Source Software 5, 2158 (2020).
  33. A. Nold, B. D. Goddard, P. Yatsyshin, N. Savva, and S. Kalliadasis, Pseudospectral methods for density functional theory in bounded and unbounded domains, J. Comput. Phys. 334, 639 (2017).
  34. H. Hansen-Goos and R. Roth, Density functional theory for hard-sphere mixtures: the White Bear version mark II, J. Phys.: Condens. Matter 18, 8413 (2006).
  35. D. Britt, H. Furukawa, B. Wang, T. G. Glover, and O. M. Yaghi, Highly efficient separation of carbon dioxide by a metal-organic framework replete with open metal sites, Proc. Natl. Acad. Sci. USA 106, 20637 (2009).
  36. S. Sharma, S. R. G. Balestra, R. Baur, U. Agarwal, E. Zuidema, M. S. Rigutto, S. Calero, T. J. H. Vlugt, and D. Dubbeldam, RUPTURA: simulation code for breakthrough, ideal adsorption solution theory computations, and fitting of isotherm models, Mol. Simul. 49, 893 (2023).
  37. M. Eddaoudi, J. Kim, N. Rosi, D. Vodak, J. Wachter, M. O'Keeffe, and O. M. Yaghi, Systematic design of pore size and functionality in isoreticular MOFs and their application in methane storage, Science 295, 469 (2002).
  38. A. Savitzky and M. J. E. Golay, Smoothing and differentiation of data by simplified least squares procedures, Anal. Chem. 36, 1627 (1964).
  39. T. Zimmermann, F. Sammüller, S. Hermann, M. Schmidt, and D. de las Heras, Neural force functional for non-equilibrium many-body colloidal systems, Mach. Learn.: Sci. Technol. 5, 035062 (2024).
  40. D. de las Heras, T. Zimmermann, F. Sammüller, S. Hermann, and M. Schmidt, Perspective: How to overcome dynamical density functional theory, J. Phys.: Condens. Matter 35, 271501 (2023).
  41. Code and data files to reproduce our results available at https://doi.org/10.5281/zenodo.17116151.
  42. A. Yoshimori, Microscopic derivation of time-dependent density functional methods, Phys. Rev. E 71, 031203 (2005).
  43. P. G. Boyd, S. M. Moosavi, M. Witman, and B. Smit, Force-field prediction of materials properties in metal-organic frameworks, J. Phys. Chem. Lett. 8, 357 (2017).

Outline

Information

Phys. Rev. E 112, 045314– Published 17 October, 2025
  • Received 14 May 2025
  • Accepted 10 September 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation