- Physical Review Journals
All JournalsPhysics Magazine

PRX Intelligence

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

Fidelity of Machine-Learned Potentials: Quantitative Assessment for Protonated Oxalate

Chen Qu1,*, Paul L. Houston2, Qi Yu3, Apurba Nandi4, Joel M. Bowman4,†, Valerii Andreichev5, Silvan Käser5, and Markus Meuwly5,‡

  • 1Independent Researcher, Toronto, Ontario M9B0E3, Canada
  • 2Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
  • 3Department of Chemistry, Fudan University, Shanghai 200438, People’s Republic of China
  • 4Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, USA
  • 5Department of Chemistry, University of Basel, Basel, Switzerland

  • *Contact author: szquchen@gmail.com
  • †Contact author: jmbowma@emory.edu
  • ‡Contact author: m.meuwly@unibas.ch
PDF

PRX Intelligence 1, 023006 – Published 21 October, 2026

DOI: https://doi.org/10.1103/jfjx-djk2

Abstract

There has been a veritable explosion of methods and software to perform machine-learned regression on datasets of electronic energies and forces to develop high-dimensional machine-learned potential energy surfaces (ML-PESs). A major, but not deeply studied aspect is how well different ML-PESs represent the same dataset on which they were trained, beyond the standard statistical precision measures. Here, this is examined using several “stress tests,” for two widely used machine-learned potential approaches. One is based on permutationally invariant polynomial linear least-square regression and the other is the message-passing neural network PhysNet approach. The energies from the two PESs are directly compared as are the IR spectra determined from vibrational self-consistent field/vibrational configuration interaction calculations using fitted dipole moment surfaces. In addition, tunneling splittings for the hydrogen transfer between two equivalent structures are reported using three different methods: ring polymer instanton theory, diffusion Monte Carlo simulations, and the Qim-path method. These calculations require the evaluation of on the order of one billion energies that are widely dispersed in the 15-dimensional configurational space. The two PESs yield results for these quantities in excellent agreement with each other.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (77)

  1. M. Meuwly, Reactive molecular dynamics: From small molecules to proteins, WIREs Comput. Mol. Sci. 9, e1386 (2019).
  2. S. Manzhos and T. J. Carrington, Neural network potential energy surfaces for small molecules and reactions, Chem. Rev. 121, 10187 (2021).
  3. B. J. Braams and J. M. Bowman, Permutationally invariant potential energy surfaces in high dimensionality, Int. Rev. Phys. Chem. 28, 577 (2009).
  4. O. T. Unke, S. Chmiela, H. E. Sauceda, M. Gastegger, I. Poltavsky, K. T. Schütt, A. Tkatchenko, and K.-R. Müller, Machine learning force fields, Chem. Rev. 121, 10142 (2021).
  5. D. Lu, H. Wang, M. Chen, L. Lin, R. Car, E. Weinan, W. Jia, and L. Zhang, 86 PFLOPS deep potential molecular dynamics simulation of 100 million atoms with ab initio accuracy, Comput. Phys. Commun. 259, 107624 (2021).
  6. L. Zhang, J. Han, H. Wang, R. Car, and W. E., Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics, Phys. Rev. Lett. 120, 143001 (2018).
  7. O. T. Unke et al., Biomolecular dynamics with machine-learned quantum-mechanical force fields trained on diverse chemical fragments, Sci. Adv. 10, eadn4397 (2024).
  8. V. Zaverkin, M. Ferraz, F. Alesiani, and M. Niepert, Performance of universal machine-learned potentials with explicit long-range interactions in biomolecular simulations, J. Chem. Theory Comput. 22, 2074 (2026).
  9. A. Musaelian, S. Batzner, A. Johansson, L. Sun, C. J. Owen, M. Kornbluth, and B. Kozinsky, Learning local equivariant representations for large-scale atomistic dynamics, Nat. Commun. 14, 579 (2023).
  10. B. Kozinsky, A. Musaelian, A. Johansson, and S. Batzner, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC’23 (ACM Press, New York, 2023).
  11. K. Kadau, T. C. Germann, and P. S. Lomdahl, Molecular dynamics comes of age: 320 billion atom simulation on BlueGene/L, Int. J. Mod. Phys. C 17, 1755 (2006).
  12. T. C. Germann and K. Kadau, Trillion-atom molecular dynamics becomes a reality, Int. J. Mod. Phys. C 19, 1315 (2008).
  13. Y. Shibuta, S. Sakane, E. Miyoshi, S. Okita, T. Takaki, and M. Ohno, Heterogeneity in homogeneous nucleation from billion-atom molecular dynamics simulation of solidification of pure metal, Nat. Commun. 8, 10 (2017).
  14. L. Casalino, C. Seitz, J. Lederhofer, Y. Tsybovsky, I. A. Wilson, M. Kanekiyo, and R. E. Amaro, Breathing and tilting: Mesoscale simulations illuminate influenza glycoprotein vulnerabilities, ACS Cent. Sci. 8, 1646 (2022).
  15. K. Santos et al., Breaking the molecular dynamics timescale barrier using a wafer-scale system, in SC24: International Conference for High Performance Computing, Networking, Storage and Analysis (IEEE, New York, 2024), pp. 1–13.
  16. D. Ugarte La Torre, J. Jung, and Y. Sugita, Scaling molecular dynamics for large-scale simulation of biological systems on AMD CPU/GPU supercomputers: Lessons from LUMI: Optimizing GENESIS for maximizing the computational efficiency of CPU and GPU kernels on the LUMI supercomputer, in Proceedings of the International Conference on High Performance Computing in Asia-Pacific Region (ACM ICPS, New York, 2025), pp. 1–12.
  17. K. Lek Chaton and M. Meuwly, Enhancing empirical energy functions using physics- and machine learning-based extensions: Structure, dynamics and spectroscopy of modified benzenes, J. Comput. Chem. 46, e70162 (2025).
  18. S.-L. J. Lahey and C. N. Rowley, Simulating protein–ligand binding with neural network potentials, Chem. Sci. 11, 2362 (2020).
  19. M. Gastegger, K. T. Schütt, and K.-R. Müller, Machine learning of solvent effects on molecular spectra and reactions, Chem. Sci. 12, 11473 (2021).
  20. T. J. Inizan, T. Plé, O. Adjoua, P. Ren, H. Gökcan, O. Isayev, L. Lagardère, and J.-P. Piquemal, Scalable hybrid deep neural networks/polarizable potentials biomolecular simulations including long-range effects, Chem. Sci. 14, 5438 (2023).
  21. K. Song, S. Käser, K. Töpfer, L. I. Vazquez-Salazar, and M. Meuwly, PhysNet meets CHARMM: A framework for routine machine learning/molecular mechanics simulations, J. Chem. Phys. 159, 024125 (2023).
  22. K. Zinovjev, Electrostatic embedding of machine learning potentials, J. Chem. Theory Comput. 19, 1888 (2023).
  23. K. Zinovjev, L. Hedges, R. Montagud Andreu, C. Woods, I. Tuñón, and M. W. van der Kamp, emle-engine: A flexible electrostatic machine learning embedding package for multiscale molecular dynamics simulations, J. Chem. Theory Comput. 20, 4514 (2024).
  24. J. Kalayan, I. Ramzan, C. D. Williams, R. A. Bryce, and N. A. Burton, A neural network potential based on pairwise resolved atomic forces and energies, J. Comput. Chem. 45, 1143 (2024).
  25. I. Poltavsky et al., Crash testing machine learning force fields for molecules, materials, and interfaces: Model analysis in the TEA challenge 2023, Chem. Sci. 16, 3720 (2025).
  26. I. Poltavsky et al., Crash testing machine learning force fields for molecules, materials, and interfaces: Molecular dynamics in the TEA challenge 2023, Chem. Sci. 16, 3738 (2025).
  27. J. Xia, Y. Zhang, and B. Jiang, The evolution of machine learning potentials for molecules, reactions and materials, Chem. Soc. Rev. 54, 4790 (2025).
  28. W. G. Stark, C. van der Oord, I. Batatia, Y. Zhang, B. Jiang, G. Csányi, and R. J. Maurer, Benchmarking of machine learning interatomic potentials for reactive hydrogen dynamics at metal surfaces, Mach. Learn.: Sci. Technol. 5, 030501 (2024).
  29. O. T. Unke and M. Meuwly, PhysNet: A neural network for predicting energies, forces, dipole moments, and partial charges, J. Chem. Theory Comput. 15, 3678 (2019).
  30. J. M. Bowman, C. Qu, R. Conte, A. Nandi, P. L. Houston, and Q. Yu, A perspective marking 20 years of using permutationally invariant polynomials for molecular potentials, J. Chem. Phys. 162, 180901 (2025).
  31. S. Käser, L. I. Vazquez-Salazar, M. Meuwly, and K. Töpfer, Neural network potentials for chemistry: Concepts, applications and prospects, Digital Discovery 2, 28 (2023).
  32. W. Hwang et al., CHARMM at 45: Enhancements in accessibility, functionality, and speed, J. Phys. Chem. B 128, 9976 (2024).
  33. J. M. Bowman, S. Carter, and X. Huang, MULTIMODE: A code to calculate rovibrational energies of polyatomic molecules, Int. Rev. Phys. Chem. 22, 533 (2003).
  34. Q. Yu et al., MULTIMODE, the n-mode representation of the potential and illustrations to IR spectra of glycine and two protonated water clusters, in Vibrational Dynamics of Molecules edited by J. M. Bowman (World Scientific, Singapore, 2022), pp. 296–339.
  35. C. Qu, T. C. Allison, P. L. Houston, R. Conte, A. Nandi, and J. M. Bowman, Computational spectroscopy using MULTIMODE and machine-learned potentials, J. Chem. Phys. 164, 134116 (2026).
  36. C. Qu, P. L. Houston, and J. M. Bowman, Does the diffuse OH-stretch band in the IR spectrum of protonated oxalate exhibit quantum chaos? J. Chem. Phys. 164, 061103 (2026).
  37. J. B. Anderson, A random-walk simulation of the Schrödinger equation: H+3, J. Chem. Phys. 63, 1499 (1975).
  38. I. Kosztin, B. Faber, and K. Schulten, Introduction to the diffusion Monte Carlo method, Am. J. Phys. 64, 633 (1996).
  39. J. B. Anderson, Quantum chemistry by random walk. H2P,H3+D3h1A1′,H23Σu+,H41Σg+,Be1S, J. Chem. Phys. 65, 4121 (1976).
  40. Y. Wang, B. J. Braams, J. M. Bowman, S. Carter, and D. P. Tew, Full-dimensional quantum calculations of ground-state tunneling splitting of malonaldehyde using an accurate ab initio potential energy surface, J. Chem. Phys. 128, 224314 (2008).
  41. J. O. Richardson and S. C. Althorpe, Ring-polymer instanton method for calculating tunneling splittings, J. Chem. Phys. 134, 054109 (2011).
  42. J. O. Richardson, Ring-polymer instanton theory, Int. Rev. Phys. Chem. 37, 171 (2018).
  43. S. Käser, J. O. Richardson, and M. Meuwly, Transfer learning for affordable and high-quality tunneling splittings from instanton calculations, J. Chem. Theory Comput. 18, 6840 (2022).
  44. J. E. Lawrence, J. Dušek, and J. O. Richardson, Perturbatively corrected ring-polymer instanton theory for accurate tunneling splittings, J. Chem. Phys. 159, 014111 (2023).
  45. E. Kamarchik, Y. Wang, and J. Bowman, Reduced-dimensional quantum approach to tunneling splittings using saddle-point normal coordinates, J. Phys. Chem. A 113, 7556 (2009).
  46. D. T. Colbert and W. H. Miller, A novel discrete variable representation for quantum mechanical reactive scattering via the S -matrix Kohn method, J. Chem. Phys. 96, 1982 (1992).
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jfjx-djk2 for methods for the spectroscopic studies [33, 34, 35, 36], diffusion Monte Carlo calculations [37, 38, 39, 40], and computing tunneling splittings [41, 42, 43, 44, 45, 46], and reports a comparison of IR spectra for 3MR and 4MR VSCF/VCI calculations and details of the CCSD(T)/CBS energies along the Qim path.
  48. M. E. Taylor and P. Stone, Transfer learning for reinforcement learning domains: A survey, J. Mach. Learn. Res. 10, 1633 (2009).
  49. S. J. Pan and Q. Yang, A survey on transfer learning, IEEE Trans. Knowl. Data Eng. 22, 1345 (2009).
  50. J. S. Smith, B. T. Nebgen, R. Zubatyuk, N. Lubbers, C. Devereux, K. Barros, S. Tretiak, O. Isayev, and A. E. Roitberg, Approaching coupled cluster accuracy with a general-purpose neural network potential through transfer learning, Nat. Commun. 10, 2903 (2019).
  51. S. Käser and M. Meuwly, Transfer learned potential energy surfaces: Accurate anharmonic vibrational dynamics and dissociation energies for the formic acid monomer and dimer, Phys. Chem. Chem. Phys. 24, 5269 (2022).
  52. H.-J. Werner, P. J. Knowles, et al., MOLPRO, version 2019.2, a package of ab initio programs (2019), https://www.molpro.net/.
  53. V. Andreichev, S. Käser, E. L. Bocanegra, M. Salik, M. A. Johnson, and M. Meuwly, Dynamics of protonated oxalate from machine-learned simulations and experiment: Infrared signatures, proton transfer dynamics and tunneling splittings, Phys. Chem. Chem. Phys. 27, 23288 (2025).
  54. S. Käser, D. Koner, and M. Meuwly, Compact kernel/neural network representation for accurate, fast, and global reactive molecular potential energy surfaces, Precis. Chem. 4, 571 (2026).
  55. Z. Xie and J. M. Bowman, Permutationally invariant polynomial basis for molecular energy surface fitting via monomial symmetrization, J. Chem. Theory Comput. 6, 26 (2010).
  56. MSA Software with Gradients, 2029, https://github.com/szquchen/MSA-2.0, accessed 20 January 2019.
  57. Intel Corporation, Intel® oneAPI Math Kernel Library, 2020, https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl.html.
  58. P. L. Houston, C. Qu, Q. Yu, R. Conte, A. Nandi, J. K. Li, and J. M. Bowman, PESPIP: Software to fit complex molecular and many-body potential energy surfaces with permutationally invariant polynomials, J. Chem. Phys. 158, 044109 (2023).
  59. R. Ramakrishnan, P. Dral, M. Rupp, and O. A. von Lilienfeld, Big data meets quantum chemistry approximations: The Δ-machine learning approach, J. Chem. Theory Comput. 11, 2087 (2015).
  60. A. Nandi, C. Qu, P. L. Houston, R. Conte, and J. M. Bowman, Δ-machine learning for potential energy surfaces: A PIP approach to bring a DFT-based PES to CCSD(T) level of theory, J. Chem. Phys. 154, 051102 (2021).
  61. A. Nandi et al., Ring-polymer instanton tunneling splittings of tropolone and isotopomers using a Δ-machine learned CCSD(T) potential: Theory and experiment shake hands, J. Am. Chem. Soc. 145, 9655 (2023).
  62. J. M. Bowman, C. Qu, R. Conte, A. Nandi, P. L. Houston, and Q. Yu, Δ-machine learned potential energy surfaces and force fields, J. Chem. Theory Comput. 19, 1 (2023).
  63. Y. Liu and J. Li, Permutation-invariant-polynomial neural-network-based Δ-machine learning approach: A case for the HO2 self-reaction and its dynamics study, J. Phys. Chem. Lett. 13, 4729 (2022).
  64. P. L. Houston, C. Qu, Q. Yu, P. Pandey, R. Conte, A. Nandi, J. M. Bowman, and S. G. Kukolich, Formic acid–ammonia heterodimer: A new Δ-machine learning CCSD(T)-level potential energy surface allows investigation of the double proton transfer, J. Chem. Theory Comput. 20, 1821 (2024).
  65. X. Huang, B. J. Braams, and J. M. Bowman, Ab initio potential energy and dipole moment surfaces for H5O2+, J. Chem. Phys. 122, 044308 (2005).
  66. A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008), 2nd ed.
  67. D. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980.
  68. S. J. Reddi, S. Kale, and S. Kumar, On the convergence of adam and beyond, arXiv:1904.09237.
  69. E. Boittier, K. Töpfer, M. Devereux, and M. Meuwly, Kernel-based minimal distributed charges: A conformationally dependent ESP-model for molecular simulations, J. Chem. Theory Comput. 20, 8088 (2024).
  70. E. D. Boittier and M. Meuwly, Efficient and equivariant prediction of distributed charges for accurate molecular electrostatics, J. Chem. Theory Comput. 22, 4532 (2026).
  71. S. Käser, D. Koner, A. S. Christensen, O. A. von Lilienfeld, and M. Meuwly, Machine learning models of vibrating H2CO: Comparing reproducing kernels, FCHL, and PhysNet, J. Phys. Chem. A 124, 8853 (2020).
  72. S. Käser, E. D. Boittier, M. Upadhyay, and M. Meuwly, Transfer learning to CCSD(T): Accurate anharmonic frequencies from machine learning models, J. Chem. Theory Comput. 17, 3687 (2021).
  73. C. T. Wolke, A. F. DeBlase, C. M. Leavitt, A. B. McCoy, and M. A. Johnson, Diffuse vibrational signature of a single proton embedded in the oxalate scaffold, HO2 CCO2−, J. Phys. Chem. A 119, 13018 (2015).
  74. S. Käser, J. O. Richardson, and M. Meuwly, Transfer learning for predictive molecular simulations: Data-efficient potential energy surfaces at CCSD(T) accuracy, J. Chem. Theory Comput. 21, 6633 (2025).
  75. C. Qu, A. Nandi, P. L. Houston, and J. M. Bowman, Tests demonstrate the accuracy of the Qim -reaction path Hamiltonian for double-well tunneling splittings for H-atom transfer, J. Phys. Chem. A 130, 3027 (2026).
  76. J. Wang, J. C. S. V. Veliz, M. Upadhyay, and M. Meuwly, High-energy reaction dynamics of O3, J. Chem. Phys. 163, 074109 (2025).
  77. C. Qu, P. L. Houston, Q. Yu, A. Nandi, J. M. Bowman, V. Andreichev, S. Käser, and M. Meuwly, oxalate-2, 2026, https://github.com/MMunibas/oxalate-2.

Outline

Information

PRX Intelligence 1, 023006– Published 21 October, 2026
  • Received 20 April 2026
  • Accepted 2 September 2026
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