- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Materials

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
  • Access by Xinjiang University

Interpretability of linear regression models of glassy dynamics

Anand Sharma

Chen Liu

Misaki Ozawa*

Daniele Coslovich†

  • Innovation and Research Division, Ge-Room Inc., 93160 Noisy le Grand, France

  • *Contact author: misaki.ozawa@univ-grenoble-alpes.fr
  • †Contact author: dcoslovich@units.it
PDF

Phys. Rev. Materials 10, 035602 – Published 2 March, 2026

DOI: https://doi.org/10.1103/q6pd-7trs

Abstract

Data-driven models can accurately describe and predict the dynamical properties of glass-forming liquids from structural data. Accurate predictions, however, do not guarantee an understanding of the underlying physical phenomena and the key factors that control them. In this paper, we illustrate the merits and limitations of linear regression models of glassy dynamics built on high-dimensional structural descriptors. By analyzing data for a two-dimensional glass model, we show that several descriptors commonly used in glass-transition studies display multicollinearity, which hinders the interpretability of linear models. Ridge regression suppresses some of the shortcomings of multicollinearity, but its solutions are not concise enough to be physically interpretable. Only by using dimensional reduction techniques we do eventually obtain linear models that strike a balance between prediction accuracy and interpretability. Our analysis points to a key role of local packing and composition fluctuations in the glass model under study.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (83)

  1. L. Berthier and G. Biroli, Theoretical perspective on the glass transition and amorphous materials, Rev. Mod. Phys. 83, 587 (2011).
  2. K. Binder and W. Kob, Glassy Materials and Disordered Solids: An Introduction to Their Statistical Mechanics (World Scientific, Singapore, 2011).
  3. M. D. Ediger, C. A. Angell, and S. R. Nagel, Supercooled liquids and glasses, J. Phys. Chem. 100, 13200 (1996).
  4. M. D. Ediger, Spatially heterogeneous dynamics in supercooled liquids, Annu. Rev. Phys. Chem. 51, 99 (2000).
  5. L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, and W. van Saarloos, Dynamical Heterogeneities in Glasses, Colloids, and Granular Media (Oxford University Press, Oxford, 2011).
  6. S. Karmakar, C. Dasgupta, and S. Sastry, Growing length scales and their relation to timescales in glass-forming liquids, Annu. Rev. Condens. Matter Phys. 5, 255 (2014).
  7. A. Widmer-Cooper, P. Harrowell, and H. Fynewever, How reproducible are dynamic heterogeneities in a supercooled liquid? Phys. Rev. Lett. 93, 135701 (2004).
  8. E. R. Weeks, Introduction to the colloidal glass transition, ACS Macro Lett. 6, 27 (2017).
  9. D. Coslovich and G. Pastore, Understanding fragility in supercooled Lennard-Jones mixtures. I. Locally preferred structures, J. Chem. Phys. 127, 124504 (2007).
  10. C. Patrick Royall, S. R. Williams, T. Ohtsuka, and H. Tanaka, Direct observation of a local structural mechanism for dynamic arrest, Nat. Mater. 7, 556 (2008).
  11. A. Widmer-Cooper, H. Perry, P. Harrowell, and D. R. Reichman, Irreversible reorganization in a supercooled liquid originates from localized soft modes, Nat. Phys. 4, 711 (2008).
  12. G. M. Hocky, D. Coslovich, A. Ikeda, and D. R. Reichman, Correlation of local order with particle mobility in supercooled liquids is highly system dependent, Phys. Rev. Lett. 113, 157801 (2014).
  13. R. L. Jack, A. J. Dunleavy, and C. P. Royall, Information-theoretic measurements of coupling between structure and dynamics in glass formers, Phys. Rev. Lett. 113, 095703 (2014).
  14. F. Turci, C. P. Royall, and T. Speck, Nonequilibrium phase transition in an atomistic glassformer: The connection to thermodynamics, Phys. Rev. X 7, 031028 (2017).
  15. C. P. Royall and S. R. Williams, The role of local structure in dynamical arrest, Phys. Rep. 560, 1 (2015).
  16. H. Tanaka, H. Tong, R. Shi, and J. Russo, Revealing key structural features hidden in liquids and glasses, Nat. Rev. Phys. 1, 333 (2019).
  17. G. Kapteijns, D. Richard, E. Bouchbinder, T. B. Schrøder, J. C. Dyre, and E. Lerner, Does mesoscopic elasticity control viscous slowing down in glassforming liquids? J. Chem. Phys. 155, 074502 (2021).
  18. M. Sharma, M. K. Nandi, and S. M. Bhattacharyya, Identifying structural signature of dynamical heterogeneity via the local softness parameter, Phys. Rev. E 105, 044604 (2022).
  19. M. L. Manning and A. J. Liu, Vibrational modes identify soft spots in a sheared disordered packing, Phys. Rev. Lett. 107, 108302 (2011).
  20. D. Richard, M. Ozawa, S. Patinet, E. Stanifer, B. Shang, S. Ridout, B. Xu, G. Zhang, P. Morse, J.-L. Barrat, et al., Predicting plasticity in disordered solids from structural indicators, Phys. Rev. Mater. 4, 113609 (2020).
  21. M. Baggioli, I. Kriuchevskyi, T. W. Sirk, and A. Zaccone, Plasticity in amorphous solids is mediated by topological defects in the displacement field, Phys. Rev. Lett. 127, 015501 (2021).
  22. Z. W. Wu, Y. Chen, W.-H. Wang, W. Kob, and L. Xu, Topology of vibrational modes predicts plastic events in glasses, Nat. Commun. 14, 2955 (2023).
  23. E. D. Cubuk, S. S. Schoenholz, J. M. Rieser, B. D. Malone, J. Rottler, D. J. Durian, E. Kaxiras, and A. J. Liu, Identifying structural flow defects in disordered solids using machine-learning methods, Phys. Rev. Lett. 114, 108001 (2015).
  24. G. Jung, R. M. Alkemade, V. Bapst, D. Coslovich, L. Filion, F. P. Landes, A. J. Liu, F. S. Pezzicoli, H. Shiba, G. Volpe, et al., Roadmap on machine learning glassy dynamics, Nat. Rev. Phys. 7, 91 (2025).
  25. S. S. Schoenholz, E. D. Cubuk, D. M. Sussman, E. Kaxiras, and A. J. Liu, A structural approach to relaxation in glassy liquids, Nat. Phys. 12, 469 (2016).
  26. G. Jung, G. Biroli, and L. Berthier, Predicting dynamic heterogeneity in glass-forming liquids by physics-inspired machine learning, Phys. Rev. Lett. 130, 238202 (2023).
  27. V. Bapst, T. Keck, A. Grabska-Barwińska, C. Donner, E. D. Cubuk, S. S. Schoenholz, A. Obika, A. W. Nelson, T. Back, D. Hassabis, et al., Unveiling the predictive power of static structure in glassy systems, Nat. Phys. 16, 448 (2020).
  28. H. Shiba, M. Hanai, T. Suzumura, and T. Shimokawabe, BOTAN: BOnd TArgeting Network for prediction of slow glassy dynamics by machine learning relative motion, J. Chem. Phys. 158, 084503 (2023).
  29. X. Jiang, Z. Tian, K. Li, and W. Hu, A geometry-enhanced graph neural network for learning the smoothness of glassy dynamics from static structure, J. Chem. Phys. 159, 144504 (2023).
  30. F. S. Pezzicoli, G. Charpiat, and F. P. Landes, Rotation-equivariant graph neural networks for learning glassy liquids representations, SciPost Phys. 16, 136 (2024).
  31. E. Boattini, F. Smallenburg, and L. Filion, Averaging local structure to predict the dynamic propensity in supercooled liquids, Phys. Rev. Lett. 127, 088007 (2021).
  32. D. Teney, M. Peyrard, and E. Abbasnejad, Predicting is not understanding: Recognizing and addressing underspecification in machine learning, in European Conference on Computer Vision (Springer, Berlin, 2022), p. 458.
  33. A. Swain, S. A. Ridout, and I. Nemenman, Machine learning that predicts well may not learn the correct physical descriptions of glassy systems, Phys. Rev. Res. 6, 033091 (2024).
  34. K. Swanson, S. Trivedi, J. Lequieu, K. Swanson, and R. Kondor, Deep learning for automated classification and characterization of amorphous materials, Soft Matter 16, 435 (2020).
  35. F. Font-Clos, M. Zanchi, S. Hiemer, S. Bonfanti, R. Guerra, M. Zaiser, and S. Zapperi, Predicting the failure of two-dimensional silica glasses, Nat. Commun. 13, 2820 (2022).
  36. N. Oyama, S. Koyama, and T. Kawasaki, What do deep neural networks find in disordered structures of glasses? Front. Phys. 10, 1007861 (2023).
  37. S. Ciarella, D. Khomenko, L. Berthier, F. C. Mocanu, D. R. Reichman, C. Scalliet, and F. Zamponi, Finding defects in glasses through machine learning, Nat. Commun. 14, 4229 (2023).
  38. G. Janzen, X. L. Smeets, V. E. Debets, C. Luo, C. Storm, L. M. Janssen, and S. Ciarella, Dead or alive: Distinguishing active from passive particles using supervised learning, Europhys. Lett. 143, 17004 (2023).
  39. G. Janzen, C. Smit, S. Visbeek, V. E. Debets, C. Luo, C. Storm, S. Ciarella, and L. M. C. Janssen, Classifying the age of a glass based on structural properties: A machine learning approach, Phys. Rev. Mater. 8, 025602 (2024).
  40. M. Liu, N. Oyama, T. Kawasaki, and H. Mizuno, Classification of solid and liquid structures using a deep neural network unveils origin of dynamical heterogeneities in supercooled liquids, J. Appl. Phys. 136, 144702 (2024).
  41. G. Jung, G. Biroli, and L. Berthier, Dynamic heterogeneity at the experimental glass transition predicted by transferable machine learning, Phys. Rev. B 109, 064205 (2024).
  42. Q. Wang, L.-F. Zhang, Z.-Y. Zhou, and H.-B. Yu, Predicting the pathways of string-like motions in metallic glasses via path-featurizing graph neural networks, Sci. Adv. 10, eadk2799 (2024).
  43. K. Yoshikawa, K. Yano, S. Goto, K. Kim, and N. Matubayasi, Graph neural network-based structural classification of glass-forming liquids and its interpretation via self-attention mechanism, J. Chem. Phys. 163, 024508 (2025).
  44. Q. Wang, J. Ding, L. Zhang, E. Podryabinkin, A. Shapeev, and E. Ma, Predicting the propensity for thermally activated β events in metallic glasses via interpretable machine learning, npj Comput. Mater. 6, 194 (2020).
  45. C. Liu, Y. Wang, Y. Wang, M. Islam, J. Hwang, Y. Wang, and Y. Fan, Concurrent prediction of metallic glasses' global energy and internal structural heterogeneity by interpretable machine learning, Acta Mater. 259, 119281 (2023).
  46. R. M. Alkemade, E. Boattini, L. Filion, and F. Smallenburg, Comparing machine learning techniques for predicting glassy dynamics, J. Chem. Phys. 156, 204503 (2022).
  47. C. Rudin, Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead, Nat. Mach. Intell. 1, 206 (2019).
  48. D. C. Montgomery, E. A. Peck, and G. G. Vining, Introduction to Linear Regression Analysis (John Wiley & Sons, Hoboken, NJ, 2021).
  49. A. Widmer-Cooper and P. Harrowell, Predicting the long-time dynamic heterogeneity in a supercooled liquid on the basis of short-time heterogeneities, Phys. Rev. Lett. 96, 185701 (2006).
  50. D. A. Broniatowski, Psychological Foundations of Explainability and Interpretability in Artificial Intelligence (US Department of Commerce, National Institute of Standards and Technology, Gaithersburg, MD, 2021), Vol. 4.
  51. S. J. Wetzel, S. Ha, R. Iten, M. Klopotek, and Z. Liu, Interpretable machine learning in physics: A review, arXiv:2503.23616.
  52. C. Rowan and A. Doostan, On the definition and importance of interpretability in scientific machine learning, arXiv:2505.13510.
  53. S. Kivelson and S. Kivelson, Understanding complexity, Nat. Phys. 14, 426 (2018).
  54. A. Sharma, C. Liu, and M. Ozawa, Selecting relevant structural features for glassy dynamics by information imbalance, J. Chem. Phys. 161, 184506 (2024).
  55. M. L. Falk and J. S. Langer, Dynamics of viscoplastic deformation in amorphous solids, Phys. Rev. E 57, 7192 (1998).
  56. A. Barbot, M. Lerbinger, A. Hernandez-Garcia, R. García-García, M. L. Falk, D. Vandembroucq, and S. Patinet, Local yield stress statistics in model amorphous solids, Phys. Rev. E 97, 033001 (2018).
  57. M. Lerbinger, A. Barbot, D. Vandembroucq, and S. Patinet, Relevance of shear transformations in the relaxation of supercooled liquids, Phys. Rev. Lett. 129, 195501 (2022).
  58. A. D. Parmar, M. Ozawa, and L. Berthier, Ultrastable metallic glasses in silico, Phys. Rev. Lett. 125, 085505 (2020).
  59. D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Elsevier, Amsterdam, 2023).
  60. L. Berthier and W. Kob, The Monte Carlo dynamics of a binary Lennard-Jones glass-forming mixture, J. Phys.: Condens. Matter 19, 205130 (2007).
  61. H. Shiba, Y. Yamada, T. Kawasaki, and K. Kim, Unveiling dimensionality dependence of glassy dynamics: 2D infinite fluctuation eclipses inherent structural relaxation, Phys. Rev. Lett. 117, 245701 (2016).
  62. J. Behler and M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy surfaces, Phys. Rev. Lett. 98, 146401 (2007).
  63. J. W. Rocks, S. A. Ridout, and A. J. Liu, Learning-based approach to plasticity in athermal sheared amorphous packings: Improving softness, APL Mater. 9, 021107 (2021).
  64. J. Behler, Constructing high-dimensional neural network potentials: A tutorial review, Int. J. Quantum Chem. 115, 1032 (2015).
  65. T. Kawasaki, T. Araki, and H. Tanaka, Correlation between dynamic heterogeneity and medium-range order in two-dimensional glass-forming liquids, Phys. Rev. Lett. 99, 215701 (2007).
  66. C. F. Schreck, C. S. O'Hern, and L. E. Silbert, Tuning jammed frictionless disk packings from isostatic to hyperstatic, Phys. Rev. E 84, 011305 (2011).
  67. H. Tong and H. Tanaka, Revealing hidden structural order controlling both fast and slow glassy dynamics in supercooled liquids, Phys. Rev. X 8, 011041 (2018).
  68. C. Rycroft, VORO++: A three-dimensional Voronoi cell library in C++, Chaos 19, 041111 (2009).
  69. F. H. Stillinger and T. A. Weber, Packing structures and transitions in liquids and solids, Science 225, 983 (1984).
  70. G. James, D. Witten, T. Hastie, and R. Tibshirani, An Introduction to Statistical Learning: With Applications in R (Springer, New York, 2013).
  71. The oscillations are not due to limited statistics or dataset-to-dataset fluctuations, since the error bars are relatively small compared to the typical absolute value of the largest weights.
  72. D. W. Marquardt and R. D. Snee, Ridge regression in practice, Am. Stat. 29, 3 (1975).
  73. H. Zou and T. Hastie, Regularization and variable selection via the elastic net, J. R. Stat. Soc. B 67, 301 (2005).
  74. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, et al., Scikit-learn: Machine learning in Python, J. Mach. Learn. Res. 12, 2825 (2011).
  75. H. Peng, F. Long, and C. Ding, Feature selection based on mutual information criteria of max-dependency, max-relevance, and min-redundancy, IEEE Trans. Pattern Anal. Mach. Intell. 27, 1226 (2005).
  76. L. Berthier and R. L. Jack, Structure and dynamics of glass formers: Predictability at large length scales, Phys. Rev. E 76, 041509 (2007).
  77. J. Rottler and C. Ortner, Analysis of local structure of mechanical and thermal rearrangements in glasses with the atomic cluster expansion, J. Phys. Chem. B 128, 11492 (2024).
  78. H. Tanaka, A simple physical model of liquid-glass transition: Intrinsic fluctuating interactions and random fields hidden in glass-forming liquids, J. Phys.: Condens. Matter 10, L207 (1998).
  79. H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19, 716 (1974).
  80. G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6, 461 (1978).
  81. A. Sharma, C. Liu, M. Ozawa, and D. Coslovich, Interpretability of linear regression models of glassy dynamics, Zenodo (2025), https://doi.org/10.5281/zenodo.16942436.
  82. G. H. Golub and C. F. Van Loan, Matrix Computations (John Hopkins University Press, Baltimore, MD, 2013).
  83. L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, Philadelphia, 2022).

Outline

Information

Phys. Rev. Materials 10, 035602– Published 2 March, 2026
  • Received 25 August 2025
  • Revised 23 December 2025
  • Accepted 6 February 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