REFERENCES

1. Tang, Y.; Li, B. Belted and ensembled neural network for linear and nonlinear sufficient dimension reduction. J. Am. Stat. Assoc. 2026, 1-12.

2. Wang, S.; Shang, S.; Liu, Z.; Hao, W. ZENN: A thermodynamics-inspired computational framework for heterogeneous data-driven modeling. Proc. Natl. Acad. Sci. U.S.A. 2026, 123, e2511227122.

3. Azevedo, F. A.; Carvalho, L. R.; Grinberg, L. T.; et al. Equal numbers of neuronal and nonneuronal cells make the human brain an isometrically scaled‐up primate brain. J. Comp. Neurol. 2009, 513, 532-41.

4. Lent, R.; Azevedo, F. A. C.; Andrade‐Moraes, C. H.; Pinto, A. V. O. How many neurons do you have? Some dogmas of quantitative neuroscience under revision. Eur. J. Neurosci. 2011, 35, 1-9.

5. Kande, E. R.; Koester, J. D.; Mack, S. H.; Siegelbaum, S. A. Principles of neural science, 6th ed.; McGraw-Hill, 2021.

6. Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. 1964, 136, B864-71.

7. Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133-8.

8. Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865-8.

9. Liu, Z.; Wang, Y.; Shang, S. Zentropy theory for positive and negative thermal expansion. J. Phase. Equilib. Diffus. 2022, 43, 598-605.

10. Wang, Y.; Hector, L. G.; Zhang, H.; Shang, S. L.; Chen, L. Q.; Liu, Z. K. Thermodynamics of the Ceγ-αtransition: density-functional study. Phys. Rev. B. 2008, 78, 104113.

11. Wang, Y.; Hector Jr, L. G.; Zhang, H.; Shang, S. L.; Chen, L. Q.; Liu, Z. K. A thermodynamic framework for a system with itinerant-electron magnetism. J. Phys:. Condens. Matter. 2009, 21, 326003.

12. Wang, Y.; Shang, S.; Zhang, H.; Chen, L.; Liu, Z. Thermodynamic fluctuations in magnetic states: Fe3Pt as a prototype. Philos. Mag. Lett. 2010, 90, 851-9.

13. Perdew, J. P. SCAN meta-GGA, strong correlation, symmetry breaking, self-interaction correction, and semi-classical limit in density functional theory: Hidden connections and beneficial synergies? APL. Computational. Physics. 2025, 1, 010903.

14. Minsky, M., Papert, S. A. Perceptrons: an introduction to computational geometry. The MIT Press, 2017.

15. Rumelhart, D. E.; Hinton, G. E.; Williams, R. J. Learning representations by back-propagating errors. Nature 1986, 323, 533-6.

16. Lecun, Y.; Boser, B.; Denker, J. S.; et al. Backpropagation applied to handwritten zip code recognition. Neural. Comput. 1989, 1, 541-51.

17. Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-based learning applied to document recognition. Proc. IEEE. 1998, 86, 2278-324.

18. Hochreiter, S.; Schmidhuber, J. Long short-term memory. Neural. Comput. 1997, 9, 1735-80.

19. Scarselli, F.; Gori, M.; Tsoi, A. C.; Hagenbuchner, M.; Monfardini, G. The graph neural network model. IEEE. Trans. Neural. Netw. 2009, 20, 61-80.

20. Vaswani, A.; Brain, G.; Shazeer, N.; et al. Attention is all you need. arXiv 2017;arXiv:1706.03762. Available online: https://arxiv.org/abs/1706.03762. [accessed 9 July 2026].

21. Radford, A., Narasimhan, K., Salimans, T. & Sutskever, I. Improving language understanding by generative pre-training. 2018. Available from: https://www.mikecaptain.com/resources/pdf/GPT-1.pdf.

22. Raissi, M.; Perdikaris, P.; Karniadakis, G. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 2019, 378, 686-707.

23. Lewis, P.; Perez, E.; Piktus, A.; et al. Retrieval-augmented generation for knowledge-intensive NLP tasks. Adv. Neural. Inf. Process. Syst. 2021, 33, 9459-74. https://proceedings.neurips.cc/paper/2020/hash/6b493230205f780e1bc26945df7481e5-Abstract.html.

24. Cobbe, K.; Kosaraju, V.; Bavarian, M.; et al. Training verifiers to solve math word problems. arXiv 2021;arXiv:2110.14168. Available online: http://arxiv.org/abs/2110.14168. [accessed 9 July 2026].

25. Ouyang, L.; Wu, J.; Jiang, X.; et al. Training language models to follow instructions with human feedback. 2022, 35, 27730-44.

26. Wei, J.; Wang, X.; Schuurmans, D.; et al. Chain-of-thought prompting elicits reasoning in large language models. Adv. Neural. Inf. Process. Syst. 2023, 35, 24824-37.

27. Schick, T.; Dwivedi-Yu, J.; Dessì, R.; et al. Toolformer: language models can teach themselves to use tools. Adv. Neural Inf. Process. Syst. 2023, 36, 68539-51.

28. Huang, Y.; Hao, W.; Lin, G. HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations. Comput. Math. Appl. 2022, 121, 62-73.

29. Zheng, H.; Huang, Y.; Huang, Z.; Hao, W.; Lin, G. HomPINNs: homotopy physics-informed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions. J. Comput. Phys. 2024, 500, 112751.

30. E, W.; Yu, B. The Deep Ritz Method: A deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 2018, 6, 1-12.

31. Lu, L.; Jin, P.; Pang, G.; Zhang, Z.; Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. 2021, 3, 218-29.

32. Li, Z.; Huang, D. Z.; Liu, B.; Anandkumar, A. Fourier neural operator with learned deformations for PDEs on general geometries. J. Mach. Learn. Res. 2023, 24, 1-26. http://jmlr.org/papers/v24/23-0064.html.

33. Hao, W.; Liu, X.; Yang, Y. Newton informed neural operator for solving nonlinear partial differential equations. Adv. Neural. Inf. Process. Syst. 2024, 37, 120832. PMC11973962.

34. Wang, J.; Hao, W. Laplacian eigenfunction-based neural operator for learning nonlinear reaction-diffusion dynamics. J. Comput. Phys. 2025, 543, 114400.

35. Clausius, R. Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Annalen. der. Physik. 2006, 201, 353-400. (in German).

36. Boltzmann, L. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte der Kais. Akad. der. Wissenschaften. 1872, 66, 275-370. (in German).

37. Gibbs, J. W. Elementary principles in statistical mechanics: developed with especial reference to the rational foundation of thermodynamics; Cambridge University Press, 2011.

38. Shannon, C. E. A mathematical theory of communication. Bell. System. Technical. Journal. 1948, 27, 379-423.

39. Kullback, S.; Leibler, R. A. On Information and Sufficiency. Ann. Math. Statist. 1951, 22, 79-86.

40. Jaynes, E. T. Information theory and statistical mechanics. Phys. Rev. 1957, 106, 620-30.

41. Bridle, J. S. Probabilistic Interpretation of Feedforward Classification Network Outputs, with Relationships to Statistical Pattern Recognition. In Neurocomputing; Soulié, F. F., Hérault, J., Eds.; Springer Berlin Heidelberg, 1990; pp 227-36.

42. Richard, M. D.; Lippmann, R. P. Neural network classifiers estimate bayesian a posteriori probabilities. Neural. Comput. 1991, 3, 461-83.

43. Quinlan, J. R. Induction of decision trees. Mach. Learn. 1986, 1, 81-106.

44. Wainwright, M. J.; Jordan, M. I. Graphical Models, Exponential Families, and Variational Inference. Found. Trends. Mach. Learn. 2008, 1, 1-305.

45. Kingma, D. P.; Welling, M. Auto-encoding variational bayes. arXiv 2014;arXiv:1312.6114. Available online: http://arxiv.org/abs/1312.6114. [accessed 9 July 2026].

46. Haarnoja, T.; Zhou, A.; Abbeel, P.; Levine, S. Soft actor-critic: off-policy maximum entropy deep reinforcement learning with a stochastic actor. arXiv 2018;arXiv:1801.01290. Available online: http://arxiv.org/abs/1801.01290. [accessed 9 July 2026].

47. Holtzman, A.; Buys, J.; Du, L.; Forbes, M.; Choi, Y. The curious case of neural text degeneration. arXiv 2020;arXiv:1904.09751. Available online: http://arxiv.org/abs/1904.09751. [accessed 9 July 2026].

48. Bayes, T. LII. An essay towards solving a problem in the doctrine of chances. By the late Rev. Mr. Bayes, F. R. S. communicated by Mr. Price, in a letter to John Canton, A. M. F. R. S. Phil. Trans. R. Soc. A. 1763, 370-418.

49. Myers, L. A.; Hew, N. L. E.; Shang, S. L.; Liu, Z. K. Recursive entropy in thermodynamics: establishing the statistical-physics basis of the zentropy approach. arXiv 2025;arXiv:2511.04950. Available online: http://arxiv.org/abs/2511.04950. [accessed 9 July 2026].

50. Wang, Y.; Liu, Z.; Chen, L. Thermodynamic properties of Al, Ni, NiAl, and Ni3Al from first-principles calculations. Acta. Mater. 2004, 52, 2665-71.

51. Liu, Z.; Li, B.; Lin, H. Multiscale entropy and its implications to critical phenomena, emergent behaviors, and information. J. Phase. Equilib. Diffus. 2019, 40, 508-21.

52. Gibbs, J. W. The collected works of J. Willard Gibbs: Vol. II Elementary Principles in Statistical Mechanics. Yale University Press, 1948, Vol. II.

53. Liu, Z. Computational thermodynamics and its applications. Acta. Mater. 2020, 200, 745-92.

54. Liu, Z. Thermodynamics and its prediction and CALPHAD modeling: review, state of the art, and perspectives. Calphad 2023, 82, 102580.

55. Liu, Z. Quantitative predictive theories through integrating quantum, statistical, equilibrium, and nonequilibrium thermodynamics. J. Phys. Condens. Matter. 2024, 36, 343003.

56. Wang, S.; Shang, S. L.; Liu, Z. K.; Hao, W. ZENN on GitHub. 2026. https://github.com/WilliamMoriaty/ZENN. [accessed 9 July 2026].

57. Liu, Z.; Hew, N. L. E.; Shang, S. Zentropy theory for accurate prediction of free energy, volume, and thermal expansion without fitting parameters. Microstructures 2024, 4, 2024009.

58. Yuan, K.; Miao, D.; Yao, Y.; Zhang, H.; Zhao, X. Feature selection using zentropy-based uncertainty measure. IEEE. Trans. Fuzzy. Syst. 2024, 32, 2246-60.

59. Yuan, K.; Miao, D.; Pedrycz, W.; Ding, W.; Zhang, H. Ze-HFS: zentropy-based uncertainty measure for heterogeneous feature selection and knowledge discovery. IEEE. Trans. Knowl. Data. Eng. 2024, 36, 7326-39.

60. Yuan, K.; Miao, D.; Pedrycz, W.; Zhang, H.; Hu, L. Multigranularity data analysis with zentropy uncertainty measure for efficient and robust feature selection. IEEE. Trans. Cybern. 2025, 55, 740-52.

61. Yuan, K.; Miao, D.; Zhang, H.; Pedrycz, W. An efficient and robust feature selection approach based on zentropy measure and neighborhood-aware model. IEEE. Trans. Neural. Netw. Learning. Syst. 2025, 36, 16351-65.

62. Ye, W.; Xu, W. Innovative multi-granularity granular-balls rough set for feature selection: Driving generalized multi-granularity rough set evolution with Zentropy integration. Inform. Sciences. 2025, 718, 122411.

63. Dong, H.; Liu, C.; Chen, X.; Miao, D. A multi-granularity decision tree algorithm based on variable precision rough sets and Zentropy. Appl. Soft. Comput. 2025, 185, 113851.

64. Yuan, K.; Miao, D.; Ding, W.; Pedrycz, W.; Yao, Y. Robust semi-supervised feature selection with multi-granularity zentropy modeling. IEEE. Trans. Pattern. Anal. Mach. Intell. 2026, 48, 4587-604.

65. Einstein, A. Autobiographical notes. Albert Einstein: philosopher-scientist. Library of Living Philosophers, 1949. https://www.amazon.com/Albert-Einstein-Philosopher-Scientist-Philosophers-Paperback/dp/0875482864.

66. Brush, S. G. The kind of motion we call heat. North-Holland, 1976.

67. Suleyman, M.; Bhaskar, M. The coming wave: technology, power, and the twenty-first century’s greatest dilemma. Crown, an imprint of Crown Publishing Group, 2023.

68. Diamond, J. M. Guns, germs, and steel: the fates of human societies. W. W. Norton & Company, 1999.

69. Palmer, R. Broken ergodicity. Adv. Phys. 2006, 31, 669-735.

70. Binder, K.; Young, A. P. Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 1986, 58, 801-976.

71. Bouchaud, J. P. Weak ergodicity breaking and aging in disordered systems. J. Phys. I. France. 1992, 2, 1705-13.

72. Cugliandolo, L. F.; Kurchan, J. Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model. Phys. Rev. Lett. 1993, 71, 173-6.

73. Berthier, L.; Biroli, G. Theoretical perspective on the glass transition and amorphous materials. Rev. Mod. Phys. 2011, 83, 587-645.

74. Liu, Z. A Unified Thermodynamic framework: from equilibrium and nonequilibrium to zentropy, cross phenomena, and applications for AI and AI safety. J. Phase. Equilib. Diffus. 2026, 1248.