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Identifying Ising and percolation phase transitions based on KAN method

Dian Xu, Shanshan Wang, Wei Li, Weibing Deng, Feng Gao, Jianmin Shen

TL;DR

This work presents Kolmogorov-Arnold Networks (KAN) as a data-driven approach to identify critical points and universal exponents in Ising and percolation models from raw lattice configurations. By grounding the network in the Kolmogorov-Arnold Representation Theorem and employing edge activations with B-spline components, the method achieves accurate critical-point detection and data-collapse-based estimation of $\nu$ exponents, with finite-size scaling aligning results to theoretical values. The results demonstrate that KAN can recover known critical behavior directly from raw configurations, offering a non-global approximation framework that complements traditional unsupervised techniques. The approach potentially broadens ML applications in critical phenomena by enabling reliable extraction of phase information without explicit order parameters.

Abstract

Modern machine learning, grounded in the Universal Approximation Theorem, has achieved significant success in the study of phase transitions in both equilibrium and non-equilibrium systems. However, identifying the critical points of percolation models using raw configurations remains a challenging and intriguing problem. This paper proposes the use of the Kolmogorov-Arnold Network, which is based on the Kolmogorov-Arnold Representation Theorem, to input raw configurations into a learning model. The results demonstrate that the KAN can indeed predict the critical points of percolation models. Further observation reveals that, apart from models associated with the density of occupied points, KAN is also capable of effectively achieving phase classification for models where the sole alteration pertains to the orientation of spins, resulting in an order parameter that manifests as an external magnetic flux, such as the Ising model.

Identifying Ising and percolation phase transitions based on KAN method

TL;DR

This work presents Kolmogorov-Arnold Networks (KAN) as a data-driven approach to identify critical points and universal exponents in Ising and percolation models from raw lattice configurations. By grounding the network in the Kolmogorov-Arnold Representation Theorem and employing edge activations with B-spline components, the method achieves accurate critical-point detection and data-collapse-based estimation of exponents, with finite-size scaling aligning results to theoretical values. The results demonstrate that KAN can recover known critical behavior directly from raw configurations, offering a non-global approximation framework that complements traditional unsupervised techniques. The approach potentially broadens ML applications in critical phenomena by enabling reliable extraction of phase information without explicit order parameters.

Abstract

Modern machine learning, grounded in the Universal Approximation Theorem, has achieved significant success in the study of phase transitions in both equilibrium and non-equilibrium systems. However, identifying the critical points of percolation models using raw configurations remains a challenging and intriguing problem. This paper proposes the use of the Kolmogorov-Arnold Network, which is based on the Kolmogorov-Arnold Representation Theorem, to input raw configurations into a learning model. The results demonstrate that the KAN can indeed predict the critical points of percolation models. Further observation reveals that, apart from models associated with the density of occupied points, KAN is also capable of effectively achieving phase classification for models where the sole alteration pertains to the orientation of spins, resulting in an order parameter that manifests as an external magnetic flux, such as the Ising model.

Paper Structure

This paper contains 11 sections, 30 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Models
  3. percolation model
  4. Ising model
  5. KAN
  6. The KAN Learning Results
  7. Detect crital point
  8. Data collapses for critical exponent
  9. Conclusion
  10. Discussion
  11. Acknowledgements

Figures (4)

  • Figure 1: In this article, we examine two distinct configurations. a is the raw configuration of site percolation with occupation probability = 0.8. b is the raw configuration of ising model.
  • Figure 2: Neural network schematic structure of KAN.
  • Figure 3: (a)Presents the training outcomes of the Ising model under various dimensions, denoted as kan. (b) Depicts the results of site percolation. Both scenarios encompass sizes of 10, 20, 30, and 40, with a sample size of 1000 for each. (c) Illustrates the extrapolated results of the Ising model to the thermodynamic limit. (d) Represents the outcomes of percolation in the same thermodynamic limit.
  • Figure 4: The experimental results of data collapse. Panel a shows the results of Ising, and panel b shows the results of Percolation.The collapse of the average output layer as a function of $(T- T_{c}) L^{1/ \nu_{\perp}}$ and $(P- P_{c}) L^{1/ \nu_{\perp}}$.