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Comparative Study of Neural Network Methods for Solving Topological Solitons

Koji Hashimoto, Koshiro Matsuo, Masaki Murata, Gakuto Ogiwara

TL;DR

This work has developed a novel method using neural network (NN) to efficiently solve solitons, and finds that this method achieves shorter computation times while maintaining the same level of accuracy.

Abstract

Topological solitons, which are stable, localized solutions of nonlinear differential equations, are crucial in various fields of physics and mathematics, including particle physics and cosmology. However, solving these solitons presents significant challenges due to the complexity of the underlying equations and the computational resources required for accurate solutions. To address this, we have developed a novel method using neural network (NN) to efficiently solve solitons. A similar NN approach is Physics-Informed Neural Networks (PINN). In a comparative analysis between our method and PINN, we find that our method achieves shorter computation times while maintaining the same level of accuracy. This advancement in computational efficiency not only overcomes current limitations but also opens new avenues for studying topological solitons and their dynamical behavior.

Comparative Study of Neural Network Methods for Solving Topological Solitons

TL;DR

This work has developed a novel method using neural network (NN) to efficiently solve solitons, and finds that this method achieves shorter computation times while maintaining the same level of accuracy.

Abstract

Topological solitons, which are stable, localized solutions of nonlinear differential equations, are crucial in various fields of physics and mathematics, including particle physics and cosmology. However, solving these solitons presents significant challenges due to the complexity of the underlying equations and the computational resources required for accurate solutions. To address this, we have developed a novel method using neural network (NN) to efficiently solve solitons. A similar NN approach is Physics-Informed Neural Networks (PINN). In a comparative analysis between our method and PINN, we find that our method achieves shorter computation times while maintaining the same level of accuracy. This advancement in computational efficiency not only overcomes current limitations but also opens new avenues for studying topological solitons and their dynamical behavior.

Paper Structure

This paper contains 11 sections, 16 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Method
  3. NNDE
  4. Field equations
  5. $\phi^4$ theory
  6. Sine-Gordon equation
  7. Training condition
  8. Results
  9. Kink soliton of $\phi^4$ theory
  10. Kink soliton of Sine-Gordon equation
  11. Summary and discussions

Figures (4)

  • Figure 1: Structure of NNDE.
  • Figure 2: Comparison of the exact solution, NNDE, and PINN predictions with step size $10^{-5}$ and batch size 10%.
  • Figure 3: Comparison of MSE and computation time for the $\phi^4$ theory with varying step sizes and batch sizes
  • Figure 4: Comparison of MSE and computation time for the Sine-Gordon equation with varying step sizes and batch sizes