Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks

Summary

The paper discusses the application of Physics-Informed Neural Networks (PINNs) to quasi-integrable equations, specifically the Zakharov-Kuznetsov (ZK) equation and the regularized long-wave (RLW) equation. PINNs are used to solve forward and inverse problems, and the results show that PINNs can successfully identify the governing equations from given data.

Highlights

• PINNs are used to solve forward and inverse problems for quasi-integrable equations.
• The ZK equation and RLW equation are used as examples.
• PINNs can successfully identify the governing equations from given data.
• The method is tested with different initial profiles, including Gaussian and oval-shaped profiles.
• A small perturbation term is introduced to improve the predictability of PINNs.
• The conservation laws are used to improve the accuracy of PINNs.

Key Insights

• The study demonstrates the effectiveness of PINNs in solving forward and inverse problems for quasi-integrable equations, which is a significant contribution to the field of mathematical physics.
• The use of different initial profiles, such as Gaussian and oval-shaped profiles, helps to improve the predictability of PINNs, which is an important consideration in real-world applications.
• The introduction of a small perturbation term can significantly improve the accuracy of PINNs, which is a useful technique for improving the performance of the method.
• The conservation laws play a crucial role in improving the accuracy of PINNs, which highlights the importance of incorporating physical constraints into the neural network architecture.
• The study highlights the potential of PINNs for discovering new mathematical models from observational data, which is a promising area of research in the field of machine learning and mathematical physics.
• The results of the study demonstrate that PINNs can be used to identify the governing equations of complex systems from given data, which is a significant contribution to the field of mathematical modeling.
• The study provides a new perspective on the application of machine learning techniques to mathematical physics, which is an exciting area of research with many potential applications.

Mindmap

Citation

Nakamula, A., Obuse, K., Sawado, N., Shimasaki, K., Shimazaki, Y., Suzuki, Y., & Toda, K. (2024). Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks (Version 3). arXiv. https://doi.org/10.48550/ARXIV.2410.19014