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Summary

This study introduces a machine learning framework to model traveling wave dynamics using neural ordinary differential equations (ODEs). It enables empirical modeling of wave systems even when governing equations are unknown, including complex phenomena like rotating detonation waves.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Machine Learning

Background:

  • Traveling waves are ubiquitous in physical systems, exhibiting complex spatiotemporal dynamics.
  • Modeling these waves often requires knowledge of underlying governing partial differential equations (PDEs).
  • However, in many real-world scenarios, these governing equations are unknown or difficult to derive.

Purpose of the Study:

  • To develop a data-driven machine learning framework for modeling traveling wave spatiotemporal dynamics.
  • To apply this framework to systems where governing equations are unknown.
  • To demonstrate the framework's applicability to diverse physical phenomena and real-world systems.

Main Methods:

  • Utilized the steadily propagating traveling wave ansatz, u(x,t)=U(ξ=x-ct+a), to transform PDEs into ordinary differential equations (ODEs).
  • Employed neural ordinary differential equations (Neural ODEs) for empirical modeling of traveling wave dynamics when governing equations are unknown.
  • Applied the framework to model traveling wave fronts, pulses, and wavetrains in one spatial dimension.

Main Results:

  • Successfully modeled traveling wave spatiotemporal dynamics across various physical systems using the proposed machine learning framework.
  • Demonstrated the efficacy of Neural ODEs in capturing wave behavior without prior knowledge of the governing equations.
  • Showcased the framework's potential for real-world applications, including the data-driven modeling of rotating detonation waves.

Conclusions:

  • The developed data-driven framework provides a powerful tool for analyzing and predicting traveling wave phenomena.
  • Neural ODEs offer a viable approach for empirical modeling of complex dynamical systems.
  • This methodology broadens the scope of applying machine learning to understand and engineer physical systems exhibiting wave propagation.