Data-driven modeling of nonlinear traveling waves

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.