Transformer-based Koopman autoencoder for linearizing Fisher's equation.

Area of Science:

• Computational physics
• Applied mathematics
• Deep learning

Background:

• Reaction-diffusion systems exhibit complex spatiotemporal dynamics.
• Linearizing these dynamics simplifies analysis and prediction.
• Existing methods may struggle with complex patterns or require equation knowledge.

Purpose of the Study:

• To introduce a transformer-based Koopman autoencoder for linearizing Fisher's reaction-diffusion equation.
• To leverage deep learning for uncovering complex spatiotemporal patterns.
• To transform nonlinear system dynamics into a linear, comprehensible form.

Main Methods:

• Developed a transformer-based Koopman autoencoder architecture.
• Trained the autoencoder on a dataset of 60,000 initial conditions.
• Evaluated model performance on various partial differential equations (PDEs), including Kuramoto-Sivashinsky and Burger's equations.

Main Results:

• The model accurately predicts system evolution and generalizes well across datasets.
• Demonstrated superior accuracy compared to other comparable methods.
• Showcased the ability of a single architecture to solve diverse PDEs.

Conclusions:

• The transformer-based Koopman autoencoder effectively linearizes complex dynamics.
• This data-driven method offers a powerful, versatile tool for PDE analysis, even when equations are unknown.
• The approach significantly advances the field of deep learning for scientific discovery.