Transformer-based Koopman Autoencoder for Linearizing Fisher's Equation

TL;DR

This paper introduces a Transformer-based Koopman autoencoder that linearizes Fisher's reaction-diffusion equation, enabling better understanding and prediction of complex spatiotemporal patterns without prior knowledge of the underlying equations.

Contribution

It presents a novel deep learning architecture that transforms nonlinear PDE dynamics into linear form, outperforming existing methods across various PDEs using a single, data-driven model.

Findings

Achieves accurate prediction of PDE evolution

Demonstrates superior performance over comparable methods

Generalizes well across different PDE types

Abstract

A Transformer-based Koopman autoencoder is proposed for linearizing Fisher's reaction-diffusion equation. The primary focus of this study is on using deep learning techniques to find complex spatiotemporal patterns in the reaction-diffusion system. The emphasis is on not just solving the equation but also transforming the system's dynamics into a more comprehensible, linear form. Global coordinate transformations are achieved through the autoencoder, which learns to capture the underlying dynamics by training on a dataset with 60,000 initial conditions. Extensive testing on multiple datasets was used to assess the efficacy of the proposed model, demonstrating its ability to accurately predict the system's evolution as well as to generalize. We provide a thorough comparison study, comparing our suggested design to a few other comparable methods using experiments on various PDEs, such as the Kuramoto-Sivashinsky equation and the Burger's equation. Results show improved accuracy, highlighting the capabilities of the Transformer-based Koopman autoencoder. The proposed architecture in is significantly ahead of other architectures, in terms of solving different types of PDEs using a single architecture. Our method relies entirely on the data, without requiring any knowledge of the underlying equations. This makes it applicable to even the datasets where the governing equations are not known.