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.

翻译：本文提出了一种基于Transformer的Koopman自编码器，用于线性化Fisher反应-扩散方程。本研究的主要焦点在于利用深度学习技术发现反应-扩散系统中的复杂时空模式，其重点不仅在于求解方程，更在于将系统动力学转化为更易于理解的线性形式。全局坐标变换通过自编码器实现，该自编码器通过在包含60,000个初始条件的数据集上进行训练，学习捕捉底层动力学特性。我们通过在多组数据集上的广泛测试来评估所提出模型的有效性，证明了其能够准确预测系统演化并具有良好的泛化能力。我们提供了一项全面的对比研究，通过在多种偏微分方程（如Kuramoto-Sivashinsky方程和Burger方程）上的实验，将我们提出的架构与其他几种可比方法进行比较。结果表明，该模型具有更高的精度，凸显了基于Transformer的Koopman自编码器的强大能力。在利用单一架构求解不同类型偏微分方程方面，所提出的架构显著优于其他架构。我们的方法完全依赖于数据，无需任何关于底层方程的先验知识，这使得该方法即使对于控制方程未知的数据集也适用。