用Transformer实现反应扩散方程的线性化，无需知道方程也能预测演化。
Transformer-based Koopman Autoencoder for Linearizing Fisher's Equation
- 用Transformer结构的自编码器学习系统全局坐标变换。
- 在6万组初始条件下训练，可准确预测并泛化到不同PDE。
- 不依赖方程知识，适合未知动力学系统的建模。
提出一种基于Transformer的Koopman自编码器，用于线性化Fisher反应-扩散方程。研究重点在于利用深度学习捕捉反应-扩散系统中的复杂时空模式，不仅求解方程，更将系统动力学转化为更易理解的线性形式。通过自编码器在包含60,000个初始条件的数据集上训练，实现全局坐标变换，学习底层动态。在多个数据集上进行广泛测试，验证了模型在系统演化预测和泛化能力上的优异表现。与多种可比方法（如柯尔莫哥洛夫-希瓦辛斯基方程、伯格斯方程）的实验对比显示，该方法精度显著提升。所提架构在单一框架下解决多种类型PDE方面明显优于其他架构。本方法完全基于数据，无需了解底层方程，适用于动力学方程未知的数据集。
原文摘要 · Abstract (English)
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.
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