基于Transformer的Koopman自编码器用于线性化Fisher方程
Transformer-based Koopman Autoencoder for Linearizing Fisher's Equation
AI总结：
本文提出基于Transformer的Koopman自编码器，用于线性化Fisher方程，通过数据驱动方法实现复杂时空模式的捕捉与预测。
摘要：
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。本研究的重点是利用深度学习技术在反应扩散系统中发现复杂的时空模式。重点不仅在于解决方程，还在于将系统的动态转换为更易理解的线性形式。通过自编码器实现全局坐标变换，该自编码器通过在60,000个初始条件的数据集上训练，学习捕捉底层动态。对多个数据集进行了广泛测试，以评估所提模型的有效性，证明其能够准确预测系统的发展并具有良好的泛化能力。我们提供了一项全面的比较研究，通过在各种PDE上进行实验，将所提设计与几种其他类似方法进行比较，如Kuramoto-Sivashinsky方程和Burger方程。结果表明，精度有所提高，突显了基于Transformer的Koopman自编码器的能力。所提出的架构在解决不同类型PDE时，使用单一架构方面显著优于其他架构。我们的方法完全依赖于数据，不需要任何关于底层方程的知识。这使其适用于即使在已知控制方程的 datasets 中也适用。
英文摘要：
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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