We develop a deep autoencoder architecture that can be used to find a coordinate transformation which turns a non-linear partial differential equation (PDE) into a linear PDE. Our architecture is motivated by the linearising transformations provided by the Cole–Hopf transform for Burgers’ equation and the inverse scattering transform for completely integrable PDEs. By leveraging a residual network architecture, a near-identity transformation can be exploited to encode intrinsic coordinates in which the dynamics are linear. The resulting dynamics are given by a Koopman operator matrix K. The decoder allows us to transform back to the original coordinates as well. Multiple time step prediction can be performed by repeated multiplication by the matrix K in the intrinsic coordinates. We demonstrate our method on a number of examples, including the heat equation and Burgers’ equation, as well as the substantially more challenging Kuramoto–Sivashinsky equation, showing that our method provides a robust architecture for discovering linearising transforms for non-linear PDEs.

中文翻译：

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用于线性化 PDE 的全局坐标变换的深度学习模型

我们开发了一种深度自动编码器架构，可用于找到将非线性偏微分方程 (PDE) 转换为线性 PDE 的坐标变换。我们的架构受到 Burgers 方程的 Cole-Hopf 变换和完全可积 PDE 的逆散射变换提供的线性化变换的启发。通过利用残差网络架构，可以利用近似恒等变换来编码动态线性的内在坐标。产生的动态由 Koopman 算子矩阵给出ķ. 解码器也允许我们转换回原始坐标。可以通过矩阵的重复乘法来执行多时间步预测ķ在固有坐标中。我们在许多示例上演示了我们的方法，包括热方程和 Burgers 方程，以及更具挑战性的 Kuramoto-Sivashinsky 方程，表明我们的方法为发现非线性 PDE 的线性化变换提供了一个强大的架构。