Hochrainer Thomas, Kar Gurudas
Institute of Strength of Materials, Graz University of Technology, Kopernikusgasse 24/I, 8010 Graz, Austria.
Chaos. 2024 Aug 1;34(8). doi: 10.1063/5.0212620.
Many physical systems exhibit translational invariance, meaning that the underlying physical laws are independent of the position in space. Data driven approximations of the infinite dimensional but linear Koopman operator of non-linear dynamical systems need to be physically informed in order to respect such physical symmetries. In the current work, we introduce a translation invariant extended dynamic mode decomposition (tieDMD) for coupled non-linear systems on periodic domains. This is achieved by exploiting a block-diagonal structure of the Koopman operator in Fourier space. Variants of tieDMD are applied to data obtained on one-dimensional periodic domains from the non-linear phase-diffusion equation, the Burgers equation, the Korteweg-de Vries equation, and a coupled FitzHugh-Nagumo system of partial differential equations. The reconstruction capability of tieDMD is compared to existing linear and non-linear variants of the dynamic mode decomposition applied to the same data. For the regarded data, tieDMD consistently shows superior capabilities in data reconstruction.
许多物理系统表现出平移不变性，这意味着基本物理定律与空间位置无关。为了尊重这种物理对称性，非线性动力系统的无限维但线性的库普曼算子的数据驱动近似需要有物理依据。在当前工作中，我们为周期域上的耦合非线性系统引入了一种平移不变扩展动态模态分解（tieDMD）。这是通过利用傅里叶空间中库普曼算子的块对角结构来实现的。tieDMD的变体被应用于从非线性相位扩散方程、伯格斯方程、科特韦格 - 德弗里斯方程以及一个耦合的菲茨休 - 纳古莫偏微分方程组在一维周期域上获得的数据。将tieDMD的重构能力与应用于相同数据的动态模态分解的现有线性和非线性变体进行了比较。对于所考虑的数据，tieDMD在数据重构方面始终表现出卓越的能力。
