偏微分方程的Koopman理论
Koopman Theory for Partial Differential Equations
摘要：
考虑Koopman理论在非线性偏微分方程中的应用。我们证明，选择用于构建Koopman算子的观测量是关键，使非线性动力学的准确近似。如果可以找到这样的可观测量，则可以执行动态模式分解算法来计算库普曼算子的有限维近似，包括其本征函数、本征值和库普曼模式。明智地选择观测导致物理上可解释的时空特征的复杂系统正在考虑和提供连接到多方面的学习方法。我们展示了可观察的选择，包括核方法，和建设的Koopman算子上的两个典型的，非线性偏微分方程：伯格斯方程和非线性薛定谔方程的影响。这些例子突出了库普曼理论最紧迫和最关键的挑战：选择适当的可观测量的原则性方法。
英文摘要：
We consider the application of Koopman theory to nonlinear partial differential equations. We demonstrate that the observables chosen for constructing the Koopman operator are critical for enabling an accurate approximation to the nonlinear dynamics. If such observables can be found, then the dynamic mode decomposition algorithm can be enacted to compute a finite-dimensional approximation of the Koopman operator, including its eigenfunctions, eigenvalues and Koopman modes. Judiciously chosen observables lead to physically interpretable spatio-temporal features of the complex system under consideration and provide a connection to manifold learning methods. We demonstrate the impact of observable selection, including kernel methods, and construction of the Koopman operator on two canonical, nonlinear PDEs: Burgers' equation and the nonlinear Schrödinger equation. These examples serve to highlight the most pressing and critical challenge of Koopman theory: a principled way to select appropriate observables.
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