论文标题
随机Koopman操作员的强大近似
Robust Approximation of the Stochastic Koopman Operator
论文作者
论文摘要
我们分析了动态模式分解（DMD）基于随机动力学系统的随机Koopman操作员的近似值，在这些动力学系统中，动态或可观察到受噪声影响。对于许多DMD算法，噪声的存在会在DMD操作员中引入偏置，从而导致动力学的近似值差。特别地，当动力学是随机的时，使用时间延迟可观察到的方法（例如Hankel DMD）会偏差。我们引入了一种新的，可靠的DMD算法，尽管存在噪声，该算法仍可以近似随机的Koopman操作员。然后，我们演示了如何将此算法应用于时间延迟的可观测值，这使我们能够从单个可观察的可观察到的Krylov子空间生成Krylov子空间。这使我们能够使用在单个轨迹上测量的单个可观察到的随机Koopman操作员来实现。我们在几个示例中测试了算法的性能。
We analyze the performance of Dynamic Mode Decomposition (DMD)-based approximations of the stochastic Koopman operator for random dynamical systems where either the dynamics or observables are affected by noise. For many DMD algorithms, the presence of noise can introduce a bias in the DMD operator, leading to poor approximations of the dynamics. In particular, methods using time delayed observables, such as Hankel DMD, are biased when the dynamics are random. We introduce a new, robust DMD algorithm that can approximate the stochastic Koopman operator despite the presence of noise. We then demonstrate how this algorithm can be applied to time delayed observables, which allows us to generate a Krylov subspace from a single observable. This allows us to compute a realization of the stochastic Koopman operator using a single observable measured over a single trajectory. We test the performance of the algorithms over several examples.
