标题： 基于动态启发的Koopman算子和转移算子逼近不变子空间的学习 显示英文标题

标题： Learning dynamically inspired invariant subspaces for Koopman and transfer operator approximation

摘要： 转移算子和Koopman算子方法提供了一个框架，可以通过线性变换来表示复杂的非线性动力系统，从而更深入地理解其底层动力学。 这些算子的谱提供了关于系统可预测性和涌现行为的重要见解，尽管从数据中高效估计它们可能具有挑战性。 我们通过广义算子学习和表征学习的视角来解决这个问题，使用有效的有限维表示来近似这些线性算子。 具体来说，我们利用机器学习生成正交且局部支持的基函数，这些基函数根据系统的动态特性进行调整。 这种学习得到的基不仅能够非常准确地逼近算子的作用，还能提供几乎不变的有限维子空间。 我们通过示例展示我们的方法，突出显示从估计的算子中提取光谱性质的能力，并强调机器学习基的动态自适应特性。 显示英文摘要

摘要： Transfer and Koopman operator methods offer a framework for representing complex, nonlinear dynamical systems via linear transformations, enabling for a deeper understanding of the underlying dynamics. The spectrum of these operators provide important insights into system predictability and emergent behaviour, although efficiently estimating them from data can be challenging. We tackle this issue through the lens of general operator and representational learning, in which we approximate these linear operators using efficient finite-dimensional representations. Specifically, we machine-learn orthonormal, locally supported basis functions that are dynamically tailored to the system. This learned basis provides a particularly accurate approximation of the operator's action as well as a nearly invariant finite-dimensional subspace. We illustrate our approach with examples that showcase the retrieval of spectral properties from the estimated operator, and emphasise the dynamically adaptive quality of the machine-learned basis.