Transfer and Koopman operator methods offer a framework for representing complex, nonlinear dynamical systems via linear transformations, enabling a deeper understanding of the underlying dynamics. The spectra of these operators provide important insights into system predictability and emergent behaviour, although efficiently estimating them from data can be challenging. We approach 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 basis functions that are dynamically tailored to the system. This learned basis provides a particularly accurate approximation of the operator's action and enables efficient recovery of eigenfunctions and invariant measures. 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.

翻译：迁移算子和Koopman算子方法提供了通过线性变换表示复杂非线性动力系统的框架，从而能够深入理解潜在动力学机制。这些算子的谱可为系统可预测性和涌现行为提供重要见解，尽管从数据中有效估计这些谱具有挑战性。我们通过通用算子学习与表征学习的视角来处理该问题，利用高效的有限维表示来近似这些线性算子。具体而言，我们通过机器学习获得正交归一的基函数，这些基函数动态适配系统特性。这种学习到的基函数能够特别精确地逼近算子的作用，并支持高效恢复本征函数和不变量测度。我们通过实例展示了如何从估计的算子中提取谱特性，并强调机器学习基函数所具有的动态自适应特性。