用机器学习动态适配基函数，更准地捕捉系统演化规律
Learning dynamically inspired bases for Koopman and transfer operator approximation
- 通过机器学习生成动态适配的正交基函数
- 显著提升转移算子与科普曼算子的谱特性估计精度
- 适合研究非线性动力系统的科研人员参考
转移算子与科普曼算子方法为复杂非线性动力系统提供了线性化表征框架，有助于深入理解系统可预测性与涌现行为。然而，从数据中高效估算其谱特性仍具挑战。本文从泛化算子与表征学习视角出发，利用高效有限维表示近似这些线性算子。具体而言，我们机器学习得到一组动态定制的正交基函数，该基能精确逼近算子作用，实现特征函数与不变测度的高效恢复。通过实例展示了所提方法对谱性质的有效重构，并凸显了机器学习基函数的动态适应能力。
原文摘要 · Abstract (English)
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.
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