AI 中文总结
针对EDMD中非不变字典易引入伪特征值的问题，提出投影Koopman算子逼近框架构建滤波EDMD算子，分析两种投影几何并开发相关算法，实验验证其能有效减少谱污染。
AI 中文摘要
Koopman算子为通过谱特性分析非线性动力系统提供了线性框架。扩展动态模态分解（Extended Dynamic Mode Decomposition, EDMD）可从数据中逼近该算子，但非不变字典可能引入伪特征值。\n 我们提出了投影Koopman算子逼近框架，用于构建滤波EDMD算子。该框架将Koopman作用投影到无需满足不变性的容许字典子空间上，同时精确保留所有已表示的、具有非零特征值的Koopman特征对。前向-交链提供了相容子空间的规范层级，将完整字典与其最大不变核心连接起来，同时保留了有用的中间模型。\n 我们分析了两种投影几何结构：一种是坐标正交投影子，它无需估计函数空间Gram矩阵，但依赖于基；另一种是函数空间正交投影子，它在未滤波层级下可恢复总体EDMD。我们刻画了它们与EDMD及现有子空间选择方法的关系。我们还开发了基于SVD的算法，用于构建采样前向-交空间并实现坐标投影子。在独立无噪声采样和精确秩可识别性条件下，所得经验算子几乎必然收敛到其总体对应形式。\n 在Kronecker流、多项式系统以及Van der Pol振子上的实验表明，该方法减少了谱污染。对于Van der Pol振子，坐标投影子可独立于采样测度恢复局部平衡谱，而$L^2(μ)$投影子则在同一经认证的子空间上逼近极限环谱。
英文摘要
The Koopman operator provides a linear framework for analyzing nonlinear dynamical systems through spectral properties. Extended Dynamic Mode Decomposition (EDMD) approximates this operator from data, but non-invariant dictionaries can introduce spurious eigenvalues. We introduce the Projected Koopman Operator Approximation framework for constructing Filtered EDMD operators. The framework projects the Koopman action onto admissible dictionary subspaces that need not be invariant, while exactly preserving every represented Koopman eigenpair with nonzero eigenvalue. A forward--intersection chain provides a canonical hierarchy of compatible subspaces, connecting the full dictionary to its maximal invariant core while retaining useful intermediate models. We analyze two projection geometries: a coordinate-orthogonal projector, which requires no function-space Gram-matrix estimate but is basis-dependent, and a function-space orthogonal projector, which recovers population EDMD at the unfiltered level. We characterize their relationship to EDMD and existing subspace-selection methods. We also develop SVD-based algorithms for constructing sampled forward--intersection spaces and implementing the coordinate projector. Under independent noiseless sampling and exact-rank identifiability, the resulting empirical operators converge almost surely to their population counterparts. Experiments on a Kronecker flow, a polynomial system, and the Van~der~Pol oscillator demonstrate reduced spectral pollution. For Van~der~Pol, the coordinate projector recovers the local equilibrium spectrum independently of the sampling measure, whereas the $L^2(μ)$ projector approximates the limit-cycle spectrum on the same certified subspace.
