用托普利茨矩阵构建深层动力系统模型，提升非自治系统分析能力。
Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices
- 基于托普利茨矩阵设计可学习的深层柯尔莫哥洛夫层，捕捉时序动态变化。
- 在多个非自治系统上，特征值估计精度优于现有方法。
- 融合数值线性代数中的克里洛夫子空间方法，兼具理论深度与计算效率。
我们提出一种基于托普利茨矩阵的可学习参数深层柯尔莫哥洛夫层模型，用于分析时序数据的动力学演化。该模型兼具理论严谨性与灵活性。得益于托普利茨矩阵的通用性质及模型背后的再生性，我们证明了其泛化能力与普遍适用性。此外，模型灵活性使其能够拟合来自非自治动力系统的时序数据。训练过程中，采用克里洛夫子空间方法实现高效计算，建立柯尔莫哥洛夫算子与数值线性代数的新关联。实验表明，该模型在多个非自治系统上对多个柯尔莫哥洛夫算子的特征值估计性能超越现有方法。
原文摘要 · Abstract (English)
We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlying the model, we show its universality and generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations, which establish a new connection between Koopman operators and numerical linear algebra. We also empirically demonstrate that the proposed model outperforms existing methods on eigenvalue estimation of multiple Koopman operators for nonautonomous systems.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。