更改

=数据驱动算法=

=数据驱动算法=

在分析复杂的[[非线性系统]]时，传统的分析方法只能在不动点附近很小的领域内进行线性化处理，大大限制了我们对[[非线性系统]]的解释和预测。Koopman[[算子]]理论通过将系统的状态映射到由可观测函数构成的无穷维空间，从而在'''全局'''上将非线性动力学问题转化为一个无穷维线性算子问题。不过，这个理想的Koopman算子本身是无穷维的，无法直接计算。实践中，我们要通过数据拟合出有限维的线性迭代系统，尽可能地逼近[[无限]]维的Koopman线性[[算子]]。

在分析复杂的[[非线性系统]]时，传统的分析方法只能在不动点附近很小的领域内进行线性化处理，大大限制了我们对[[非线性系统]]的解释和预测。Koopman[[算子]]理论通过将系统的状态映射到由可观测函数构成的无穷维空间，从而在'''全局'''上将非线性动力学问题转化为一个无穷维线性算子问题。不过，这个理想的Koopman算子本身是无穷维的，无法直接计算。实践中，我们要利用Koopman不变子空间的概念（参见第3.3节），通过数据拟合出有限维的线性迭代系统，尽可能地逼近[[无限]]维的Koopman线性[[算子]]。

那么如何用有限的、来自现实的数据，去逼近[[无限]]维的Koopman线性[[算子]]呢？最开始的方法是'''[[动力学]][[模式分解]]（DMD，Dynamic Mode Decomposition）'''<ref name=Schmid_2008>P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, in 61st Annual Meeting of the APS Division of Fluid Dynamics, American Physical Society, 2008. (Cited on pp. 249, 258)</ref><ref name=Schmid_2010>P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), pp. 5--28. (Cited on pp. 231, 237, 238, 249, 250, 255, 258, 284, 292)</ref>，它直接把“[[状态变量]]本身”当作观测函数。然而，对于真正的[[非线性系统]]，DMD的近似就太粗糙了。故此后学者们围绕着“如何更有效地拓展观测空间”与“如何优化求解过程”展开。'''扩展动态[[模式分解]]（EDMD, Extended Dynamic Mode Decomposition）'''<ref name=william_2015>M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 6 (2015), pp. 1307--1346. (Cited on pp. 246, 247, 256, 284, 285, 292)</ref>通过预定义的[[非线性]]函数显式拓展；'''残差动态[[模式分解]]（ResDMD, Residual DMD）'''<ref name=Colbrook_2023>Colbrook, M. J., Ayton, L. J. & Szőke, M. Residual dynamic mode decomposition: Robust and verified koopmanism. J. Fluid Mech. 955, A21 (2023).</ref><ref name=Colbrook_2024>Colbrook, M. J. & Townsend, A. Rigorous data‐driven computation of spectral properties of koopman operators for dynamical systems. Commun. Pure Appl. Math. 77, 221–283 (2024).</ref>致力于严格评估逼近精度并筛选可靠模态；'''核动态[[模式分解]]（Kernel DMD）'''<math>\mathbf{A}_\phi</math>为<ref name=william_2015_kernal>M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, A kernel approach to data-driven Koopman spectral analysis, J. Comput. Dynam., 2 (2015), pp. 247--265. (Cited on pp. 255, 256, 284, 285, 294, 318)</ref><ref name=Gonzalez_2023_kernal>Gonzalez, E., Abudia, M., Jury, M., Kamalapurkar, R. & Rosenfeld, J. A. The kernel perspective on dynamic mode decomposition. Preprint at https://doi.org/10.48550/arXiv.2106.00106 (2023).</ref>利用[[核技巧]]隐式拓展；'''基于[[神经网络]]的动态[[模式分解]]（[[Neural Network]] DMD）'''<ref name=CW2018>C. Wehmeyer and F. Noe\', Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics, J. Chem. Phys., 148 (2018), pp. 1--9. (Cited on pp. 284, 290, 291)</ref><ref name=NT2017>N. Takeishi, Y. Kawahara, and T. Yairi, Learning Koopman invariant subspaces for dynamic mode decomposition, in Advances in Neural Information Processing Systems, 2017, pp. 1130--1140. (Cited on pp. 284, 290)</ref>用[[深度学习]]自动学习最优的拓展方式；'''时间延迟动态[[模式分解]]（Time-delay DMD/Hankel DMD）'''<ref name=brunton_2017>S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz, Chaos as an intermittently forced linear system, Nature Commun., 8 (2017), pp. 1--9. (Cited on pp. 257, 275, 284, 287, 288, 289, 294)</ref>通过历史数据构造[[嵌入空间]]；'''[[随机]]动态[[模式分解]]（Randomized DMD）'''<ref name=ranDMD>Erichson, N. B., Mathelin, L., Kutz, J. N. & Brunton, S. L. Randomized dynamic mode decomposition. Siam J. Appl. Dyn. Syst. 18, 1867–1891 (2019)</ref>专注于利用[[随机]]算法提升高维问题的计算效率；'''最优[[模式分解]]（OPD/Optimal DMD）'''<ref name=opdmd>T. Askham and J. N. Kutz, Variable projection methods for an optimized dynamic mode decomposition, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 380--416, https://doi.org/10. 1137/M1124176. (Cited on pp. 253, 254)</ref>将问题重构为全局[[非线性]]优化以求取最佳拟合；'''多分辨率动态[[模式分解]]（mrDMD, Multiresolution DMD）'''<ref name=mrdmd>J. N. Kutz, X. Fu, and S. L. Brunton, Multiresolution dynamic mode decomposition, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 713--735, https://doi.org/10.1137/15M1023543. (Cited on pp. 257, 278)</ref>则能分离系统在不同时间尺度或层次上的[[动力学]]；'''前向-后向动态[[模式分解]]（fbDMD, forward-backward DMD）'''<ref name=fbdmd>S. T. Dawson, M. S. Hemati, M. O. Williams, and C. W. Rowley, Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Exp. Fluids, 57 (2016), pp. 1--19. (Cited on p. 253)</ref>通过平均正反向数据提升稳定性；'''张量动态[[模式分解]]（Tensor DMD）'''<ref name=tensordmd>S. Klus, P. Gel{\ss}, S. Peitz, and C. Sch\"utte, Tensor-based dynamic mode decomposition, Nonlinearity, 31 (2018), pp. 3359--3380. (Cited on p. 257)</ref>处理具有空间结构的多维数据；'''受控动态[[模式分解]]（Controlled DMD）'''<ref name=DMDc>J. L. Proctor, S. L. Brunton, and J. N. Kutz, Dynamic mode decomposition with control, SIAM J. Appl. Dyn. Syst., 15 (2016), pp</ref>则旨在辨识含外部输入系统的Koopman模型。DMD算法现在已经扩展出许多不同的版本，在这里只是罗列的部分情况，下文会对主流的几种方法进行更详细的介绍，其他算法请参考文献。

那么如何用有限的、来自现实的数据，去逼近[[无限]]维的Koopman线性[[算子]]呢？最开始的方法是'''[[动力学模式分解]]（DMD，Dynamic Mode Decomposition）'''<ref name=Schmid_2008>P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, in 61st Annual Meeting of the APS Division of Fluid Dynamics, American Physical Society, 2008. (Cited on pp. 249, 258)</ref><ref name=Schmid_2010>P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), pp. 5--28. (Cited on pp. 231, 237, 238, 249, 250, 255, 258, 284, 292)</ref>，它直接把“[[状态变量]]本身”当作观测函数。然而，对于真正的[[非线性系统]]，DMD的近似就太粗糙了。故此后学者们围绕着“如何更有效地拓展观测空间”与“如何优化求解过程”展开。'''扩展动态[[模式分解]]（EDMD, Extended Dynamic Mode Decomposition）'''<ref name=william_2015>M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 6 (2015), pp. 1307--1346. (Cited on pp. 246, 247, 256, 284, 285, 292)</ref>通过预定义的[[非线性]]函数显式拓展；'''残差动态[[模式分解]]（ResDMD, Residual DMD）'''<ref name=Colbrook_2023>Colbrook, M. J., Ayton, L. J. & Szőke, M. Residual dynamic mode decomposition: Robust and verified koopmanism. J. Fluid Mech. 955, A21 (2023).</ref><ref name=Colbrook_2024>Colbrook, M. J. & Townsend, A. Rigorous data‐driven computation of spectral properties of koopman operators for dynamical systems. Commun. Pure Appl. Math. 77, 221–283 (2024).</ref>致力于严格评估逼近精度并筛选可靠模态；'''核动态[[模式分解]]（Kernel DMD）'''<math>\mathbf{A}_\phi</math>为<ref name=william_2015_kernal>M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, A kernel approach to data-driven Koopman spectral analysis, J. Comput. Dynam., 2 (2015), pp. 247--265. (Cited on pp. 255, 256, 284, 285, 294, 318)</ref><ref name=Gonzalez_2023_kernal>Gonzalez, E., Abudia, M., Jury, M., Kamalapurkar, R. & Rosenfeld, J. A. The kernel perspective on dynamic mode decomposition. Preprint at https://doi.org/10.48550/arXiv.2106.00106 (2023).</ref>利用[[核技巧]]隐式拓展；'''基于[[神经网络]]的动态[[模式分解]]（[[Neural Network]] DMD）'''<ref name=CW2018>C. Wehmeyer and F. Noe\', Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics, J. Chem. Phys., 148 (2018), pp. 1--9. (Cited on pp. 284, 290, 291)</ref><ref name=NT2017>N. Takeishi, Y. Kawahara, and T. Yairi, Learning Koopman invariant subspaces for dynamic mode decomposition, in Advances in Neural Information Processing Systems, 2017, pp. 1130--1140. (Cited on pp. 284, 290)</ref>用[[深度学习]]自动学习最优的拓展方式；'''时间延迟动态[[模式分解]]（Time-delay DMD/Hankel DMD）'''<ref name=brunton_2017>S. L. Brunton, B. W. Brunton, J. L. Proctor, E. Kaiser, and J. N. Kutz, Chaos as an intermittently forced linear system, Nature Commun., 8 (2017), pp. 1--9. (Cited on pp. 257, 275, 284, 287, 288, 289, 294)</ref>通过历史数据构造[[嵌入空间]]；'''[[随机]]动态[[模式分解]]（Randomized DMD）'''<ref name=ranDMD>Erichson, N. B., Mathelin, L., Kutz, J. N. & Brunton, S. L. Randomized dynamic mode decomposition. Siam J. Appl. Dyn. Syst. 18, 1867–1891 (2019)</ref>专注于利用[[随机]]算法提升高维问题的计算效率；'''最优[[模式分解]]（OPD/Optimal DMD）'''<ref name=opdmd>T. Askham and J. N. Kutz, Variable projection methods for an optimized dynamic mode decomposition, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 380--416, https://doi.org/10. 1137/M1124176. (Cited on pp. 253, 254)</ref>将问题重构为全局[[非线性]]优化以求取最佳拟合；'''多分辨率动态[[模式分解]]（mrDMD, Multiresolution DMD）'''<ref name=mrdmd>J. N. Kutz, X. Fu, and S. L. Brunton, Multiresolution dynamic mode decomposition, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 713--735, https://doi.org/10.1137/15M1023543. (Cited on pp. 257, 278)</ref>则能分离系统在不同时间尺度或层次上的[[动力学]]；'''前向-后向动态[[模式分解]]（fbDMD, forward-backward DMD）'''<ref name=fbdmd>S. T. Dawson, M. S. Hemati, M. O. Williams, and C. W. Rowley, Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition, Exp. Fluids, 57 (2016), pp. 1--19. (Cited on p. 253)</ref>通过平均正反向数据提升稳定性；'''张量动态[[模式分解]]（Tensor DMD）'''<ref name=tensordmd>S. Klus, P. Gel{\ss}, S. Peitz, and C. Sch\"utte, Tensor-based dynamic mode decomposition, Nonlinearity, 31 (2018), pp. 3359--3380. (Cited on p. 257)</ref>处理具有空间结构的多维数据；'''受控动态[[模式分解]]（Controlled DMD）'''<ref name=DMDc>J. L. Proctor, S. L. Brunton, and J. N. Kutz, Dynamic mode decomposition with control, SIAM J. Appl. Dyn. Syst., 15 (2016), pp</ref>则旨在辨识含外部输入系统的Koopman模型。DMD算法现在已经扩展出许多不同的版本，在这里只是罗列的部分情况，下文会对主流的几种方法进行更详细的介绍，其他算法请参考文献。

 

==[[动力学模式分解]]（DMD）==

==[[动力学]][[模式分解]]（DMD）==

[[动力学模式分解]]（DMD，Dynamic Mode Decomposition）最初由 Schmid

[[动力学模式分解]]（DMD，Dynamic Mode Decomposition）最初由 Schmid