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==基于[[神经网络]]的DMD==

==基于[[神经网络]]的DMD==

尽管Koopman数据驱动算法前景广阔，但要选取合适的观测函数仍然是一个核心挑战。即使对于相对简单的[[动力系统]]，Koopman [[算子]]的[[特征函数]]也可能任意复杂，并且在有限基中只能被近似地表示。[[深度学习]]非常适合表示这种任意复杂的函数，并且最近在Koopman 预测方面显示出了巨大的潜力<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>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>。除了利用[[神经网络]]来学习Koopman嵌入之外，[[Koopman理论]]也被应用于理解[[神经网络]]<ref>I. Manojlovic\', M. Fonoberova, R. Mohr, A. Andrejc\v uk, Z. Drmac\v , Y. Kevrekidis, and I. Mezic\', Applications of Koopman Mode Analysis to Neural Networks, preprint, https://arxiv.org/abs/2006.11765, 2020. (Cited on p. 290)</ref><ref>A. S. Dogra and W. T. Redman, Optimizing neural networks via Koopman operator theory, in Advances in Neural Information Processing Systems, H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, eds., Curran Associates, 2020, pp. 2087--2097. (Cited on p. 290)</ref>以及更广泛的算法<ref>F. Dietrich, T. N. Thiem, and I. G. Kevrekidis, On the Koopman operator of algorithms, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 860--885, https://doi.org/10.1137/19M1277059. (Cited on p. 290)</ref>。   

尽管Koopman数据驱动算法前景广阔，但要选取合适的观测函数仍然是一个核心挑战。即使对于相对简单的[[动力系统]]，Koopman [[算子]]的[[特征函数]]也可能任意复杂，并且在有限基中只能被近似地表示。[[深度学习]]非常适合表示这种任意复杂的函数，并且最近在Koopman 预测方面显示出了巨大的潜力<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>。除了利用[[神经网络]]来学习Koopman嵌入之外，[[Koopman理论]]也被应用于理解[[神经网络]]<ref>I. Manojlovic\', M. Fonoberova, R. Mohr, A. Andrejc\v uk, Z. Drmac\v , Y. Kevrekidis, and I. Mezic\', Applications of Koopman Mode Analysis to Neural Networks, preprint, https://arxiv.org/abs/2006.11765, 2020. (Cited on p. 290)</ref><ref>A. S. Dogra and W. T. Redman, Optimizing neural networks via Koopman operator theory, in Advances in Neural Information Processing Systems, H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, eds., Curran Associates, 2020, pp. 2087--2097. (Cited on p. 290)</ref>以及更广泛的算法<ref>F. Dietrich, T. N. Thiem, and I. G. Kevrekidis, On the Koopman operator of algorithms, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 860--885, https://doi.org/10.1137/19M1277059. (Cited on p. 290)</ref>。   

[[文件:Neuralkoop.png|500x500px|替代=]]

[[文件:Neuralkoop.png|500x500px|替代=]]