稀疏观测数据驱动的非线性动力学重构
Data-driven Reconstruction of Nonlinear Dynamics from Sparse Observation
摘要：
我们提出了一个数据驱动的模型，从一个非常稀疏的时间序列数据重建非线性动力学，这依赖于回声状态网络（ESN）在学习数据的非线性表示的强度。在假设ESN具有泛函数逼近能力的情况下，证明了重构问题可以转化为不动点问题，其中动力系统的轨迹是ESN的不动点.提出了一种欠松弛不动点迭代法，用于从稀疏观测值重构非线性动力学。建议的固定点ESN测试对单变量和多变量混沌动力系统随机删除高达95%的数据。结果表明，定点ESN仅用5 ~ 10%的数据就能重建复杂的动力学过程。对于一个相对简单的非混沌动力系统，对一个受迫van der Pol振子的数值实验表明，仅用1~2%的数据就可以重建非线性动力系统。
英文摘要：
We present a data-driven model to reconstruct nonlinear dynamics from a very sparse times series data, which relies on the strength of the echo state network (ESN) in learning nonlinear representation of data. With an assumption of the universal function approximation capability of ESN, it is shown that the reconstruction problem can be formulated as a fixed-point problem, in which the trajectory of the dynamical system is a fixed point of the ESN. An under-relaxed fixed-point iteration is proposed to reconstruct the nonlinear dynamics from a sparse observation. The proposed fixed-point ESN is tested against both univariate and multivariate chaotic dynamical systems by randomly removing up to 95% of the data. It is shown that the fixed-point ESN is able to reconstruct the complex dynamics from only 5 ~ 10% of the data. For a relatively simple non-chaotic dynamical system, the numerical experiments on a forced van der Pol oscillator show that it is possible to reconstruct the nonlinear dynamics from only 1~2% of the data.
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