Discovering governing equations from partial measurements with deep delay autoencoders
Discovering governing equations from partial measurements with deep delay autoencoders
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DOI：
10.1098/rspa.2023.0422
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
Joseph Bakarji;Kathleen P. Champion;J. Nathan Kutz;S. Brunton
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文献类型：
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作者：
Joseph Bakarji;Kathleen P. Champion;J. Nathan Kutz;S. Brunton
数据驱动模型发现的一个核心挑战是隐藏的或潜在的变量的存在，这些变量不能直接测量，但动态重要。塔肯斯定理提供了条件，当它是可能的，以增加部分测量与时间延迟的信息，导致在一个吸引子，这是同构的原始全状态系统。这种同构通常是未知的，学习嵌入空间中的动态几十年来一直是一个开放的挑战。在这里，我们设计了一个深度自动编码器网络来学习从延迟嵌入空间到新空间的坐标变换，在新空间中可以以稀疏的封闭形式表示动态。我们证明了这种方法的洛伦兹，罗斯勒和洛特卡-Volterra系统，以及洛伦兹模拟从视频的混沌水车实验。该框架结合了深度学习和非线性动力学方法的稀疏识别，以发现有效坐标内的可解释模型。
A central challenge in data-driven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens’ theorem provides conditions for when it is possible to augment partial measurements with time delayed information, resulting in an attractor that is diffeomorphic to that of the original full-state system. This diffeomorphism is typically unknown, and learning the dynamics in the embedding space has remained an open challenge for decades. Here, we design a deep autoencoder network to learn a coordinate transformation from the delay embedded space into a new space, where it is possible to represent the dynamics in a sparse, closed form. We demonstrate this approach on the Lorenz, Rössler and Lotka–Volterra systems, as well as a Lorenz analogue from a video of a chaotic waterwheel experiment. This framework combines deep learning and the sparse identification of nonlinear dynamics methods to uncover interpretable models within effective coordinates.