数据驱动的Koopman模态近似：一种神经幂迭代算法

中文摘要 AI 辅助

本文提出了一种新颖的数据驱动算法，用于使用神经网络近似非线性动力系统Koopman算子的主导本征函数（又称模态）。学习主导Koopman模态的意义在于在提升空间中用线性模型近似非线性动力学，从而实现简化的控制与分析。为应对使用表达性模板（此处为神经网络）进行模态近似时产生的维度灾难问题，所提方法采用幂迭代方案，直接学习主导Koopman模态，无需显式构建Koopman算子在函数模板上的投影。该方法与文献中其他通过学习小型函数字典避免维度灾难的方法相关，但区别在于，我们无需“反坍塌机制”来确保学习到的字典具有足够的表达性以近似Koopman算子，因为我们的幂迭代方案旨在收敛到投影Koopman算子的主导模态。该方法完全数据驱动，仅需采样得到的状态转移数据。理论保证表明，在样本量和网络宽度增加时（与神经正切核定理相关），算法可收敛。数值实验表明，该方法能实现主导模态的精确且平滑的近似，同时避免了扩展动态模式分解等传统技术的局限性。

英文摘要

This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.

发表机构

• ICTEAM, UCLouvain（比利时鲁汶大学ICTEAM学院）

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