Koopman框架中的强迫项建模用于非线性动力系统的线性算子学习
Forced-Term Modeling in the Koopman Framework for Linear Operator Learning in Nonlinear Dynamical Systems
- University of California, Los Angeles（加州大学洛杉矶分校）
- Universität Bonn（波恩大学）
- Institut für Numerische Simulation, Universität Bonn（波恩大学数值模拟研究所）
- Fraunhofer SCAI（弗劳恩霍夫SCA）
- Fraunhofer SCAI, Sankt Augustin（圣奥古斯丁弗劳恩霍夫SCA）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对非线性周期系统，提出基于缺陷空间建模的CB-DMD方法，通过强迫项吸收非线性成分，学习更可靠的自主线性算子，实验验证优于标准DMD。
AI中文摘要：
我们研究了从具有周期行为的非线性动力系统中提取可靠且可解释的线性模型的问题。动态模态分解（DMD）是一种广泛使用的数据驱动方法，用于通过线性系统近似动力学，通常通过Koopman算子理论进行解释。然而，由于标准DMD假设所选可观测量的线性演化，它可能在非线性系统上产生误导性结果。最近的扩展通过将动力学建模为由非线性强迫项驱动的线性系统来解决这一局限性，但它们与Koopman理论的关系及其可靠性仍不清楚。我们观察到，Koopman理论框架允许对动力系统进行显式的、有限维的、强迫线性表示，且独立于所选择的可观测量空间。因此，主要问题在于如何为强迫项选择一个有用、可计算且可解释的模型，因为强迫线性表示通常是非唯一的。在此考虑的框架中，该建模问题与缺陷空间的选择有关：缺陷空间是一个补充空间，用于闭合Koopman算子对所选择可观测量的作用。现有的强迫DMD方法可关联到该缺陷空间的不同建模选择，其中学习到的线性算子的结构取决于缺陷空间的选择。作为缺陷空间建模用于算法设计的一个示例，我们引入了基于相关基增强的DMD（CB-DMD），这是一种数据驱动方法，其目标是学习周期非线性动力学的有效自主线性算子。强迫项用于吸收无法由学习到的线性算子表示的非线性成分。在三个非线性周期系统上的实验表明，在测试示例中，CB-DMD比标准DMD及相关变体产生更可靠的线性模型。
英文摘要：
We study the problem of extracting reliable and interpretable linear models from nonlinear dynamical systems with periodic behavior. Dynamic Mode Decomposition (DMD) is a widely used data-driven method for approximating dynamics by a linear system and is commonly interpreted through Koopman operator theory. However, because standard DMD assumes linear evolution in the chosen observables, it can produce misleading results on nonlinear systems. Recent extensions address this limitation by modeling the dynamics as a linear system driven by a nonlinear forcing term, but their relationship to Koopman theory and their reliability remain unclear. We observe that the Koopman theoretical framweork allows for explicit, finite-dimensional, forced linear representations of dynamical systems, independently of the chosen observable space. The main question is therefore how to choose a useful, computable, and interpretable model for the forcing term, since forced linear representations are generally nonunique. In the framework considered here, this modeling question is related to the choice of a defect space: a complementary space that closes the action of the Koopman operator on the chosen observables. Existing forced-DMD approaches can be related to different modeling choices for this defect space, where the structure of the learned linear operator depends on the choice of the defect space. As an example of defect-space modeling for algorithm design, we introduce Correlation-Basis enhanced DMD (CB-DMD), a data-driven method whose goal is to learn an effective autonomous linear operator for periodic nonlinear dynamics. The forcing term is used to absorb nonlinear components that cannot be represented by the learned linear operator. Experiments on three nonlinear periodic systems demonstrate that CB-DMD yields more reliable linear models than standard DMD and related variants in the tested examples.
