AI 中文总结
本研究提出NSDMD方法，利用非线性项数据快照构建回归问题，求解线性算子低秩近似，提升了混沌等类型动力学的数据驱动线性分析能力。
AI 中文摘要
动态模态分解（Dynamic Mode Decomposition, DMD）已成为数据驱动动力系统分析的基础工具，可从时间分辨测量中同时识别相干结构及其动力学。然而，由于核心是线性回归，DMD无法从本质非线性动力学的记录中生成准确模型，例如对大扰动的响应和混沌吸引子上的演化。近期方法尝试通过对具有物理动机的模型结构进行回归，同时拟合动力学的线性和非线性贡献。但尽管所得非线性模型可产生准确的短期预测，其线性化未必与原系统的线性化一致。本研究提出一种新颖的数据驱动方法——非线性项减去动态模态分解（nonlinearity-subtracted DMD, NSDMD），该方法聚焦于当动力学的非线性贡献已知而线性部分未知时，生成系统的准确线性化。此类场景例如：控制方程中的非线性项已知，但线性算子包含不确定的材料属性或需考虑未 resolves 动力学的闭合；或数据由黑箱模拟代码生成，该代码可输出非线性项，但无法输出线性算子对快照的作用。NSDMD利用非线性项的数据快照，明确考虑动力学中纯非线性贡献，并构建回归问题以寻找底层线性算子的低秩近似。我们在多个数值示例中验证该方法，展示其在混沌、部分观测、平流主导及高维动力学的数据驱动线性分析中的改进能力。
英文摘要
The Dynamic Mode Decomposition (DMD) has been consolidated as a basic tool for data-driven analysis of dynamical systems, allowing simultaneous identification of coherent structures and their dynamics from time-resolved measurements. However, with a linear regression at its core, DMD is unable to produce accurate models from recordings of dynamics that are inherently nonlinear, such as the response to large perturbations and the evolution on chaotic attractors. Recent approaches attempt to simultaneously fit the linear and nonlinear contributions to the dynamics by performing a regression onto a physically motivated model structure. However, although the resulting nonlinear models can produce accurate short-term predictions, their linearization does not necessarily agree with that of the original system. In this work, we introduce a novel data-driven method --- nonlinearity-subtracted DMD (NSDMD) --- that focuses on producing an accurate linearization of a system when the nonlinear contribution to its dynamics are available while the linear part is not. This scenario is encountered, for example, when the nonlinear terms in the governing equations are known, while the linear operator contains uncertain material properties or it accounts for the closure of unresolved dynamics. This also arises when the data is generated by a black-box simulation code that is able to output the nonlinearity, but not the action of the linear operator on the snapshots. NSDMD leverages data snapshots of the nonlinearity to explicitly account for the purely nonlinear contributions to the dynamics and formulate a regression problem that finds a low-rank approximation of the underlying linear operator. We demonstrate the approach on several numerical examples, showcasing its improved capabilities for data-driven linear analysis of chaotic, partially observed, advection-dominated, and high-dimensional dynamics.
