AI 中文总结
研究针对非线性动力系统，提出正分配伴随构造法预测，通过特定组合确定递归系数定义伴随矩阵，证明其谱特性，开发诊断方法，经数值实验展示该方法在不同模型中的行为、可解释性及局限性。
AI 中文摘要
我们引入了一种用于受柯普曼启发的非线性动力系统有限维预测的正分配伴随构造。该方法通过将目标可观测量快照表示为早期训练快照的非负归一化组合来确定递归系数。这些系数定义了一个伴随矩阵，其谱结构由构造阶段的分配约束诱导。我们证明归一化正分配将伴随谱置于闭单位圆盘内，且因系数和为1，包含1作为特征值。此外，我们为所得模型轨迹开发了模态和非模态诊断。当伴随矩阵可对角化时，模态表示表明一阶和二阶有限差分通过\(\lambda_\ell - 1\)和\((\lambda_\ell - 1)^2\)因子充当谱滤波器。我们还推导了基于\(C\)的有限差分界，避免对角化且可直接从训练数据和伴随矩阵评估。在FitzHugh - Nagumo和易感 - 感染 - 恢复（SIR）模型上的数值实验说明了该构造在振荡和瞬态耗散设置中的行为。这些例子展示了伴随递归的可解释性及其局限性，特别是当逐点轨迹一致性下降而有限差分和模态诊断仍然有用时。
英文摘要
We introduce a positive-allocation companion construction for Koopman-inspired finite-dimensional prediction of nonlinear dynamical systems. The method determines recurrence coefficients by representing a target observable snapshot as a nonnegative, normalized combination of earlier training snapshots. These coefficients define a companion matrix whose spectral structure is induced by the allocation constraints at the construction stage. We prove that normalized positive allocation places the companion spectrum in the closed unit disk and, because the coefficients sum to one, includes $1$ as an eigenvalue. Additionally, we develop modal and non-modal diagnostics for the resulting model trajectory. When the companion matrix is diagonalizable, the modal representation shows that first and second finite differences act as spectral filters through factors of $λ_\ell-1$ and $(λ_\ell-1)^2$. We also derive $C$-based finite-difference bounds that avoid diagonalization and can be evaluated directly from the training data and companion matrix. Numerical experiments on the FitzHugh--Nagumo and susceptible--infectious--recovered (SIR) models illustrate the behavior of the construction in oscillatory and transient dissipative settings. The examples demonstrate both the interpretability of the companion recurrence and its limitations, particularly when pointwise trajectory agreement degrades while finite-difference and modal diagnostics remain informative.
Comments27 pages, 7 figures
