深度学习动力学系统发现中线性多步法的核定位与全轨迹泛化
Kernel Localization and Whole-Trajectory Generalization for Linear Multistep Methods in Deep Learning-Based Discovery of Dynamical Systems
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中文总结 AI 辅助
本文针对深度学习动力学系统发现中线性多步法的未决问题，通过核分析解释非辅助型格式的零残差解特性，推导两类公式的全轨迹泛化估计，经数值实验验证收敛行为。
中文摘要 AI 辅助
线性多步法（LMM）结合神经网络近似，为从离散轨迹数据中学习动力学系统的控制向量场提供了高阶框架。本文研究基于LMM的动力学发现中现有网格级收敛理论未解决的两个问题：其一，在非辅助型Adams–Bashforth（A-B）与Adams–Moulton（A-M）发现系统中，零残差网格解非唯一但沿轨迹一致，差异仅局限于边界层；我们通过对非辅助型发现矩阵的核分析解释该现象，在相应的发现稳定性条件下，A-B格式的零残差网格解差异指数定位于初始索引附近，而A-M格式的差异则在初始与终端索引附近形成双侧边界层。其二，我们推导了辅助型与非辅助型公式的全轨迹泛化估计：当学习到的向量场被限制于固定观测轨迹时，误差的每个分量成为时间的标量函数；对于辅助型公式，在对应网格精度、近似与迹正则性假设下，轨迹误差为O(h^p)；对于非辅助型公式，全局估计包含额外的边界层项，在固定内部子区间上这些项呈指数衰减；数值实验验证了收敛行为。
英文摘要
Linear multistep methods (LMMs) combined with neural-network approximation provide a high-order framework for learning governing vector fields of dynamical systems from discrete trajectory data. This paper studies two issues in LMM-based discovery that are not resolved by existing grid-level convergence theory. First, in non-auxiliary Adams--Bashforth (A-B) and Adams--Moulton (A-M) discovery systems, we observe that zero-residual grid solutions are nonunique but consistent along the trajectory, with differences limited to the boundary layer. We explain this phenomenon through a kernel analysis of the non-auxiliary discovery matrices. Under the corresponding discovery-stability conditions, the differences between zero-residual grid solutions are exponentially localized near the initial indices for A-B schemes, whereas they form two-sided boundary layers near the initial and terminal indices for A-M schemes. Second, we derive whole-trajectory generalization estimates for both auxiliary and non-auxiliary formulations. Once the learned vector field is restricted to a fixed observed trajectory, each component of the error becomes a scalar function of time. For the auxiliary formulation, the trajectory error is $O(h^p)$ under the corresponding grid accuracy, approximation, and trace regularity assumptions. For non-auxiliary formulations, the global estimates contain additional boundary-layer terms. On fixed interior subintervals, these terms are exponentially damped. Numerical experiments illustrate the convergence behavior.
