We propose a system identification method, Non-Markovian Optimization-based Modeling for Approximate Dynamics with Spatially-homogeneous memory (NOMADS), for identifying linear dynamical systems from a set of multi-dimensional time-series data obtained through multiple partially excited experiments. NOMADS formulates model identification as a convex optimization problem, in which the state-space coefficient matrices and a memory kernel are estimated jointly under physically motivated constraints using projected gradient descent. The proposed framework models memory effects through a spatially homogeneous kernel, enabling scalable identification of non-Markovian dynamics while keeping the number of free parameters moderate. This structure allows NOMADS to integrate information from multiple multi-dimensional time-series data even when no single experiment provides full excitation. In the Markovian setting, physical constraints can be incorporated to enforce conservation laws. Numerical experiments on synthetic data demonstrate that NOMADS achieves substantially improved generalization accuracy compared to existing DMD-based methods even for noisy train data, and reproduces energy conservation in the Markovian case.

翻译：我们提出了一种系统辨识方法——基于优化的非马尔可夫建模方法用于具有空间齐次记忆的近似动力学（NOMADS），用于从多个部分激励实验获得的多维时间序列数据集中辨识线性动力系统。NOMADS将模型辨识构建为一个凸优化问题，其中状态空间系数矩阵与记忆核在物理驱动的约束下通过投影梯度下降法进行联合估计。所提出的框架通过空间齐次核建模记忆效应，能够在保持自由参数数量适中的同时，实现非马尔可夫动力学的可扩展辨识。这一结构使得NOMADS能够整合来自多个多维时间序列数据的信息，即使没有任何单一实验能提供完全激励。在马尔可夫情形下，可以引入物理约束以强制满足守恒定律。在合成数据上的数值实验表明，即使在训练数据存在噪声的情况下，NOMADS相比现有基于DMD的方法仍能显著提升泛化精度，并在马尔可夫情形下重现能量守恒特性。