NOMADS: Non-Markovian Optimization-based Modeling for Approximate Dynamics with Spatially-homogeneous Memory

TL;DR

NOMADS is a scalable system identification method that models non-Markovian dynamics with spatially-homogeneous memory kernels, improving accuracy and energy conservation in multi-experiment data analysis.

Contribution

It introduces a convex optimization framework for joint estimation of system matrices and memory kernels, enabling scalable modeling of non-Markovian dynamics from multiple datasets.

Findings

Outperforms existing DMD-based methods in accuracy, especially with noisy data.

Successfully enforces physical constraints like energy conservation.

Handles multiple partially excited experiments effectively.

Abstract

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.