NOMADS: Non-Markovian Optimization-based Modeling for Approximate Dynamics with Spatially-homogeneous Memory
This work addresses the problem of identifying non-Markovian dynamics from limited and noisy experimental data, which is important for engineers and scientists working with complex systems.
This paper introduces NOMADS, a system identification method that models linear dynamical systems with spatially-homogeneous memory from multi-dimensional time-series data. It formulates model identification as a convex optimization problem, jointly estimating state-space matrices and a memory kernel. Numerical experiments show NOMADS significantly improves generalization accuracy over existing DMD-based methods, even with noisy training data.
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.