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

TL;DR

NOMADS introduces a convex optimization framework for identifying linear non‑Markovian systems with spatially homogeneous memory from multiple partially excited trajectories. By jointly estimating $(A,B,D)$ with memory in a low‑dimensional kernel $D\in\mathbb{T}_m^{q}(Q)$ and enforcing physical constraints via projection, it achieves better generalization than DMD‑based methods and preserves energy in Markovian cases. The method comes with convergence guarantees for the projected gradient descent and demonstrates robust performance on synthetic high‑dimensional grids under noise and partial excitation. The work extends to fractional dynamics in a memory‑kernel form and analyzes uniqueness conditions for NOMADS and related methods, highlighting practical considerations for identifiability in multi‑trajectory settings.

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.

Figures (4)

• Figure 1: NOMADS with A2 and B constraints for the non-Markovian dynamical system. Comparison between the ground truth (solid lines) and reconstructed trajectories (dotted lines), evaluated on $\mathcal{S}_{\mathrm{ts}}$. Left column: models trained on noiseless training data. Right column: models trained on noisy training data.
• Figure 2: Existing methods. Ground truth (solid) and reconstructed trajectories (dotted) for the Markovian dynamical system. Models are identified from noiseless train data $\mathcal{S}_{\mathrm{tr}}$ and evaluated on $\mathcal{S}_{\mathrm{ts}}$. Left: DMDm. Right: DMDc trained on horizontally concatenated trajectories.
• Figure 3: Running maximum of the absolute energy error in the Markovian settings, where the energy $E(t)$ is defined in \ref{['eq:num:energy']}. The initial condition is set to $\boldsymbol{x}(0)=\mathbf{1}_{n\times 1}$. All models are trained on noiseless training data.
• Figure 4: Learning curves of the training loss for NOMADS under different constraints. All models are trained on noiseless training data.

Theorems & Definitions (27)

• Definition 1: Causal matrix
• Definition 2: Time-invariant causal matrix
• Definition 3
• Lemma 4
• Theorem 5
• Definition 6
• Remark 7: Constraint
• Remark 8
• Remark 9
• Theorem 10