Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute “most powerful unfalsified models” (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.

💡 Research Summary

The paper addresses the problem of exact linear modeling in system identification, where one seeks a linear differential‑ or difference‑equation model that reproduces a given set of observed trajectories without error. While the case of operators with constant coefficients has been extensively studied—often using Gröbner bases for modules over commutative polynomial rings—the authors turn to the more general setting of Ore algebras, which accommodate both difference and differential operators with non‑constant (variable) coefficients.

The authors introduce two central notions: the “most powerful unfalsified model” (MPUM) for constant‑coefficient operators and its variable‑coefficient counterpart (VMPUM). An MPUM is defined as a linear model that (i) exactly fits the data, (ii) is unfalsified by any additional observation, and (iii) has minimal module dimension (or, equivalently, minimal number of independent equations). The VMPUM extends this definition by allowing the coefficients of the operators to be polynomial or polynomial‑exponential functions of the independent variables, thereby capturing a much richer class of signals.

To compute MPUMs and VMPUMs the paper develops an algorithmic framework that blends three strands of algebraic computation:

1. Ore algebra representation – The authors formalize the ambient algebra of operators as an Ore algebra ( \mathcal{O} = K