Dynamics, data and reconstruction

TL;DR

This work casts dynamical systems, their measurements, and time-series data in a unified category-theoretic framework. By treating data generation as a functorial process across categories such as DSMO, Seq, and TSD, the authors show reconstruction can be analyzed as a functor, with inner/outer approximations arising as Kan extensions. They establish that, under mild consistency conditions, observation-based inner reconstructions are exact, linking data, measurements, and dynamics through universal categorical constructions. The framework clarifies the limits of universal reconstruction, highlights the role of discretization and delay coordinates, and offers a principled lens for comparing data-driven reconstruction methods. This category-theoretic viewpoint provides a high-level, structural understanding of reconstruction in dynamical systems with measurements, with potential implications for designing data-driven algorithms and assessing their theoretical consistency.

Abstract

The goal of data-driven learning of dynamical systems is to interpret time series as a continuous observation of an underlying dynamical system. This task is not well-posed for a variety of reasons - such as multiple co-existing sub-systems, topologically inter-weaving of these sub-systems; and more importantly, the non-injectivity of the correspondence between dynamical systems and time series. We show how these ambiguities are circumvented if one considers dynamical systems and measurement maps collectively. Dynamical systems, observed dynamical systems, and time series data - each of these three collections have an extensive network of relations within them, which gives them the mathematical structure of a category. One of the new concepts proposed is a rigorous definition of time series data as a chain of measurement sequences with decreasing information content. This definition subsumes the familiar notions of sequences, time series and even subshifts. Using these notions it is shown that the entire process of converting an observed dynamical systems into a time series object is functorial, and passes through a number of phases each bearing its own categorical structure. This discovery sheds new light on the nature of reconstruction algorithms. Under mild conditions of consistency, reconstruction itself is shown to be functorial operation. This provides a new category theoretic perspective on the nature and limits of reconstruction.

Figures (4)

• Figure 1: General scheme for devising algorithms for functorial reconstruction of dynamics.
• Figure 2: Representing semi-groups as 1-object categories -- $\left[ \mathbb{N}_0, + \right]$, $\left[ \mathbb{Z}, + \right]$ and $\left[ \mathbb{R}, + \right]$.
• Figure 3: Dynamics as a functor. Section \ref{['sec:cat']} presents a dynamical system as a functor between a semigroup category $\mathcal{T}$ representing time, and a category $\mathcal{C}$ representing the context. Shown here is such a functor $\Phi$ transforming the additive semigroup $\left[ \mathbb{N}_0, + \right]$ of non-negative integers into self-maps on a space $\Omega$. Since $\left[ \mathbb{N}_0, + \right]$ is generated by the unit element $1$, the image $F$ of this element generates all iterations of the dynamics.
• Figure 4: Algorithms and their limiting behavior. A $\mathop{\mathrm{ \textbf{TSD} }}\nolimits$-object \ref{['eqn:def:TSD']}$X$ is represented as a sequence of sets of $N$-sequences $X_N$. Each such dataset $X_N$ is mapped by the algorithm $\mathcal{A}$ into an estimate of the dynamical system. This mapping depends on the state of the design and performance parameters $\lambda$. Their limiting behavior is indicated in blue by $\bar{A}$. Although $\mathcal{A}$ is not functorial, $\bar{\mathcal{A}}$ is. In topological reconstructions, by pointwise convergence, the limit $\bar{A}(X)$ of the estimates $\bar{A}(X_N)$ have a functorial dependence on $X$. The reconstruction functor $\mathop{\mathrm{\mathfrak{R}}}\nolimits(X)$ is precisely this limit. It does not represent the output of an algorithm but its idealized operation.

Theorems & Definitions (81)

• Example 1: Pre-order valued observations
• Definition 1
• Definition 2: Dynamical systems with measurements along orbits
• Definition 3
• Lemma 5.1
• Lemma 5.2
• Definition 4: Shift morphism
• Definition 5: Subshift spaces
• Definition 6: Category of subshifts
• Example 2