arXiv subjects

Suddhasattwa Das

Publications and source records attributed to Suddhasattwa Das.

At least 19 recordsLinked to original sources

arXiv2026

Joinings in Markov categories

Dynamical systems theory primarily concerns the study of different forms of invariance, such as invariant sets, densities, measures, and observables. Ergodic systems are a special class of dynamical systems which are measure theoretically irreducible. In spite of this specialized property they dominate most discussions on ergodic theory because of the ergodic decomposition theorem. The viewpoint that any dynamical system is a composite of multiple ergodic components enables us to partition the face space into the basins of the different measures. The coexistence and mutual connections between these various coexisting subsystems are illuminated very effectively using the language of Category theory (CT). Some recent advancements have shown how the essence of measure theoretic dynamical systems can be captured through the formalism of Markov categories. This formalism captures the essential features of invariance and ergodicity through the language of limits and colimits. This article continues that formalism by studying the concept of joins using the language of spans and push-outs. A classical result is re-proven which establishes the connection between joins, ergodicity and mixing. In the process, the categorical language is refined to capture the notion of sub sigma-algebras and their invariance.

math.CT↗

arXiv2026

Cells, convexity and contractibility in general categories

The two pillars of Algebraic topology - homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and contractibility. The first two cells, namely the line and point lead to the concept of homotopy. The collection of maps from the cells and the redundancies among them determine the homology of objects. This article presents a procedure by which such cells can be built in general categories satisfying some simple axioms. The cells satisfy the categorical analogs of convexity and contractibility. This enables a cellular theory for the general category, carrying notions of homotopy, homology, cellular approximation and homotopy equivalence which are mutually compatible in the same way as in the familiar context of Topology.

math.CT↗

arXiv2026

Discrete-time dynamics, step-skew products, and pipe-flows

Dynamical processes can be classified in various ways as deterministic or stochastic, and continuous or discrete time. All these types can be studied by the path-spaces they generate, and stationary measures on that path-space. Such measures are called the law of the dynamics. This article presents how a general ergodic dynamical system may be approximated in terms of their law, by a simple and restricted family of deterministic continuous-time skew-product systems. In these systems, a deterministic, mixing flow intermittently drives a deterministic flow through a topological space created by gluing cylinders. The resulting orbits mimic the law of the original dynamics. This comparison is made possible by introducing a secondary intermediary approximation of the ergodic dynamics. This third system is a step-skew dynamical system, in which a finite state Markov process drives a dynamics on topological disk. Each of these three representations have their advantages. It is proved that the distribution induced on the space of paths by these three dynamics can be made arbitrarily close to each other. This analysis reconfirms the old principle that it is impossible to decide whether a general timeseries is generated by a deterministic or stochastic process, and is of continuous or discrete time.

math.DS↗

arXiv2026

Physics-informed machine learning for reconstruction of dynamical systems with invariant measure score matching

In this paper, we develop a novel mesh-free framework, termed physics-informed neural networks with invariant measure score matching (PINN-IMSM), for reconstructing dynamical systems from unlabeled point-cloud data that capture the system's invariant measure. The invariant density satisfies the steady-state Fokker-Planck (FP) equation. We reformulate this equation in terms of its score function (the gradient of the log-density), which is estimated directly from data via denoising score matching, thereby bypassing explicit density estimation. This learned score is then embedded into a physics-informed neural network (PINN) to reconstruct the drift velocity field under the resulting score-based FP equation. The mesh-free nature of PINNs allows the framework to scale to higher dimensions, avoiding the curse of dimensionality inherent in mesh-based methods. To address the ill-posedness of high-dimensional inverse problems, we recast the problem as a PDE-constrained optimization that seeks the minimal-energy velocity field. Under suitable conditions, we prove that this problem admits a unique solution that depends continuously on the score function. The constrained formulation is solved using a stochastic augmented Lagrangian method. Numerical experiments on representative dynamical systems, including the Van der Pol oscillator, an active swimmer in an anharmonic trap, and the chaotic Lorenz-63 and Lorenz-96 systems, demonstrate that PINN-IMSM accurately recovers invariant measures and reconstructs faithful dynamical behavior for problems in up to five dimensions.

math.NA↗

arXiv2026

Dynamics, data and reconstruction

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.

math.DS↗

arXiv2025

The concept of null in general spaces and contexts

The notions of null-sets and nullity are present in all discourses of mathematics. They are based on the dual-pair of notions of "almost-every" and "almost none". A notion of nullity corresponds to a choice of subsets that one interprets as null or empty. The rationale behind this choice depends on the context, such as Topology or Measure theory. One also expects that the morphisms or transformations within the contexts preserve the nullity structures. To formalize this idea a generalized notion of nullity is presented as a functor between categories. A constructive procedure is presented by which an existing notion of nullity can be extended functorially to categories with richer structure. Nullity is thus presented as an arbitrary construct, which can be extended to broader contexts using well defined rules. These rules are succinctly expressed by right and left Kan extensions.

math.CT↗

arXiv2025

Homology and homotopy for arbitrary categories

One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was discovered with the development of Homology theory. Some of the deepest results in topology are about the connections between Homotopy and Homology. These results are proved using intricate constructions. This paper re-proves these connections via an axiomatic approach that provides a common ground for homotopy and homology in arbitrary categories. One of the main contributions is a re-interpretation of convexity as an extrinsic rather than intrinsic property. All the axioms and results are applicable for the familiar context of topological spaces. At the same time it provides a complete framework for an algebraic characterization of objects in a general category, which also preserves a notion of Homotopy.

math.CT↗

arXiv2025

A probabilistic approach to drift estimation from stochastic data

Timeseries generated from a dynamical source can often be modeled as sample paths of a stochastic differential equation (SDE). The timeseries thus reflects the motion of a particle which flows along the direction provided by a drift / vector field, and is simultaneously scattered by the effect of white noise. The resulting motion can only be described as a random process instead of a solution curve. Due to the non-deterministic nature of this motion, the task of determining the drift from data is quite challenging, since the data does not directly represent the directional information of the flow. This paper describes an interpretation of a drift as a conditional expectation, which makes its estimation feasible via kernel-integral methods. In addition, some techniques are proposed to overcome the challenge of dimensionality if the SDE's carry some structure enabling sparsity. The technique is shown to be convergent, consistent and permits a wide choice of kernels.

math.DS↗

arXiv2025

Functors induced by comma categories

Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the collection is provided by its associated left or right slice. The assignment of slices to objects extends to a functor from the base category, into the category of categories. Slice categories are a special case of the more general notion of comma categories. Comma categories are created when two categories $\mathcal{A}$ and $\mathcal{B}$ transform into a common third category $\mathcal{C}$, via functors $F,G$. Such arrangements denoted as $\Comma{F}{G}$ abound in mathematics, and provide a categorical interpretation of many constructions in Mathematics. Objects in this category are morphisms between objects of $\mathcal{A}$ and $\mathcal{B}$, via the functors $F,G$. We show that these objects also have a natural interpretation as functors between slice categories of $\mathcal{A}$ and $\mathcal{B}$. Thus even though $\mathcal{A}$ and $\mathcal{B}$ may have completely disparate structures, some morphisms in $\mathcal{C}$ lead to functors between their respective slices. We present this relation in the form of a functor from $\mathcal{C}$ into the category of left slices. The proof of our main result requires a deeper look into associated categories, in which the objects themselves are various commuting diagrams.

math.CT↗

arXiv2025

Dynamical systems as enriched functors

This article presents a general description of dynamical systems using the language of enriched functors and enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a semigroup action on the domain; a parameterized family of endomorphisms; and a transformation of time-space into the collection of endomorphisms. A collection of categorical axioms are presented that provides a complete categorical language to develop dynamical systems theory. None of the assumptions are rooted in specific contexts such as topology and measure spaces. The equivalence of the three descriptions is further used to construct other related notions,such as transfer operators, orbits and shift-spaces. All of these objects are defined by their structural role and universal properties, instead of their usual pointwise definitions.

math.CT↗

arXiv2025

Image denoising as a conditional expectation

All techniques for denoising involve a notion of a true (noise-free) image, and a hypothesis space. The hypothesis space may reconstruct the image directly as a grayscale valued function, or indirectly by its Fourier or wavelet spectrum. Most common techniques estimate the true image as a projection to some subspace. We propose an interpretation of a noisy image as a collection of samples drawn from a certain probability space. Within this interpretation, projection based approaches are not guaranteed to be unbiased and convergent. We present a data-driven denoising method in which the true image is recovered as a conditional expectation. Although the probability space is unknown apriori, integrals on this space can be estimated by kernel integral operators. The true image is reformulated as the least squares solution to a linear equation in a reproducing kernel Hilbert space (RKHS), and involving various kernel integral operators as linear transforms. Assuming the true image to be a continuous function on a compact planar domain, the technique is shown to be convergent as the number of pixels goes to infinity. We also show that for a picture with finite number of pixels, the convergence result can be used to choose the various parameters for an optimum denoising result.

eess.IV↗

arXiv2025

Learning Dynamical Systems with the Spectral Exterior Calculus

We present a data-driven framework for learning dynamical systems on compact Riemannian manifolds based on the spectral exterior calculus (SEC). This approach represents vector fields as linear combinations of frame elements constructed using the eigenfunctions of the Laplacian on smooth functions, along with their gradients. Such reconstructed vector fields generate dynamical flows that consistently approximate the true system, while being compatible with the nonlinear geometry of the manifold. The data-driven implementation of this framework utilizes embedded data points and tangent vectors as training data, along with a graph-theoretic approximation of the Laplacian. In this paper, we prove the convergence of the SEC-based reconstruction in the limit of large data. Moreover, we illustrate the approach numerically with applications to dynamical systems on the unit circle and the 2-torus. In these examples, the reconstructed vector fields compare well with the true vector fields, in terms of both pointwise estimates and generation of orbits.

math.DS↗

arXiv2025

Smooth Koopman eigenfunctions

Any dynamical system, whether it is generated by a differential equation or a transformation map on a manifold, induces a dynamics on functional-spaces. The choice of functional-space may vary, but the induced dynamics is always linear, and codified by the Koopman operator. The eigenfunctions of the Koopman operator are of extreme importance in the study of the dynamics. They provide a clear distinction between the mixing and non-mixing components of the dynamics, and also reveal embedded toral rotations. The usual choice of functional-space is $L^2$, a class of square integrable functions. A fundamental problem with eigenfunctions in $L^2$ is that they are often extremely discontinuous, particularly if the system is chaotic. There are some prototypical systems called skew-product dynamics in which $L^2$ Koopman eigenfunctions are also smooth. The article shows that under general assumptions on an ergodic system, these prototypical examples are the only possibility. Moreover, the smooth eigenfunctions can be used to create a change of variables which explicitly characterizes the weakly mixing component too.

math.DS↗

arXiv2024

Limits of Learning Dynamical Systems

A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system. It is the object of focus of many learning techniques. Yet there are many secondary aspects of dynamical systems - invariant sets, the Koopman operator, and Markov approximations, which provide alternative objectives for learning techniques. Crucially, while many learning methods are focused on the transformation law, we find that forecast performance can depend on how well these other aspects of the dynamics are approximated. These different facets of a dynamical system correspond to objects in completely different spaces - namely interpolation spaces, compact Hausdorff sets, unitary operators and Markov operators respectively. Thus learning techniques targeting any of these four facets perform different kinds of approximations. We examine whether an approximation of any one of these aspects of the dynamics could lead to an approximation of another facet. Many connections and obstructions are brought to light in this analysis. Special focus is put on methods of learning of the primary feature - the dynamics law itself. The main question considered is the connection of learning this law with reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and reveal how these properties determine the limits of forecasting techniques.

math.DS↗

arXiv2024

Reconstructing dynamical systems as zero-noise limits

A dynamical system may be defined by a simple transition law - such as a map or a vector field. The objective of most learning techniques is to reconstruct this dynamic transition law. This is a major shortcoming, as most dynamic properties of interest are asymptotic properties such as an attractor or invariant measure. Thus approximating the dynamical law may not be sufficient to approximate these asymptotic properties. This article presents a method of representing a discrete-time deterministic dynamical system as the zero-noise limit of a Markov process. The Markov process approximation is completely data-driven. Besides proving a low-noise approximation of the dynamics the process also approximates the invariant set, via the support of its stationary measures. Thus invariant sets of arbitrary dynamical systems, even with complicated non-smooth topology, can be approximated by this technique. Under further assumptions, we show that the technique performs a convergent statistical approximation as well as approximations of true orbits.

math.DS↗

arXiv2024

Data-driven discovery of quasiperiodically driven dynamics

The analysis of a timeseries can provide many new perspectives if it is accompanied by the assumption that the timeseries is generated from an underlying dynamical system. For example, statistical properties of the data can be related to measure theoretic aspects of the dynamics, and one can try to recreate the dynamics itself. The underlying dynamics could represent a natural phenomenon or a physical system, where the timeseries represents a sequence of measurements. In this paper, we present a completely data-driven framework to identify and model quasiperiodically driven dynamical systems (Q.P.D.) from the timeseries it generates. Q.P.D. are a special class of systems that are driven by a periodic source with multiple base frequencies. Such systems abound in nature, e.g., astronomy and traffic flow. Our framework reconstructs the dynamics into two components - the driving quasiperiodic source with generating frequencies; and the driven nonlinear dynamics. We make a combined use of a kernel-based harmonic analysis, kernel-based interpolation technique, and Koopman operator theory. Our framework provides accurate reconstructions and frequency identification for three real-world case studies.

math.DS↗

arXiv2024

The categorical basis of dynamical entropy

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.

math.CT↗

arXiv2024

Conditional expectation using compactification operators

The separate tasks of denoising, least squares expectation, and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that allow numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems are also shown.

stat.ML↗