arXiv subjects

Álvaro Rodríguez Abella

Publications and source records attributed to Álvaro Rodríguez Abella.

15 recordsLinked to original sources

arXiv2026

Deep learning from a control vantage point

This article offers an overview of recent developments in deep learning from a control vantage point. It reviews several classes of models including feedforward and residual neural networks, transformers, large-language models, and selective state space models while explaining how control theory plays an important role in understanding their training, properties, or architecture design. The article explains how residual neural networks can be interpreted as the discretization of a continuous-time control system thereby enabling the use of control theoretic techniques for its analysis. To illustrate the power of this bridge between control and deep learning, two technical results are discussed: 1) the ability of a residual neural network to approximate arbitrary continuous functions is a non-standard controllability problem that can be solved using geometric nonlinear control and 2) transformers, the key architectural component of most large language-models, can be seen the discretization of a consensus algorithm and thus large language-models exhibit consensus. Another point of contact between control theory and deep learning is the connection between backpropagation, the algorithm used to train neural networks, and optimal control. The article addresses how the older connection between these problems, established in the 80's using discrete-time models can be reconciled with the more recent approach using continuous-time models. Several open problems and directions for future research are discussed, including selective state-space models, a recent class of models inspired by the models used in control.

eess.SY↗

arXiv2026

Covariant Euler-Poincaré Reduction for Field Theories on Centered Semidirect Products

Let $G$ be a Lie group equipped with commuting left and right linear representations on a vector space $V$, and let $G\Join V$ be the associated centered semidirect product. We develop a reduction scheme for covariant field theories on principal bundles with structure group $G\Join V$ from both Lagrangian and Hamiltonian perspectives, obtaining the reduced Euler-Poincaré and Lie-Poisson field equations decomposed into their $\mathfrak g$- and $V$-components, where $\mathfrak g$ is the Lie algebra of $G$. We also derive the reduced multisymplectic form formula, as well as the Noether and energy-momentum balance laws. Our reduction results are compared with those for principal bundles with standard semidirect-product structure groups, and the role of the centered variables is investigated: they retain the original two-sided geometric data and separate the left and right contributions via the heart and diamond operators. The framework is then applied to a two-sided matrix microstructural model and to the second-order jet group on $\mathbb R^r$, $G_r^2$. For the matrix model, the conjugation representation decomposes $\operatorname{Mat}(3)$ into scalar, vector, and symmetric-traceless sectors, clarifying its precise relation to director and alignment-tensor variables without identifying it with standard nematodynamics. For the one-dimensional jet group on a $(1+1)$-dimensional Minkowski base, we obtain a semilinear symmetric-hyperbolic system and an exact traveling conversion front with explicit energy-momentum exchange between the linear- and quadratic-jet sectors. Its hyperbolic-target interpretation yields a conserved positive energy for the linearized perturbations. A nonlinear numerical experiment illustrates the reduced evolution and reconstruction of the group-valued field, with consistency assessed through mesh refinement and an independent wave-map computation.

math-ph↗

arXiv2026

The Attention Within: Consensus Dynamics in Selective State Space Models

Selective state space models (SSMs) have recently emerged as a compelling alternative to transformers, combining competitive performance with substantially improved inference efficiency. At each SSM layer, a sequence of hidden states are propagated by a recurrence, mixing information of different tokens. Despite using a different mechanism, this mixing plays a role analogous to attention in transformers. In fact, recent works have shown that the two architectures may be closer than they first appear, as this recurrence admits a formulation akin to linear attention. In transformers, attention is known to drive the tokens to cluster, i.e., to reach consensus, collapsing in the limit to a single direction. Thus, we ask: does the recurrence at the core of SSMs drive the tokens to consensus, as attention does in transformers? To answer this question, we take a dynamical systems perspective on SSMs, modeling the evolution of tokens across layers as an ordinary differential equation. By exploiting input-to-state stability arguments, we establish local exponential stability of the consensus equilibria and characterize their domain of attraction for time-varying weight matrices, a setting not addressed by previous results. We thereby show that the resemblance between SSMs and transformers does run deeper: the recurrence at the core of SSMs aggregates tokens just as attention does. Numerical experiments on a pretrained Mamba-2 model point to the output gate as the component that regulates the extent of this consensus, preventing the tokens from reaching it in full.

cs.LG↗

arXiv2026

Stable Walking for Bipedal Locomotion under Foot-Slip via Virtual Nonholonomic Constraints

Foot slip is a major source of instability in bipedal locomotion on low-friction or uncertain terrain. Standard control approaches typically assume no-slip contact and therefore degrade when slip occurs. We propose a control framework that explicitly incorporates slip into the locomotion model through virtual nonholonomic constraints, which regulate the tangential stance-foot velocity while remaining compatible with the virtual holonomic constraints used to generate the walking gait. The resulting closed-loop system is formulated as a hybrid dynamical system with continuous swing dynamics and discrete impact events. A nonlinear feedback law enforces both classes of constraints and yields a slip-compatible hybrid zero dynamics manifold for the reduced-order locomotion dynamics. Stability of periodic walking gaits is characterized through the associated Poincaré map, and numerical results illustrate stabilization under slip conditions.

eess.SY↗

arXiv2025

Infinite-dimensional Lagrange-Dirac systems with boundary energy flow I: Foundations

A new geometric approach to systems with boundary energy flow is developed using infinite-dimensional Dirac structures within the Lagrangian formalism. This framework satisfies a list of consistency criteria with the geometric setting of finite-dimensional mechanics. In particular, the infinite-dimensional Dirac structure can be constructed from the canonical symplectic form on the system's phase space; the system's evolution equations can be derived equivalently from either a variational perspective or a Dirac structure perspective; the variational principle employed is a direct extension of Hamilton's principle in classical mechanics; and the approach allows for a process of system interconnection within its formulation. This is achieved by developing an appropriate infinite dimensional version of the previously developed Lagrange-Dirac dynamical systems. A key step in this construction is the careful choice of a suitable dual space to the configuration space, specifically, a subspace of the topological dual that captures the system's behavior in both the interior and the boundary, while allowing for a natural extension of the canonical geometric structures of mechanics. This paper focuses on systems where the configuration space consists of differential forms on a smooth manifold with a boundary. To illustrate our theory, several examples, including nonlinear wave equations, the telegraph equation, and the Maxwell equations are presented.

math.SG↗

arXiv2025

Infinite-dimensional Lagrange-Dirac systems with boundary energy flow II: Field theories with bundle-valued forms

Part I of this paper introduced the infinite dimensional Lagrange-Dirac theory for physical systems on the space of differential forms over a smooth manifold with boundary. This approach is particularly well-suited for systems involving energy exchange through the boundary, as it is built upon a restricted dual space -- a vector subspace of the topological dual of the configuration space -- that captures information about both the interior dynamics and boundary interactions. Consequently, the resulting dynamical equations naturally incorporate boundary energy flow. In this second part, the theory is extended to encompass vector-bundle-valued differential forms and non-Abelian gauge theories. To account for two commonly used forms of energy flux and boundary power densities, we introduce two distinct but equivalent formulations of the restricted dual. The results are derived from both geometric and variational viewpoints and are illustrated through applications to matter and gauge field theories. The interaction between gauge and matter fields is also addressed, along with the associated boundary conditions, applied to the case of the Yang-Mills-Higgs equations.

math.SG↗

arXiv2025

The Asymptotic Behavior of Attention in Transformers

The transformer architecture has become the foundation of modern Large Language Models (LLMs), yet its theoretical properties are still not well understood. As with classic neural networks, a common approach to improve these models is to increase their size and depth. However, such strategies may be suboptimal, as several works have shown that adding more layers yields increasingly diminishing returns. More importantly, prior studies have shown that increasing depth may lead to model collapse, i.e., all the tokens converge to a single cluster, undermining the ability of LLMs to generate diverse outputs. Building on differential equation models for the transformer dynamics, we prove that all the tokens in a transformer asymptotically converge to a cluster as depth increases. At the technical level we leverage tools from control theory, including consensus dynamics on manifolds and input-to-state stability (ISS). We then extend our analysis to autoregressive models, exploiting their structure to further generalize the theoretical guarantees.

cs.AI↗

arXiv2025

Variational Collision Integrators for Nonholonomic Lagrangian Systems

A discrete theory for implicit nonholonomic Lagrangian systems undergoing elastic collisions is developed. It is based on the discrete Lagrange-d'Alembert-Pontryagin variational principle and the dynamical equations thus obtained are the discrete nonholonomic implicit Euler-Lagrange equations together with the discrete conditions for the elastic impact. To illustrate the theory, variational integrators with collisions are built for several examples, including a bouncing ellipse and a nonholonomic spherical pendulum evolving inside a cylinder.

math.DS↗

arXiv2023

Higher-order Euler-Poincaré field equations

We develop a reduction theory for $G$-invariant Lagrangian field theories defined on the higher-order jet bundle of a principal $G$-bundle, thus obtaining the higher-order Euler-Poincaré field equations. To that end, we transfer the Hamilton's principle to the reduced configuration bundle, which is identified with the bundle of flat connections (up to a certain order) of the principal $G$-bundle. As a result, the reconstruction condition is always satisfied and, hence, every solution of the reduced field equations locally comes from a solution of the original (unreduced) equations. Furthermore, the reduced equations are shown to be equivalent to the conservation of the Noether current. Lastly, we illustrate the theory by investigating multivariate higher-order splines on Lie groups.

math.DG↗

arXiv2023

Nonholonomic reduction for mechanical systems with collisions

This paper studies nonsmooth variational problems on principal bundles for nonholonomic systems with collisions taking place in the boundary of the manifold configuration space of the nonholonopmic system. In particular, we first extended to a nonsmooth context appropriate for collisions the variational principle for nonholonomic implicit Lagrangian systems, to obtain implicit Lagrange--d'Alembert--Pontryagin equations for nonholonomic systems with collisions, and after introducing the notion of connection on a principal bundle we consider Lagrange--Poincaré--Pointryagin reduction by symmetries for systems with collisions.

math-ph↗

arXiv2023

Implicit Nonholonomic Mechanics with Collisions

In this paper, variational techniques are used to analyze the dynamics of nonholonomic mechanical systems with impacts. Implicit nonholonomic smooth Lagrangian and Hamiltonian systems are extended to a nonsmooth context appropriate for collisions. In particular, we provide a variational formulation for implicit nonholonomic mechanical systems with collisions, for those collisions that preserve energy and momentum at the impact. Lastly, the theoretical results are illustrated by examining the example of a rolling disk hitting a wall.

math-ph↗

arXiv2023

Gauge reduction in covariant field theory

In this work, we develop a Lagrangian reduction theory for covariant field theories with gauge symmetries. These symmetries are modeled by a Lie group fiber bundle acting fiberwisely on a configuration bundle. In order to reduce the variational principle, we utilize generalized principal connections, a type of Ehresmann connections that are equivariant by the fiberwise action. After obtaining the reduced equations, we give the reconstruction condition and we relate the vertical reduced equation with the Noether theorem. Lastly, we illustrate the theory with several examples, including the classical case (Lagrange-Poincaré reduction), Electromagnetism, symmetry-breaking and non-Abelian gauge theories.

math.DG↗

arXiv2022

Higher order jet bundles of Lie group-valued functions

For each positive integer $k$, the bundle of $k$-jets of functions from a smooth manifold, $X$, to a Lie group, $G$, is denoted by $J^k(X,G)$ and it is canonically endowed with a Lie groupoid structure over $X$. In this work, we utilize a linear connection to trivialize this bundle, i.e., to build an injective bundle morphism from $J^k(X,G)$ into a vector bundle over $G$. Afterwards, we give the explicit expression of the groupoid multiplication on the trivialized space, as well as the formula for the inverse element. In the last section, a coordinated chart on $X$ is considered and the local expression of the trivialization is computed.

math.DG↗

arXiv2022

Discrete Dirac reduction of implicit Lagrangian systems with abelian symmetry groups

This paper develops the theory of discrete Dirac reduction of discrete Lagrange-Dirac systems with an abelian symmetry group acting on the configuration space. We begin with the linear theory and, then, we extend it to the nonlinear setting using retraction compatible charts. We consider the reduction of both the discrete Dirac structure and the discrete Lagrange-Pontryagin principle, and show that they both lead to the same discrete Lagrange-Poincaré-Dirac equations. The coordinatization of the discrete reduced spaces relies on the notion of discrete connections on principal bundles. At last, we demonstrate the method obtained by applying it to a charged particle in a magnetic field, and to the double spherical pendulum.

math-ph↗

arXiv2022

Principal bundles and connections modelled by Lie group bundles

In this work, generalized principal bundles modelled by Lie group bundle actions are investigated. In particular, the definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article.

math.DG↗