arXiv subjects

Juan-Pablo Ortega

Publications and source records attributed to Juan-Pablo Ortega.

At least 19 recordsLinked to original sources

arXiv2026

A kernel method for the learning of Wasserstein geometric flows

Wasserstein gradient and Hamiltonian flows have emerged as essential tools for modeling complex dynamics in the natural sciences, providing a unifying geometric formulation of many partial differential equations (PDEs) and finding applications in fields ranging from optimal transport to quantum mechanics and information geometry. Despite their significance, the inverse identification of potential functions and interaction kernels underlying these flows remains relatively unexplored. In this work, we tackle this challenge by addressing the inverse problem of simultaneously recovering the potential function and interaction kernel from discretized observations of the density flow. We formulate the problem as an optimization task that minimizes a loss function specifically designed to enforce the underlying variational structure of Wasserstein flows, ensuring consistency with the geometric properties of the density manifold. Our framework employs a kernel-based operator approach using the associated {reproducing kernel Hilbert space (RKHS)}, which provides a closed-form representation of the unknown components. Furthermore, we conduct a comprehensive error analysis, providing convergence rates under adaptive regularization parameters as the temporal and spatial discretization mesh sizes tend to zero. Moreover, a stability analysis is presented to bridge the gap between discrete trajectory data and continuous-time flow dynamics for the Wasserstein Hamiltonian flow. {Finally, numerical experiments on both Wasserstein gradient and Hamiltonian flows demonstrate accurate and robust recovery from discrete density observations, while comparisons with sparse-learning approaches illustrate the strong dependence of sparse recovery on the choice of dictionary.}

math.NA↗

arXiv2026

Stability of input-output maps and their minimal realizations in state-linear, state-affine, LPV, and linear switched systems

Stability is often assumed in learning and identification, yet it is rarely characterized directly from input--output data. We show that an input--output family admits a stable finite-dimensional state-linear realization iff it has finite Hankel-rank and its response decays uniformly with time; for state-linear realizable maps this decay is necessarily exponential. We extend these results to state-affine, LPV, and linear switched systems via suitable input-forgetting notions, and relate forgetting to decay of impulse responses (sub-Markov parameters). In all cases, the decay/forgetting rate determines the decay rate of every minimal realization.

math.OC↗

arXiv2026

Data-Specific Hyper-Parameter Design: A Paradigm Shift in Reservoir Computing

Reservoir computing typically relies on large, randomly generated reservoirs, enabling simple, often linear readouts. Over the past two decades, most constructions have exploited the freedom to select the reservoir, constrained primarily by stability conditions based on state contraction or memory capacity. However, these designs are largely independent of the input data and learning objective, resulting in a trial-and-error methodology driven by randomness. In high dimensions, the reservoir acts as a random embedding of the input history, implicitly relying on Johnson--Lindenstrauss--type concentration phenomena to preserve information. In contrast, we develop reservoir design principles from a geometric perspective for inputs generated by deterministic dynamical systems. Rather than relying on random embeddings, we require reservoir state increments to align within a cone around an input-determined vector subspace, and prove that such a cone concentration reduces ridge-regression training error. When the cone angle is small, the variance of reservoir states concentrates in the input-determined subspace, improving conditioning of the empirical second-moment matrix and strengthening alignment between dominant covariance directions and the state-target cross-covariance. For echo state networks, we provide a constructive approach to reservoir design. The reservoir matrix is chosen so that associated Krylov-chain directions remain nearly closed within an input-determined subspace while permitting controlled mixing in its orthogonal complement. We also provide a spectral diagnostic for ridge regression training that identifies when reservoir geometry concentrates predictive information into a few dominant covariance modes and when ``spectral pollution'' inhibits forecasting. Numerical experiments demonstrate consistent performance gains over arbitrary reservoir constructions.

math.DS↗

arXiv2026

Stochastic dynamics learning with state-space systems

This work advances the theoretical foundations of reservoir computing (RC) by providing a unified treatment of fading memory and the echo state property (ESP) in both deterministic and stochastic settings. We investigate state-space systems, a central model class in time series learning, and establish that fading memory and solution stability hold generically -- even in the absence of the ESP -- offering a robust explanation for the empirical success of RC models without strict contractivity conditions. In the stochastic case, we critically assess stochastic echo states, proposing a novel distributional perspective rooted in attractor dynamics on the space of probability distributions, which leads to a rich and coherent theory. Our results extend and generalize previous work on non-autonomous dynamical systems, offering new insights into causality, stability, and memory in RC models. This lays the groundwork for reliable generative modeling of temporal data in both deterministic and stochastic regimes.

stat.ML↗

arXiv2026

Learning the climate of dynamical systems with state-space systems

State-space systems encompass a broad class of algorithms used for modeling and forecasting time series. For such systems to be effective, two objectives must be met: (i) accurate point forecasts of the time series must be produced, and (ii) the long-term statistical behaviour of the underlying data-generating process must be replicated. The latter objective, often referred to as learning the climate, is closely related to the task of producing accurate distribution forecasts. Empirical evidence shows that distribution forecasts are far more stable than point forecasts, which are sensitive to initial conditions. In this work, we rigorously study this phenomenon for state-space systems. The main result shows that, if the underlying data-generating process is structurally stable and possesses a mixing or an attracting measure, then a sufficiently regular initial probability distribution remains close to the true future distribution at arbitrarily long time horizons when forecasted by a $C^1-$close state-space proxy. Thus, under these conditions, learning the climate of a dynamic process with a universal family of state-space systems is feasible with arbitrarily high accuracy.

nlin.CD↗

arXiv2026

Kernel Learning of PDE Solution Operators

A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a kernel ridge regression framework, and employs a regularization-based formulation to construct an operator learner. This yields a closed-form estimator that is independent of the input functions that determine the underlying PDE. From the perspective of regularization theory, the resulting estimator induces a well-defined operator that links input and output spaces, which contain the functions that define a Dirichlet problem and its solution, respectively. Consequently, it effectively shifts from a PDE solver to an operator-based solver. In contrast to standard supervised learning methods, it does not rely on paired input--output training data and enables systematic extrapolation beyond observed regimes. A full error analysis is conducted, providing convergence rates for the operator-based solver under suitable choices of regularization parameters. Extensive numerical experiments, including Darcy flow and Helmholtz equations, demonstrate that the proposed method achieves high accuracy and efficiency across a range of problem settings, and compares favorably with operator learning approaches in both approximation quality and computational cost.

math.NA↗

arXiv2026

Echoes of the Past: A Unified Perspective on Fading memory and Echo States

Recurrent neural networks (RNNs) have become increasingly popular in information processing tasks involving time series and temporal data. A fundamental property of RNNs is their ability to create reliable input/output responses, often linked to how the network handles its memory of the information it processed. Various notions have been proposed to conceptualize the behavior of memory in RNNs, including steady states, echo states, state forgetting, input forgetting, and fading memory. Although these notions are often used interchangeably, their precise relationships remain unclear. This work aims to unify these notions in a common language, derive new implications and equivalences between them, and provide alternative proofs to some existing results. By clarifying the relationships between these concepts, this research contributes to a deeper understanding of RNNs and their temporal information processing capabilities.

stat.ML↗

arXiv2026

Feedback-driven recurrent quantum neural network universality

Quantum reservoir computing uses the dynamics of quantum systems to process temporal data, making it particularly well-suited for machine learning with noisy intermediate-scale quantum devices. Recent developments have introduced feedback-based quantum reservoir systems, which process temporal information with comparatively fewer components and enable real-time computation while preserving the input history. Motivated by their promising empirical performance, in this work, we study the approximation capabilities of feedback-based quantum reservoir computing. More specifically, we are concerned with recurrent quantum neural networks, which are quantum analogues of classical recurrent neural networks. Our results show that regular state-space systems can be approximated using quantum recurrent neural networks without the curse of dimensionality and with the number of qubits only growing logarithmically in the reciprocal of the prescribed approximation accuracy. Notably, our analysis demonstrates that quantum recurrent neural networks are universal with linear readouts, making them both powerful and experimentally accessible. These results pave the way for practical and theoretically grounded quantum reservoir computing with real-time processing capabilities.

quant-ph↗

arXiv2025

Reservoir kernels and Volterra series

A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir functional associated with a state-space representation of the Volterra series expansion available for any analytic fading memory filter, and it is hence called the Volterra reservoir kernel. Even though the state-space representation and the corresponding reservoir feature map are defined on an infinite-dimensional tensor algebra space, the kernel map is characterized by explicit recursions that are readily computable for specific data sets when employed in estimation problems using the representer theorem. The empirical performance of the Volterra reservoir kernel is showcased and compared to other standard static and sequential kernels in a multidimensional and highly nonlinear learning task for the conditional covariances of financial asset returns.

cs.LG↗

arXiv2025

Memory Capacity of Nonlinear Recurrent Networks: Is it Informative?

The total memory capacity (MC) of linear recurrent neural networks (RNNs) has been proven to be equal to the rank of the corresponding Kalman controllability matrix, and it is almost surely maximal for connectivity and input weight matrices drawn from regular distributions. This fact questions the usefulness of this metric in distinguishing the performance of linear RNNs in the processing of stochastic signals. This work shows that the MC of random nonlinear RNNs yields arbitrary values within established upper and lower bounds depending exclusively on the scale of the input process. This confirms that the existing definition of MC in linear and nonlinear cases has no practical value.

cs.LG↗

arXiv2025

Fading memory and the convolution theorem

Several topological and analytical notions of continuity and fading memory for causal and time-invariant filters are introduced, and the relations between them are analyzed. A significant generalization of the convolution theorem that establishes the equivalence between the fading memory property and the availability of convolution representations of linear filters is proved. This result extends a previous similar characterization to a complete array of weighted norms in the definition of the fading memory property. Additionally, the main theorem shows that the availability of convolution representations can be characterized, at least when the codomain is finite-dimensional, not only by the fading memory property but also by the reunion of two purely topological notions that are called minimal continuity and minimal fading memory property. Finally, when the input space and the codomain of a linear functional are Hilbert spaces, it is shown that minimal continuity and the minimal fading memory property guarantee the existence of interesting embeddings of the associated reproducing kernel Hilbert spaces.

math.OC↗

arXiv2025

Input-dependence in quantum reservoir computing

Quantum reservoir computing is an emergent field in which quantum dynamical systems are exploited for temporal information processing. In previous work, it was found a feature that makes a quantum reservoir valuable: contractive dynamics of the quantum reservoir channel toward input-dependent fixed points. These results are enhanced in this paper by finding conditions that guarantee a crucial aspect of the reservoir's design: distinguishing between different input sequences to ensure a faithful representation of temporal input data. This is implemented by finding a condition that guarantees injectivity in reservoir computing filters, with a special emphasis on the quantum case. We provide several examples and focus on a family of quantum reservoirs that is much used in the literature; it consists of an input-encoding quantum channel followed by a strictly contractive channel that enforces the echo state and the fading memory properties. This work contributes to analyzing valuable quantum reservoirs in terms of their input dependence.

quant-ph↗

arXiv2025

Quantum reservoir computing in finite dimensions

Most existing results in the analysis of quantum reservoir computing (QRC) systems with classical inputs have been obtained using the density matrix formalism. This paper shows that alternative representations can provide better insights when dealing with design and assessment questions. More explicitly, system isomorphisms are established that unify the density matrix approach to QRC with the representation in the space of observables using Bloch vectors associated with Gell-Mann bases. It is shown that these vector representations yield state-affine systems (SAS) previously introduced in the classical reservoir computing literature and for which numerous theoretical results have been established. This connection is used to show that various statements in relation to the fading memory (FMP) and the echo state (ESP) properties are independent of the representation, and also to shed some light on fundamental questions in QRC theory in finite dimensions. In particular, a necessary and sufficient condition for the ESP and FMP to hold is formulated using standard hypotheses, and contractive quantum channels that have exclusively trivial semi-infinite solutions are characterized in terms of the existence of input-independent fixed points.

quant-ph↗

arXiv2025

A global structure-preserving kernel method for the learning of Poisson systems

A structure-preserving kernel ridge regression method is presented that allows the recovery of globally defined, potentially high-dimensional, and nonlinear Hamiltonian functions on Poisson manifolds out of datasets made of noisy observations of Hamiltonian vector fields. The proposed method is based on finding the solution of a non-standard kernel ridge regression where the observed data is generated as the noisy image by a vector bundle map of the differential of the function that one is trying to estimate. Additionally, it is shown how a suitable regularization solves the intrinsic non-identifiability of the learning problem due to the degeneracy of the Poisson tensor and the presence of Casimir functions. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator is illustrated with several numerical experiments.

math.NA↗

arXiv2025

A Structure-Preserving Kernel Method for Learning Hamiltonian Systems

A structure-preserving kernel ridge regression method is presented that allows the recovery of nonlinear Hamiltonian functions out of datasets made of noisy observations of Hamiltonian vector fields. The method proposes a closed-form solution that yields excellent numerical performances that surpass other techniques proposed in the literature in this setup. From the methodological point of view, the paper extends kernel regression methods to problems in which loss functions involving linear functions of gradients are required and, in particular, a differential reproducing property and a Representer Theorem are proved in this context. The relation between the structure-preserving kernel estimator and the Gaussian posterior mean estimator is analyzed. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator together with the convergence rate is illustrated with various numerical experiments.

stat.ML↗

arXiv2025

Forecasting causal dynamics with universal reservoirs

An iterated multistep forecasting scheme based on recurrent neural networks (RNN) is proposed for the time series generated by causal chains with infinite memory. This forecasting strategy contains, as a particular case, the iterative prediction strategies for dynamical systems that are customary in reservoir computing. Explicit error bounds are obtained as a function of the forecasting horizon, functional and dynamical features of the specific RNN used, and the approximation error committed by it. In particular, the growth rate of the error is shown to be exponential and controlled by the top Lyapunov exponent of the proxy system. The framework in the paper circumvents difficult-to-verify embedding hypotheses that appear in previous references in the literature and applies to new situations like the finite-dimensional observations of functional differential equations or the deterministic parts of stochastic processes to which standard embedding techniques do not necessarily apply.

math.DS↗

arXiv2025

State-space systems as dynamic generative models

A probabilistic framework to study the dependence structure induced by deterministic discrete-time state-space systems between input and output processes is introduced. General sufficient conditions are formulated under which output processes exist and are unique once an input process has been fixed, a property that in the deterministic state-space literature is known as the echo state property. When those conditions are satisfied, the given state-space system becomes a generative model for probabilistic dependences between two sequence spaces. Moreover, those conditions guarantee that the output depends continuously on the input when using the Wasserstein metric. The output processes whose existence is proved are shown to be causal in a specific sense and to generalize those studied in purely deterministic situations. The results in this paper constitute a significant stochastic generalization of sufficient conditions for the deterministic echo state property to hold, in the sense that the stochastic echo state property can be satisfied under contractivity conditions that are strictly weaker than those in deterministic situations. This means that state-space systems can induce a purely probabilistic dependence structure between input and output sequence spaces even when there is no functional relation between those two spaces.

stat.ML↗

arXiv2025

Infinite-dimensional next-generation reservoir computing

Next-generation reservoir computing (NG-RC) has attracted much attention due to its excellent performance in spatio-temporal forecasting of complex systems and its ease of implementation. This paper shows that NG-RC can be encoded as a kernel ridge regression that makes training efficient and feasible even when the space of chosen polynomial features is very large. Additionally, an extension to an infinite number of covariates is possible, which makes the methodology agnostic with respect to the lags into the past that are considered as explanatory factors, as well as with respect to the number of polynomial covariates, an important hyperparameter in traditional NG-RC. We show that this approach has solid theoretical backing and good behavior based on kernel universality properties previously established in the literature. Various numerical illustrations show that these generalizations of NG-RC outperform the traditional approach in several forecasting applications.

cs.LG↗