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.

RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks

Parallel-in-time (PinT) techniques have been proposed to solve systems of time-dependent differential equations by parallelizing the temporal domain. Among them, Parareal computes the solution sequentially using an inaccurate (fast) solver, and then "corrects" it using an accurate (slow) integrator that runs in parallel across temporal subintervals. This work introduces RandNet-Parareal, a novel method to learn the discrepancy between the coarse and fine solutions using random neural networks (RandNets). RandNet-Parareal achieves speed gains up to x125 and x22 compared to the fine solver run serially and Parareal, respectively. Beyond theoretical guarantees of RandNets as universal approximators, these models are quick to train, allowing the PinT solution of partial differential equations on a spatial mesh of up to $10^5$ points with minimal overhead, dramatically increasing the scalability of existing PinT approaches. RandNet-Parareal's numerical performance is illustrated on systems of real-world significance, such as the viscous Burgers' equation, the Diffusion-Reaction equation, the two- and three-dimensional Brusselator, and the shallow water equation.

Memory of recurrent networks: Do we compute it right?

Numerical evaluations of the memory capacity (MC) of recurrent neural networks reported in the literature often contradict well-established theoretical bounds. In this paper, we study the case of linear echo state networks, for which the total memory capacity has been proven to be equal to the rank of the corresponding Kalman controllability matrix. We shed light on various reasons for the inaccurate numerical estimations of the memory, and we show that these issues, often overlooked in the recent literature, are of an exclusively numerical nature. More explicitly, we prove that when the Krylov structure of the linear MC is ignored, a gap between the theoretical MC and its empirical counterpart is introduced. As a solution, we develop robust numerical approaches by exploiting a result of MC neutrality with respect to the input mask matrix. Simulations show that the memory curves that are recovered using the proposed methods fully agree with the theory.

Infinite-dimensional reservoir computing

Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state-space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a fully implementable recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality.

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. 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. We showcase the performance of the Volterra reservoir kernel in a popular data science application in relation to bitcoin price prediction.

Learning strange attractors with reservoir systems

This paper shows that the celebrated Embedding Theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.

Dimension reduction in recurrent networks by canonicalization

Many recurrent neural network machine learning paradigms can be formulated using state-space representations. The classical notion of canonical state-space realization is adapted in this paper to accommodate semi-infinite inputs so that it can be used as a dimension reduction tool in the recurrent networks setup. The so called input forgetting property is identified as the key hypothesis that guarantees the existence and uniqueness (up to system isomorphisms) of canonical realizations for causal and time-invariant input/output systems with semi-infinite inputs. A second result uses the notion of optimal reduction borrowed from the theory of symmetric Hamiltonian systems to construct canonical realizations out of input forgetting but not necessarily canonical ones. These two procedures are implemented and studied in detail in the framework of linear fading memory input/output systems.

Memory and forecasting capacities of nonlinear recurrent networks

The notion of memory capacity, originally introduced for echo state and linear networks with independent inputs, is generalized to nonlinear recurrent networks with stationary but dependent inputs. The presence of dependence in the inputs makes natural the introduction of the network forecasting capacity, that measures the possibility of forecasting time series values using network states. Generic bounds for memory and forecasting capacities are formulated in terms of the number of neurons of the network and the autocovariance function of the input. These bounds generalize well-known estimates in the literature to a dependent inputs setup. Finally, for linear recurrent networks and independent inputs it is proved that the memory capacity is given by the rank of the associated controllability matrix.

Approximation Bounds for Random Neural Networks and Reservoir Systems

This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized, have been successfully applied in a wide range of static and dynamic learning problems. Despite the popularity of this approach in empirical tasks, important theoretical questions regarding the relation between the unknown function, the weight distribution, and the approximation rate have remained open. In this work it is proved that, as long as the unknown function, functional, or dynamical system is sufficiently regular, it is possible to draw the internal weights of the random (recurrent) neural network from a generic distribution (not depending on the unknown object) and quantify the error in terms of the number of neurons and the hyperparameters. In particular, this proves that echo state networks with randomly generated weights are capable of approximating a wide class of dynamical systems arbitrarily well and thus provides the first mathematical explanation for their empirically observed success at learning dynamical systems.

Risk bounds for reservoir computing

We analyze the practices of reservoir computing in the framework of statistical learning theory. In particular, we derive finite sample upper bounds for the generalization error committed by specific families of reservoir computing systems when processing discrete-time inputs under various hypotheses on their dependence structure. Non-asymptotic bounds are explicitly written down in terms of the multivariate Rademacher complexities of the reservoir systems and the weak dependence structure of the signals that are being handled. This allows, in particular, to determine the minimal number of observations needed in order to guarantee a prescribed estimation accuracy with high probability for a given reservoir family. At the same time, the asymptotic behavior of the devised bounds guarantees the consistency of the empirical risk minimization procedure for various hypothesis classes of reservoir functionals.