We introduce Prob-GParareal, a probabilistic extension of the GParareal algorithm designed to provide uncertainty quantification for the Parallel-in-Time (PinT) solution of (ordinary and partial) differential equations (ODEs, PDEs). The method employs Gaussian processes (GPs) to model the Parareal correction function, as GParareal does, further enabling the propagation of numerical uncertainty across time and yielding probabilistic forecasts of system's evolution. Furthermore, Prob-GParareal accommodates probabilistic initial conditions and maintains compatibility with classical numerical solvers, ensuring its straightforward integration into existing Parareal frameworks. Here, we first conduct a theoretical analysis of the computational complexity and derive error bounds of Prob-GParareal. Then, we numerically demonstrate the accuracy and robustness of the proposed algorithm on five benchmark ODE systems, including chaotic, stiff, and bifurcation problems. To showcase the flexibility and potential scalability of the proposed algorithm, we also consider Prob-nnGParareal, a variant obtained by replacing the GPs in Parareal with the nearest-neighbors GPs, illustrating its increased performance on an additional PDE example. This work bridges a critical gap in the development of probabilistic counterparts to established PinT methods.

Making accurate predictions of chaotic time series is a complex challenge. Reservoir computing, a neuromorphic-inspired approach, has emerged as a powerful tool for this task. It exploits the memory and nonlinearity of dynamical systems without requiring extensive parameter tuning. However, selecting and optimizing reservoir architectures remains an open problem. Next-generation reservoir computing simplifies this problem by employing nonlinear vector autoregression based on truncated Volterra series, thereby reducing hyperparameter complexity. Nevertheless, the latter suffers from exponential parameter growth in terms of the maximum monomial degree. Tensor networks offer a promising solution to this issue by decomposing multidimensional arrays into low-dimensional structures, thus mitigating the curse of dimensionality. This paper explores the application of a previously proposed tensor network model for predicting chaotic time series, demonstrating its advantages in terms of accuracy and computational efficiency compared to conventional echo state networks. Using a state-of-the-art tensor network approach enables us to bridge the gap between the tensor network and reservoir computing communities, fostering advances in both fields.

Memory capacity of nonlinear recurrent networks: Is it informative? Abstract 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 note shows that the MC of random nonlinear RNNs yields arbitrary values within established upper and lower bounds depending just on the input process scale. This confirms that the existing definition of MC in linear and nonlinear cases has no practical value.

Reservoir computing (RC) [Jaeg 10, Maas 02, Jaeg 04, Maas 11] and in particular echo state networks (ESNs) [Matt 92, Matt 93, Jaeg 04] have gained much popularity in recent years due to their excellent performance in the forecasting of dynamical systems [Grig 14, Jaeg 04, Path 17, Path 18, Lu 18, Wikn 21, Arco 22] and due to the ease of their implementation. RC aims at approximating nonlinear input/output systems using randomly generated state-space systems (called reservoirs) in which only a linear readout is estimated. It has been theoretically established that this is indeed possible in a variety of deterministic and stochastic contexts [Grig 18b, Grig 18a, Gono 20c, Gono 21b, Gono 23] in which RC systems have been shown to have universal approximation properties. In this paper, we focus on deriving error bounds for a variant of the architectures that we just cited and consider as approximants randomly generated linear systems with readouts given by randomly generated neural networks in which only the output layer is trained. Thus, from a learning perspective, we combine linear echo state networks and what is referred to in the literature as random features [Rahi 07] /extreme learning machines (ELMs) [Huan 06]. We develop explicit and readily computable approximation and estimation bounds for a newly introduced concept class whose elements we refer to as recurrent (generalized) Barron functionals since they can be viewed as a dynamical analog of the (generalized) Barron functions introduced in [Barr 92, Barr 93] and extended later in [E 20b, E 20a, E 19].

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.

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.