We consider the question whether the time evolution of controlled differential equations on general state spaces can be arbitrarily well approximated by (regularized) regressions on features generated themselves through randomly chosen dynamical systems of moderately high dimension.
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.
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.
This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights.
That research is complemented in this paper with the characterization of the differentiability of reservoir filters for very general classes of discrete-time deterministic inputs.
The universal approximation properties with respect to $L ^p $-type criteria of three important families of reservoir computers with stochastic discrete-time semi-infinite inputs is shown.
This paper shows that echo state networks are universal uniform approximants in the context of discrete-time fading memory filters with uniformly bounded inputs defined on negative infinite times.
A new class of non-homogeneous state-affine systems is introduced for use in reservoir computing.
This paper characterizes the conditional distribution properties of the finite sample ridge regression estimator and uses that result to evaluate total regression and generalization errors that incorporate the inaccuracies committed at the time of parameter estimation.
This paper addresses the reservoir design problem in the context of delay-based reservoir computers for multidimensional input signals, parallel architectures, and real-time multitasking.
This paper extends the notion of information processing capacity for non-independent input signals in the context of reservoir computing (RC).