Reservoir Computing Overview

Reservoir Computing (RC) is a framework for training Recurrent Neural Networks (RNNs) where the recurrent part (the "reservoir") is fixed, and only the output weights are trained. This allows for extremely fast training compared to traditional Backpropagation Through Time (BPTT).

Echo State Networks (ESN)

The standard Echo State Network (ESN) update equation is:

\[ \mathbf{x}(t+1) = (1-\alpha)\mathbf{x}(t) + \alpha \tanh(\mathbf{W}_{in}\mathbf{u}(t+1) + \mathbf{W}_{res}\mathbf{x}(t)) \]

Where: * \(\mathbf{x}(t)\) is the reservoir state vector. * \(\mathbf{u}(t)\) is the input vector. * \(\mathbf{W}_{in}\) is the input weight matrix. * \(\mathbf{W}_{res}\) is the reservoir weight matrix. * \(\alpha\) is the leaking rate (\(\alpha \in (0, 1]\)).

The initial transient is usually discarded with a washout period before the readout is fit. In rclib, model.fit(..., washout_len=k) advances the reservoir over the full input sequence and trains only on rows k: of the collected state matrix and target matrix.

RandomSparse optionally adds a fixed random bias vector inside the reservoir update. Readouts also have their own include_bias option, which appends a constant feature to the collected reservoir state before solving or updating output weights.

Next-Generation Reservoir Computing (NVAR)

NVAR replaces the random recurrent reservoir with a deterministic feature map of time-delayed inputs. Let:

\[ \mathbf{z}(t) = [\mathbf{u}(t), \mathbf{u}(t-1), \ldots, \mathbf{u}(t-k+1)] \]

where \(k\) is num_lags. For polynomial_order=1, the state is exactly this linear delay embedding. For higher orders, rclib appends all monomials with replacement for degrees 1..polynomial_order, in deterministic lexicographic index order. For example, if \(\mathbf{z}=[z_0,z_1]\) and polynomial_order=2, the feature vector is:

\[ [z_0, z_1, z_0^2, z_0z_1, z_1^2] \]

Constant/bias features are not part of the NVAR reservoir state; use the readout's include_bias option when an intercept term is needed.