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Many neural networks use the tanh activation function, however when given aprobability distribution as input, the problem of computing the outputdistribution in neural networks with tanh activation has not yet beenaddressed. One important example is the initialization of the echo statenetwork in reservoir computing, where random initialization of the reservoirrequires time to wash out the initial conditions, thereby wasting precious dataand computational resources. Motivated by this problem, we propose a novelsolution utilizing a moment based approach to propagate uncertainty through anEcho State Network to reduce the washout time. In this work, we contribute twonew methods to propagate uncertainty through the tanh activation function andpropose the Probabilistic Echo State Network (PESN), a method that is shown tohave better average performance than deterministic Echo State Networks giventhe random initialization of reservoir states. Additionally we test single andmulti-step uncertainty propagation of our method on two regression tasks andshow that we are able to recover similar means and variances as computed byMonte-Carlo simulations.

Propagating Uncertainty through the tanh Function with Application to Reservoir Computing