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Reservoir computing has repeatedly been shown to be extremely successful inthe prediction of nonlinear time-series. However, there is no completeunderstanding of the proper design of a reservoir yet. We find that thesimplest popular setup has a harmful symmetry, which leads to the prediction ofwhat we call mirror-attractor. We prove this analytically. Similar problems canarise in a general context, and we use them to explain the success or failureof some designs. The symmetry is a direct consequence of the hyperbolic tangentactivation function. Further, four ways to break the symmetry are comparednumerically: A bias in the output, a shift in the input, a quadratic term inthe readout, and a mixture of even and odd activation functions. Firstly, wetest their susceptibility to the mirror-attractor. Secondly, we evaluate theirperformance on the task of predicting Lorenz data with the mean shifted tozero. The short-time prediction is measured with the forecast horizon while thelargest Lyapunov exponent and the correlation dimension are used to representthe climate. Finally, the same analysis is repeated on a combined dataset ofthe Lorenz attractor and the Halvorsen attractor, which we designed to revealpotential problems with symmetry. We find that all methods except the outputbias are able to fully break the symmetry with input shift and quadraticreadout performing the best overall.

Breaking Symmetries of the Reservoir Equations in Echo State Networks