  We demonstrate that matching the symmetry properties of a reservoir computer
(RC) to the data being processed dramatically increases its processing power.
We apply our method to the parity task, a challenging benchmark problem that
highlights inversion and permutation symmetries, and to a chaotic system
inference task that presents an inversion symmetry rule. For the parity task,
our symmetry-aware RC obtains zero error using an exponentially reduced neural
network and training data, greatly speeding up the time to result and
outperforming hand crafted artificial neural networks. When both symmetries are
respected, we find that the network size $N$ necessary to obtain zero error for
50 different RC instances scales linearly with the parity-order $n$. Moreover,
some symmetry-aware RC instances perform a zero error classification with only
$N=1$ for $n\leq7$. Furthermore, we show that a symmetry-aware RC only needs a
training data set with size on the order of $(n+n/2)$ to obtain such
performance, an exponential reduction in comparison to a regular RC which
requires a training data set with size on the order of $n2^n$ to contain all
$2^n$ possible $n-$bit-long sequences. For the inference task, we show that a
symmetry-aware RC presents a normalized root-mean-square error three
orders-of-magnitude smaller than regular RCs. For both tasks, our RC approach
respects the symmetries by adjusting only the input and the output layers, and
not by problem-based modifications to the neural network. We anticipate that
generalizations of our procedure can be applied in information processing for
problems with known symmetries.
