Fig. 4: Median predictive performance against the dimensionality of D.
a Lorenz63, b double-scroll, and c Mackey–Glass systems. The configurations of the product and distributed representations depicted in Fig. 3 are marked by red squares and green circles, respectively. The median performance of the corresponding product representations from Fig. 3 is shown as thick dashed red lines where the red squares additionally emphasize the exact location of the configurations with respect to model’s size that is driven by higher-order features used to form the feature space. Similarly, the performance of the multilayer perceptron is shown as a thick solid blue line with blue squares emphasizing that the size of the first hidden layer matches that of the product representation. The echo-state network (“Experiments with traditional reservoir computing networks” section) is depicted as a dotted black line. The following hyperparameters are used: β = 0.25, γ = 0.1, α = 1 × 10−7 (Lorenz63); β = 0.1, γ = 0.1, α = 1 × 10−9 (double-scroll); β = 0.25, γ = 1.00, α = 1 × 10−7 (Mackey–Glass). For each configuration, NRMSE is computed over three Lyapunov times where the reported values are obtained from 1000 randomly initialized simulations. Shaded areas show the median standard error.
