Topologically informed echo state networks via poincaré return maps for chaotic time-series.

Abstract

We present the Poincaré-Section Reservoir (PSR), a deterministic reservoir computer whose recurrent graph is learned directly from data. A single trajectory of the target system is sliced by a transverse hyperplane; the resulting sequence of crossing points is coarse-grained into symbols, and empirical transition frequencies yield a row-stochastic matrix T. After spectral rescaling, T becomes the reservoir adjacency W, so each neuron corresponds to a concrete region of the attractor and each edge weight reproduces an observed return probability. We prove that, as the partition is refined, W converges in operator norm to a scaled Perron-Frobenius operator of the true Poincaré map, providing a formal consistency guarantee absent from classical Echo-State Networks. Coupled with a once-trained quadratic read-out, a 300-node PSR extends valid prediction time by up to 1.9 ×  over the best existing reservoirs on the Lorenz, Rössler, Chen-Ueta and Chua chaotic benchmarks-without gradients, equation knowledge or hyper-parameter tuning. By fusing Ulam's operator discretisation with reservoir computing, PSR offers a lean, interpretable and fully data-driven paradigm for long-horizon forecasting of strongly chaotic dynamics.