Hodge-projected echo-state networks with topologically anchored memory for chaotic flows

Pradeep Singh, Ojjas Rajendra Madare and Balasubramanian Raman

Chaos, Solitons & Fractals, 2026, vol. 202, issue P1

Abstract: We introduce CHORD-ESN, an echo-state network that builds long memory from topology rather than from near-unstable tuning. The reservoir state lives on a simplicial complex as node potentials (0-forms), edge fluxes (1-forms), and face circulations (2-forms), and cross-degree interactions follow the laws of exterior calculus. A Hodge projection splits edge flows into exact, coexact, and harmonic components, and we assign a tiny leak only to the harmonic part. This yields a topology-anchored slow channel — with capacity set by the number of cycles — while standard components are damped by nonexpansive heat smoothing. We give a simple, verifiable echo-state (stability) condition via a small block-contraction bound, and the whole update uses sparse operators with intermittent lightweight solves. On chaotic and real-world benchmarks, CHORD-ESN improves long-horizon forecasting and attractor fidelity, and ablations that remove cycles or disable the Hodge split eliminate these gains. In short: cycles remember; CHORD-ESN makes that memory explicit, controllable, and provably stable.

Keywords: Reservoir computing; Echo state networks; Echo-state property; Chaotic time-series forecasting (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:202:y:2026:i:p1:s0960077925014730

DOI: 10.1016/j.chaos.2025.117460

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