Contemporary reservoir computing relies heavily on smooth, globally Lipschitz continuous activation functions, limiting applications in defense, disaster response, and pharmaceutical modeling where robust operation under extreme conditions is critical. We systematically investigate non-smooth activation functions, including chaotic, stochastic, and fractal variants, in echo state networks. Through comprehensive parameter sweeps across 36,610 reservoir configurations, we demonstrate that several non-smooth functions not only maintain the Echo State Property (ESP) but outperform traditional smooth activations in convergence speed and spectral radius tolerance. Notably, the Cantor function (continuous everywhere and flat almost everywhere) maintains ESP-consistent behavior up to spectral radii of rho ~ 10, an order of magnitude beyond typical bounds for smooth functions, while achieving 2.6x faster convergence than tanh and ReLU. We introduce a theoretical framework for quantized activation functions, defining a Degenerate Echo State Property (d-ESP) that captures stability for discrete-output functions and proving that d-ESP implies traditional ESP. We identify a critical crowding ratio Q=N/k (reservoir size / quantization levels) that predicts failure thresholds for discrete activations. Our analysis reveals that preprocessing topology, rather than continuity per se, determines stability: monotone, compressive preprocessing maintains ESP across scales, while dispersive or discontinuous preprocessing triggers sharp failures. While our findings challenge assumptions about activation function design in reservoir computing, the mechanism underlying the exceptional performance of certain fractal functions remains unexplained, suggesting fundamental gaps in our understanding of how geometric properties of activation functions influence reservoir dynamics.

翻译：当代储层计算严重依赖平滑、全局Lipschitz连续的激活函数，这限制了其在国防、灾害响应和药物建模等极端条件下需稳健运行的关键应用。我们系统研究了回声状态网络中非平滑激活函数，包括混沌、随机和分形变体。通过对36,610种储层配置的全面参数扫描，我们证明多种非平滑函数不仅保持回声状态特性（ESP），且在收敛速度和谱半径容限上优于传统平滑激活函数。值得注意的是，康托尔函数（处处连续且几乎处处平坦）在谱半径高达ρ~10时仍保持ESP一致行为，较平滑函数的典型边界高出一个数量级，同时收敛速度比tanh和ReLU快2.6倍。我们提出了量化激活函数的理论框架，定义了捕获离散输出函数稳定性的退化回声状态特性（d-ESP），并证明d-ESP蕴含传统ESP。我们发现临界拥挤比Q=N/k（储层规模/量化级数）可预测离散激活函数的失效阈值。分析表明，预处理拓扑而非连续性本身决定稳定性：单调压缩型预处理能在不同尺度保持ESP，而弥散或不连续预处理会引发急剧失效。尽管本研究挑战了储层计算中激活函数设计的既有假设，但某些分形函数优异性能的机制仍未得到解释，这提示我们对激活函数几何特性如何影响储层动力学的理解存在根本性空白。