arXiv：cs.LG（机器学习，全量分类）· Hiroto Tamura, Gouhei Tanaka·· 14 小时前AI 评分33
AL-RNN 如何用不动点引导的层级约简实现最小动力学表示
From Redundancy to Minimality: Fixed-Point-Guided Hierarchical Reduction of Learned Piecewise-Linear Dynamics
AI 导读
研究提出一种不动点引导的层级约简方法，通过逐步线性化 AL-RNN 中选定的 ReLU 单元、合并相邻线性区域与图节点，将冗余非线性容量约简为最小动力学表示。作者证明重现 Q 个不同不动点至少需要 Q 个含不动点符号，并在 3-scroll Chua 系统上验证：直接训练三个 ReLU 单元仅 20% 种子成功，而先学习再约简再重训的策略将成功率提升至约 71%。
正文
Abstract:Understanding a nonlinear dynamical system from time series requires not only reproducing its trajectories, but also identifying a simple representation that preserves its essential dynamical structure. Almost-linear recurrent neural networks (AL-RNNs) are piecewise-linear RNNs in which only a subset of units use ReLU nonlinearities, so that nonlinear capacity is explicitly controlled by the number of ReLU units. Their activation patterns define linear regions, represented as symbols, whose observed transitions form a symbolic transition graph. However, directly training AL-RNNs with few ReLU units to realize minimal dynamical representations can be unreliable. We ask whether an AL-RNN with more ReLU units can instead be trained first and systematically reduced to a minimal dynamical representation. We introduce a fixed-point-guided hierarchical reduction procedure that progressively linearizes selected ReLU units, merging neighboring linear regions and graph nodes while preserving distinct symbols containing fixed points (FPs). The resulting reduction tree defines a hierarchy of progressively simpler candidates. Each reduced candidate is initialized from the parent parameters and retrained under guidance from the parent dynamics. We also prove that reproducing $Q$ distinct fixed points requires at least $Q$ FP-containing symbols, providing a certificate of symbol-level minimality when this bound is attained. On the 3-scroll Chua system, direct training with the theoretical minimum of three ReLU units achieves high-fidelity minimal realizations in only 20% of seeds, whereas our learn-reduce-retrain strategy increases the seed-macro success rate to approximately 71% at the same final nonlinear capacity. These results show that redundant nonlinear capacity can serve as a scaffold for discovering and realizing minimal dynamical representations.
| Comments: | 27 pages, 6 figures |
| Subjects: | Machine Learning (cs.LG); Chaotic Dynamics (nlin.CD) |
| Cite as: | arXiv:2610.01369 [cs.LG] |
| (or arXiv:2610.01369v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.01369 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Hiroto Tamura [view email]
[v1]
Thu, 1 Oct 2026 09:39:25 UTC (4,166 KB)
来源：arXiv：cs.LG（机器学习，全量分类） · arxiv.org