发表机构
Rutgers University; University of Toledo; University of Oklahoma（罗格斯大学; 托莱多大学; 俄克拉荷马大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在降维场景下探究组合拓扑技术，建立了将动力学代数结构从潜空间提升至原系统的边界，结合自编码器模型与Conley-Morse图计算验证了相关思路，恢复了预期拓扑不变量。
AI 中文摘要
用于表征动力学的组合拓扑方法具有严谨性、可泛化性和可计算性，且仅需近似值，但相空间的维度是其更广泛应用的计算瓶颈。受越来越多用于获取动力学低维潜表示的机器学习技术启发，我们在降维场景下开展组合拓扑技术的初步研究。我们建立了可将组织动力学的代数结构从潜空间提升至原系统的边界，由此可推论原相空间特定区域内吸引子的存在性。这些结果的假设以原动力学与潜动力学间的近似半共轭形式表达。为验证相关思路，我们将基于自编码器（autoencoder）的模型与Conley-Morse图计算结合，应用于莱斯利（Leslie）种群模型、13维地中海红珊瑚种群模型及Chafee--Infante方程。尽管Conley指标的提升仍是未决问题，但这些示例在多种场景下均恢复了预期的代数拓扑不变量。
英文摘要
Combinatorial-topological methods for characterizing dynamics are rigorous, generalizable, computable, and they only require approximations, but the dimension of the phase space is a computational bottleneck to their wider application. Motivated by the growing number of machine learning techniques for obtaining lower-dimensional latent representations of dynamics, we present an initial study of combinatorial-topological techniques in the dimensionality reduction setting. We establish bounds under which an algebraic structure that organizes dynamics can be lifted from the latent space to the original system. As a corollary, one can conclude the existence of attractors within certain regions of the original phase space. The hypothesis of these results is expressed in terms of an approximate semiconjugacy between the original and latent dynamics. To demonstrate the ideas, we combine autoencoder-based models with Conley-Morse graph computations for Leslie population models, a thirteen-dimensional Mediterranean red coral population model, and the Chafee--Infante equation. While the lift of the Conley index is still an open question, the examples recover the expected algebraic topological invariants in several settings.
