具有无限时间 horizon 保证的动态系统通用逼近定理
Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees
AI总结：
本文提出Neural ODEs在无限时间 horizon 上实现ε-δ接近性，为多稳定性动态系统提供了首个通用逼近框架。
摘要：
通用逼近定理确立了神经网络架构的表达能力。对于动态系统，现有结果局限于有限时间 horizon 或具有全局稳定平衡点的系统，未涉及多稳定性及极限环。我们证明Neural ODEs在无限时间 horizon [0,∞) 上实现ε-δ接近性——除了初始条件测度小于δ外，轨迹误差不超过ε。针对三种目标类别：(1) 具有双曲固定点的Morse-Smale系统（一个结构稳定的类别）；(2) 通过精确周期匹配的具有双曲极限环的Morse-Smale系统；(3) 通过离散化实现的具有正常双曲连续吸引子的系统。我们进一步建立了时间泛化界限：ε-δ接近性意味着对于所有t≥0，L^p误差≤ε^p + δ·D^p，将拓扑保证与训练度量联系起来。这些结果为多稳定性无限时间 horizon 动态系统提供了首个通用逼近框架。
英文摘要：
Universal approximation theorems establish the expressive capacity of neural network architectures. A trained dynamical model can pass every finite-time test yet fail in the long run in three ways: it can settle to the wrong steady state, drift out of phase, or collapse a continuum of memory states into isolated ones. We consider models whose vector field is parameterized expressively enough to approximate any target vector field together with its Jacobian, e.g., Neural ODEs. We prove that such models can avoid all three failures, matching a target system's trajectories over the infinite-time horizon $[0,\infty)$ on all but a small set of initial conditions ($\varepsilon$-$δ$ closeness: trajectory error within $\varepsilon$ except for initial conditions of measure $< δ$). We cover three target classes, the dynamical primitives of decision-making, rhythm generation, and analog memory: structurally stable systems with a finite number of hyperbolic periodic orbits (fixed points or limit cycles via exact period matching), and continuous attractors that are normally hyperbolic manifolds of fixed points or of isochronous limit cycles (via discretization). A temporal generalization bound, $\varepsilon$-$δ$ closeness implies time-averaged $L^p$ error $\leq \varepsilon^p + δ\cdot D^p$ for all $t \geq 0$, connects these guarantees to training metrics. This is the first universal approximation framework for multistable dynamics over an unbounded horizon.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
