基于SA-NODEs的长时间轨迹近似：模型预测与弗洛凯策略
Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies
- Friedrich-Alexander-Universität Erlangen-Nürnberg（弗里德里希-亚历山大-埃尔兰根-纽伦堡大学）
- School of Pedagogical & Technological Education (ASPETE)（教育与技术教育学院（ASPETE））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对SA-NODEs长时间轨迹近似的误差双指数恶化问题，提出模型预测与弗洛凯两种状态重置策略，经四个基准实验验证了误差规律。
AI中文摘要：
我们研究半自主神经常微分方程（SA-NODEs）对长时间范围内动力系统的近似问题。对于在整个时间范围内训练的单个网络，其可用误差界会随时间范围长度呈双指数恶化。我们开发了两种训练策略来规避这一障碍，每种策略都基于状态重置构建。模型预测策略自适应划分时间范围，从观测数据重启每个窗口：当训练在每个窗口上达到规定容差时，复合模型在时间上均匀满足该容差，对于具有有界、均匀正则可达管的目标，参数预算与时间范围呈线性关系。弗洛凯策略针对具有稳定极限环的自主目标，部署时无需数据：所学返回映射的可证收缩将误差限制为已用周期数的线性增长。对于我们采用的时间周期架构，标量证明退化；我们转而证明了一个时间均匀轨道保证，其假设在训练模型上进行测量，以及一个阻碍：对于完全周期的学习场，小的单周期误差和收缩频闪映射无法同时成立。在四个基准上的数值实验证实了预测的误差规律，并测量了每个保证的假设。
英文摘要：
We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.
