发表机构
Louisiana State University（路易斯安那州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明神经ODE伴随梯度具有Lyapunov分析所需结构，提出NODE-CL方法，在滑动窗口和存储数据上实现稳定在线学习，并在DeepMind控制套件上取得最优或接近最优的预测误差。
AI 中文摘要
神经ODE从轨迹损失中学习动力学，但其伴随梯度缺乏Lyapunov分析在线自适应所依赖的回归器乘以参数误差结构，因此流式数据上的训练没有稳定性保证。我们证明该结构确实存在：伴随梯度精确分解为一个正半定轨迹算子作用于参数误差，加上一个具有显式、依赖于视界界限的非线性扰动。随后，一个二次Lyapunov函数在可计算的增益和视界条件下证明了滑动窗口上的在线神经ODE训练的稳定性，并且相同的证书扩展到存储数据：其漂移分支恢复了并发学习，其轨迹分支产生了NODE-CL，一种基于批量前向灵敏度的存储段高斯-牛顿方法，无需状态导数估计。在四个DeepMind控制套件域上，NODE-CL在速度测量噪声下在三个域上取得了最低的中位预测误差，而基于观测器的并发学习性能下降高达8倍；在干净测量下，它在摆上最佳，在cartpole和reacher上优于最佳存储数据基线的1.6倍以内。
英文摘要
Neural ODEs learn dynamics from trajectory losses, but their adjoint gradients lack the regressor-times-parameter-error structure on which Lyapunov analyses of online adaptation rest, so training on streaming data comes without stability guarantees. We show that this structure is in fact present: the adjoint gradient decomposes exactly into a positive semi-definite trajectory operator acting on the parameter error plus a nonlinear perturbation with explicit, horizon-dependent bounds. A quadratic Lyapunov function then certifies online Neural ODE training over sliding windows under computable gain and horizon conditions, and the same certificate extends to stored data: its drift branch recovers concurrent learning, and its trajectory branch yields NODE-CL, a stored-segment Gauss-Newton method built on batched forward sensitivities that needs no state-derivative estimates. On four DeepMind Control Suite domains, NODE-CL attains the lowest median prediction error on three under velocity measurement noise, where observer-based concurrent learning degrades by up to 8x; with clean measurements it is best on the pendulum and within a factor of 1.6 of the best stored-data baseline on the cartpole and reacher.
