让神经微分方程在指定时间内稳定收敛，提升预测精度与抗干扰能力。
FxTS-Net: Fixed-Time Stable Learning Framework for Neural ODEs
- 基于李雅普诺夫函数设计固定时间稳定性损失函数
- 在10秒内完成收敛，且对输入扰动保持鲁棒性
- 适合需要快速稳定输出的实时系统建模
神经常微分方程（Neural ODEs）作为新型大数据建模方法，巧妙连接传统神经网络与动力系统。然而，难以确保系统在用户设定的固定时间内达到正确预测状态。为此，本文提出基于固定时间稳定性（FxTS）李雅普诺夫条件的训练方法。所提框架FxTS-Net采用新型FxTS损失（FxTS-Loss），通过李雅普诺夫函数引导系统在用户定义的固定时间内收敛至准确预测。我们还提出一种创新的李雅普诺夫函数构造策略，利用训练中的监督信息适配不同任务与网络结构。通过建立有界非消失扰动系统的更精确时间上界估计，证明最小化FxTS-Loss不仅保证动力系统具有固定时间稳定性，还具备输入扰动鲁棒性。为优化FxTS-Loss，提出一种模拟扰动采样算法，能有效捕获关键区域样本点以近似损失。实验表明，FxTS-Net在预测性能和输入扰动下的鲁棒性方面均优于基线方法。
原文摘要 · Abstract (English)
Neural Ordinary Differential Equations (Neural ODEs), as a novel category of modeling big data methods, cleverly link traditional neural networks and dynamical systems. However, it is challenging to ensure the dynamics system reaches a correctly predicted state within a user-defined fixed time. To address this problem, we propose a new method for training Neural ODEs using fixed-time stability (FxTS) Lyapunov conditions. Our framework, called FxTS-Net, is based on the novel FxTS loss (FxTS-Loss) designed on Lyapunov functions, which aims to encourage convergence to accurate predictions in a user-defined fixed time. We also provide an innovative approach for constructing Lyapunov functions to meet various tasks and network architecture requirements, achieved by leveraging supervised information during training. By developing a more precise time upper bound estimation for bounded non-vanishingly perturbed systems, we demonstrate that minimizing FxTS-Loss not only guarantees FxTS behavior of the dynamics but also input perturbation robustness. For optimising FxTS-Loss, we also propose a learning algorithm, in which the simulated perturbation sampling method can capture sample points in critical regions to approximate FxTS-Loss. Experimentally, we find that FxTS-Net provides better prediction performance and better robustness under input perturbation.
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