FxTS-Net: Fixed-Time Stable Learning Framework for Neural ODEs
This addresses a specific timing control problem in Neural ODEs for applications requiring precise convergence, though it appears incremental as it builds on existing Neural ODE and Lyapunov stability concepts.
The paper tackles the challenge of ensuring Neural ODEs reach correct predictions within a user-defined fixed time by proposing FxTS-Net, a training framework using fixed-time stability Lyapunov conditions, which experimentally shows better prediction performance and robustness under input perturbation.
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.