FxTS-Net: Fixed-time stable learning framework for Neural ODEs.

Area of Science:

• Artificial Intelligence
• Dynamical Systems Theory
• Machine Learning

Background:

• Neural Ordinary Differential Equations (Neural ODEs) integrate neural networks with dynamical systems for big data modeling.
• A key challenge is ensuring Neural ODEs reach predicted states within a specified fixed time.
• Existing methods struggle with guaranteed convergence times and robustness to perturbations.

Purpose of the Study:

• To develop a novel framework for training Neural ODEs that guarantees convergence to a predicted state within a user-defined fixed time.
• To enhance the robustness of Neural ODEs against input perturbations.
• To propose an efficient learning algorithm for optimizing the proposed training method.

Main Methods:

• Introduction of FxTS-Net, a framework utilizing fixed-time stability (FxTS) Lyapunov conditions for Neural ODE training.
• Design of a novel FxTS loss (FxTS-Loss) based on Lyapunov functions to enforce fixed-time convergence.
• Development of an innovative approach for constructing Lyapunov functions using supervised information, adaptable to various tasks and architectures.
• Proposal of a learning algorithm with simulated perturbation sampling for optimizing FxTS-Loss, ensuring robustness.

Main Results:

• FxTS-Net demonstrates improved prediction performance compared to existing methods.
• The framework guarantees fixed-time stability for the dynamical system.
• Minimizing FxTS-Loss enhances robustness against bounded non-vanishingly perturbed systems.
• The proposed learning algorithm effectively approximates FxTS-Loss by sampling critical regions.

Conclusions:

• FxTS-Net offers a robust and efficient solution for training Neural ODEs with guaranteed fixed-time convergence.
• The method enhances prediction accuracy and resilience to input perturbations, addressing critical limitations in current Neural ODEs.
• The framework provides a versatile approach for constructing Lyapunov functions, applicable to diverse machine learning tasks and architectures.