Local manifold approximation of dynamical system based on neural ordinary differential equation

In this paper, we present an innovative data-driven method that leverages the latest advancements in Neural Ordinary Differential Equations (NODEs) to learn dynamical system and perform local manifold approximation. This approach enables the capture of the underlying evolution mechanisms of complex nonlinear systems, even when only observational data is available. We first construct a NODE model, which is powered by a neural network, and then train it using the observational data to learn the vector field of the unknown nonlinear system. This allows for the approximation and prediction of its local manifold. We establish a universal approximation theorem for the local manifold approximation using NODEs, and through rigorous numerical experiments, we validate the method’s ability to approximate the local manifolds of nonlinear systems. In addition, we explore how factors such as network structure, complexity, and training data influence the approximation performance. Finally, we assess the robustness of NODE-based manifold approximation under various noisy conditions, demonstrating its generalization ability and resilience in real-world scenarios.