  Modeling dynamical systems is crucial for a wide range of tasks, but it
remains challenging due to complex nonlinear dynamics, limited observations, or
lack of prior knowledge. Recently, data-driven approaches such as Neural
Ordinary Differential Equations (NODE) have shown promising results by
leveraging the expressive power of neural networks to model unknown dynamics.
However, these approaches often suffer from limited labeled training data,
leading to poor generalization and suboptimal predictions. On the other hand,
semi-supervised algorithms can utilize abundant unlabeled data and have
demonstrated good performance in classification and regression tasks. We
propose TS-NODE, the first semi-supervised approach to modeling dynamical
systems with NODE. TS-NODE explores cheaply generated synthetic pseudo rollouts
to broaden exploration in the state space and to tackle the challenges brought
by lack of ground-truth system data under a teacher-student model. TS-NODE
employs an unified optimization framework that corrects the teacher model based
on the student's feedback while mitigating the potential false system dynamics
present in pseudo rollouts. TS-NODE demonstrates significant performance
improvements over a baseline Neural ODE model on multiple dynamical system
modeling tasks.
