Inspired by the principles of fluid dynamics, divergence-free neural ODEs represent a promising class of neural networks that can improve classification performance and robustness. In this work, we propose methods for constructing divergence-free vector fields in high-dimensional latent spaces and thus extending their application to real-world scenarios. Our experimental results show that divergence-free neural ODEs can outperform their conventional counterparts in three classification tasks. In particular, divergence-free models show remarkable resilience to adversarial perturbations. Furthermore, divergence-free models are more effective at preserving the underlying structure of the data. This research provides a physics-based perspective on machine learning applications, in particular classification tasks, by interpreting neural networks as dynamical systems.