Sequential Cascaded Hamiltonian Neural Networks

Understanding how neural networks can be enhanced to learn physically meaningful representations is fundamental to advancing scientific machine learning and ensuring AI safety in physics-critical applications where unpredictable model behavior can lead to catastrophic failures. We present a novel Sequential Cascaded Hamiltonian Neural Network (SC-HNN) approach that dramatically improves the generalization capabilities of standard neural networks through post-training with energy-conserving constraints. Our method consists of two stages: first training a baseline neural network on trajectory data $(x, t)$, then using automatic differentiation to extract velocities and train a separate Hamiltonian Neural Network that learns the underlying energy function. This approach requires only Hamiltonian structure knowledge (similar to standard HNNs) but operates on trajectory data rather than requiring direct access to phase space coordinates $(x, v)$. Using the harmonic oscillator as our test system, we demonstrate that this cascaded approach achieves comparable long-term stability and energy conservation to direct HNN training while significantly outperforming standalone baseline networks and PINNs. Most remarkably, we find that even when baseline neural networks fail to accurately predict long-term dynamics, their learned trajectory representations provide sufficient data fitting quality that can be leveraged to train separate Hamiltonian networks for improved generalization. The SC-HNN reduces energy variance by over 90\% compared to baseline approaches while maintaining computational efficiency. Our results suggest that standard neural networks may possess inherent physical understanding that can be recovered and enhanced through appropriate post-training procedures, opening new pathways for improving the reliability and physical consistency of learned dynamical systems without requiring physics knowledge during initial training.