Analysis of Neural ODE Performance in Long-term PDE Sequence Modeling

Predicting solutions to partial differential equations (PDEs) from limited snapshots is central to many scientific and engineering disciplines. However, standard autoregressive neural models often suffer from compounding errors when rolled out over large time scales. In this paper, we investigate Neural Ordinary Differential Equations (Neural ODEs) as an alternative to purely discrete, stepwise approaches for PDE sequence modeling. We propose a continuous-time neural solver architecture that combines a graph encoder to process local spatial features, a learned ODE network to integrate node embeddings forward in time, and a decoder that produces PDE field predictions at future timesteps. We compare this Neural ODE solver to a baseline autoregressive GNN on a long-horizon sequence-to-sequence prediction task for Burgers' equation without diffusion. Empirical results indicate that our Neural ODE approach significantly reduces error growth over extended time scale and achieves improved stability. These findings highlight the promise of continuous-time neural modeling for robust PDE simulation and pave the way for applying learned surrogates to complex scientific systems. Our implementation is available at https://anonymous.4open.science/r/neural-ode-pde-EBC8/.