Interpreting Learned Physics in Neural ODEs through a DEIM-based Spatio-temporal Analysis

Neural Ordinary Differential Equations (NODEs) are a powerful algorithm for learning the time-evolution of fluid dynamical systems from data. However, their black-box nature hinders their use in applications which demand inter- pretability and reliability. We propose a spatiotemporal analysis using the Discrete Empirical Interpolation Method (DEIM) to interpret and validate the physics implicitly learned by a NODE. Our method applies DEIM to the NODE's predicted time-evolution field to identify crucial spatial sampling points. Tracking these points' trajectories reveals the model's focus, enabling assessment of the NODE's modeling deficiencies in the absence of test data. We validated this on three benchmarks: 1D Burgers' equation, 1D Kuramoto-Sivashinsky equation, and 2D vortex merging. In cases of successful prediction, DEIM points from the NODE captured key dynamics like moving shock fronts, wave interactions, and vortex merging. Conversely, when the model's predictions fail, the tra- jectories of these points lose coherency, providing a clear visual indicator of the model's limitations and failure modes. Finally, we investigate how the tracked points may be utilized for effective data assimilation during deployment.