Interpretable Diagnostics and Adaptive Data Assimilation for Neural ODEs via Discrete Empirical Interpolation

Abstract

We present a framework that leverages the Discrete Empirical Interpolation Method (DEIM) for interpretable deep learning and dynamical system analysis. Although DEIM efficiently approximates nonlinear terms in projection-based reduced-order models (POD-ROM), its fixed interpolation points are repurposed for identifying dynamically representative spatial structures in learned models. We apply DEIM as an interpretability tool to examine the learned dynamics of a pre-trained Neural Ordinary Differential Equation (NODE) for two-dimensional vortex-merging and backward-facing step flows. DEIM trajectories reveal physically meaningful structures in NODE predictions and expose failure modes when extrapolating to unseen flow configurations. Building on this diagnostic capability, we further introduce a DEIM-guided data assimilation strategy that injects sparse, dynamically representative corrections into the NODE rollout. By allocating a limited nudging budget to DEIM-identified sampling locations, the framework significantly improves long-term stability and predictive accuracy in out-of-distribution scenarios for the two-dimensional vortex-merging flow. Additional experiments for a flow over a backward-facing step reveal regime-dependent gains, with alternative sampling strategies performing competitively as well. These results demonstrate that DEIM can serve as an interpretable diagnostic and control framework for understanding and enhancing neural differential equation models.