Towards Interpretable Deep Learning and Analysis of Dynamical Systems via the Discrete Empirical Interpolation Method
This work addresses interpretability and generalization issues in neural differential equation models for dynamical systems analysis, representing an incremental improvement by adapting an existing method for enhanced adaptability and diagnostic use.
The authors tackled the limitation of fixed interpolation points in the Discrete Empirical Interpolation Method (DEIM) for dynamical systems by developing a differentiable adaptive DEIM formulation for the viscous Burgers equation, enabling dynamic point selection, and applied DEIM to analyze a pre-trained Neural ODE on a vortex-merging problem, revealing physically meaningful features and exposing extrapolation limitations.
We present a differentiable 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 limit the adaptability to complex and time-varying dynamics. To address this limitation, we first develop a differentiable adaptive DEIM formulation for the one-dimensional viscous Burgers equation, which allows neural networks to dynamically select interpolation points in a computationally efficient and physically consistent manner. We then apply DEIM as an interpretable analysis tool for examining the learned dynamics of a pre-trained Neural Ordinary Differential Equation (NODE) on a two-dimensional vortex-merging problem. The DEIM trajectories reveal physically meaningful features in the learned dynamics of NODE and expose its limitations when extrapolating to unseen flow configurations. These findings demonstrate that DEIM can serve not only as a model reduction tool but also as a diagnostic framework for understanding and improving the generalization behavior of neural differential equation models.