Towards Interpretable Deep Learning and Analysis of Dynamical Systems via the Discrete Empirical Interpolation Method

2025-10-22

AI Summary

Existing discrete empirical interpolation methods (DEIM) in POD-based reduced-order models (POD-ROMs) rely on fixed interpolation points, limiting adaptability to complex, time-varying dynamics. This work proposes a differentiable, adaptive DEIM framework—first embedded as an end-to-end trainable module within neural networks—to enable dynamic optimization of interpolation points. By unifying physics-informed modeling (POD-ROM + DEIM) with deep learning, the method significantly improves prediction accuracy and physical consistency for the viscous Burgers equation and 2D vortex merger problems. Moreover, it serves as a diagnostic tool, exposing implicit dynamical biases and generalization limitations of pre-trained neural ordinary differential equations (NODEs) under extrapolation. The core contribution is the development of the first differentiable DEIM module, bridging model reduction and interpretable AI through gradient-based co-optimization of physics-driven structure and data-driven adaptation.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: Learning & Optimization for ROB
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Develops adaptive DEIM for dynamic interpolation point selection in nonlinear systems
• Applies DEIM as interpretable analysis tool for neural ODE dynamics
• Uses DEIM to diagnose generalization limitations in neural differential equations

Innovation

• Differentiable adaptive DEIM dynamically selects interpolation points
• DEIM analyzes learned dynamics of Neural ODE models
• DEIM serves as diagnostic framework for neural differential equations