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

arXiv: https://arxiv.org/abs/2510.21852 · HTML: https://mlsanity.com/paper/2510.21852/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 37 references that are in the map.
- First posted on arXiv 22 Oct 2025.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Data-Driven Sparse Sensor Placement for Reconstruction](https://mlsanity.com/paper/1701.07569/) (2017)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learning](https://mlsanity.com/paper/2302.06186/) (2023)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [Generalizable data-driven turbulence closure modeling on unstructured grids with differentiable physics](https://mlsanity.com/paper/2307.13533/) (2023)
- [Interpretable A-posteriori Error Indication for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2311.07548/) (2023)
- [Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling](https://mlsanity.com/paper/2205.12419/) (2022)
- [Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation](https://mlsanity.com/paper/2402.15656/) (2024)
- [POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation](https://mlsanity.com/paper/1402.5992/) (2014)
- [FIGNN: Feature-Specific Interpretability for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2506.11398/) (2025)

## Built on by (most cited first, 1 of 1)

- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)

## Abstract (arXiv)

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.