# xFODE: An Explainable Fuzzy Additive ODE Framework for System Identification

arXiv: https://arxiv.org/abs/2604.14883 · HTML: https://mlsanity.com/paper/2604.14883/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 21 references that are in the map.
- First posted on arXiv 16 Apr 2026.

## Builds on (most cited first, 11 of 11)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [GAMI-Net: An Explainable Neural Network based on Generalized Additive Models with Structured Interactions](https://mlsanity.com/paper/2003.07132/) (2020)
- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [Neural Ordinary Differential Equations for Nonlinear System Identification](https://mlsanity.com/paper/2203.00120/) (2022)
- [On the adaptation of recurrent neural networks for system identification](https://mlsanity.com/paper/2201.08660/) (2022)
- [Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion](https://mlsanity.com/paper/2307.07882/) (2023)
- [Deep Learning of Dynamic Systems using System Identification Toolbox(TM)](https://mlsanity.com/paper/2409.07642/) (2024)
- [Stable Linear Subspace Identification: A Machine Learning Approach](https://mlsanity.com/paper/2311.03197/) (2023)
- [dynoGP: Deep Gaussian Processes for dynamic system identification](https://mlsanity.com/paper/2502.05620/) (2025)
- [FAME: Introducing Fuzzy Additive Models for Explainable AI](https://mlsanity.com/paper/2504.07011/) (2025)

## Abstract (arXiv)

Recent advances in Deep Learning (DL) have strengthened data-driven System Identification (SysID), with Neural and Fuzzy Ordinary Differential Equation (NODE/FODE) models achieving high accuracy in nonlinear dynamic modeling. Yet, system states in these frameworks are often reconstructed without clear physical meaning, and input contributions to the state derivatives remain difficult to interpret. To address these limitations, we propose Explainable FODE (xFODE), an interpretable SysID framework with integrated DL-based training. In xFODE, we define states in an incremental form to provide them with physical meanings. We employ fuzzy additive models to approximate the state derivative, thereby enhancing interpretability per input. To provide further interpretability, Partitioning Strategies (PSs) are developed, enabling the training of fuzzy additive models with explainability. By structuring the antecedent space during training so that only two consecutive rules are activated for any given input, PSs not only yield lower complexity for local inference but also enhance the interpretability of the antecedent space. To train xFODE, we present a DL framework with parameterized membership function learning that supports end-to-end optimization. Across benchmark SysID datasets, xFODE matches the accuracy of NODE, FODE, and NLARX models while providing interpretable insights.