xFODE: An Explainable Fuzzy Additive ODE Framework for System Identification

arXiv cs.LG / 4/17/2026

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Key Points

The paper introduces xFODE, an explainable fuzzy additive ODE framework aimed at improving interpretability in data-driven system identification compared with standard NODE/FODE approaches.
xFODE defines system states incrementally to give them clearer physical meaning, addressing a key drawback where reconstructed states are often difficult to interpret physically.
It approximates state derivatives using fuzzy additive models so that the contribution of each input to the derivatives becomes more interpretable.
The method adds Partitioning Strategies that structure the fuzzy antecedent space during training so that only two consecutive rules activate per input, reducing local inference complexity and enhancing interpretability.
Experiments on benchmark system identification datasets show xFODE achieves accuracy comparable to NODE, FODE, and NLARX while also providing interpretable insights.

Abstract

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.