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This paper introduces xFODE, an explainable system identification framework that combines fuzzy additive models with deep learning to enhance the interpretability of Neural/Fuzzy Ordinary Differential Equation (NODE/FODE) models. xFODE defines system states incrementally to provide physical meaning and uses fuzzy additive models to approximate state derivatives, improving input-specific interpretability. The framework incorporates Partitioning Strategies (PSs) to reduce complexity and further enhance antecedent space interpretability, achieving comparable accuracy to NODE, FODE, and NLARX models on benchmark datasets while offering interpretable insights.
You can now get NODE/FODE-level accuracy in system identification *and* understand what the heck is going on inside.
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.