xFODE: An Explainable Fuzzy Additive ODE Framework for System Identification

TL;DR

xFODE is an interpretable system identification framework that combines fuzzy additive models with deep learning, providing physical meaning to states and input contributions while maintaining high accuracy.

Contribution

The paper introduces xFODE, a novel explainable FODE framework with incremental states, fuzzy additive models, and partitioning strategies for enhanced interpretability in system identification.

Findings

xFODE matches the accuracy of existing models like NODE and FODE.

Partitioning strategies improve interpretability and reduce local inference complexity.

xFODE provides physically meaningful states and input contributions in nonlinear dynamic modeling.

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.