提出可解释的模糊微分方程模型，同时给出预测区间与物理意义清晰的状态更新。
xFODE+: Explainable Type-2 Fuzzy Additive ODEs for Uncertainty Quantification
- 用区间型二型模糊系统构建可解释的加性微分方程模型
- 在基准数据集上保持与原模型相当的预测区间质量与精度
- 适合需要透明推理过程的物理系统建模场景
深度学习的进展推动了数据驱动的系统辨识（SysID），但可靠应用需伴随不确定性量化（UQ）。尽管具备UQ能力的模糊微分方程（FODE）能生成预测区间（PIs），但解释性不足。本文提出可解释的二型模糊加性微分方程（xFODE+），在提供点预测的同时生成预测区间，并保持物理上有意义的增量状态。xfODE+采用区间型二型模糊逻辑系统（IT2-FLSs）实现每个模糊加法模块，通过约束隶属函数仅激活相邻两规则，减少重叠，保证局部推理透明。由IT2-FLS产生的类型化集合与微分方程状态更新联合构造状态演化及预测区间。模型在深度学习框架中通过复合损失训练，联合优化预测精度与预测区间质量。在多个基准系统辨识数据集上的实验表明，xfODE+在预测区间质量上与FODE相当，精度也相近，同时显著提升了可解释性。
原文摘要 · Abstract (English)
Recent advances in Deep Learning (DL) have boosted data-driven System Identification (SysID), but reliable use requires Uncertainty Quantification (UQ) alongside accurate predictions. Although UQ-capable models such as Fuzzy ODE (FODE) can produce Prediction Intervals (PIs), they offer limited interpretability. We introduce Explainable Type-2 Fuzzy Additive ODEs for UQ (xFODE+), an interpretable SysID model which produces PIs alongside point predictions while retaining physically meaningful incremental states. xFODE+ implements each fuzzy additive model with Interval Type-2 Fuzzy Logic Systems (IT2-FLSs) and constraints membership functions to the activation of two neighboring rules, limiting overlap and keeping inference locally transparent. The type-reduced sets produced by the IT2-FLSs are aggregated to construct the state update together with the PIs. The model is trained in a DL framework via a composite loss that jointly optimizes prediction accuracy and PI quality. Results on benchmark SysID datasets show that xFODE+ matches FODE in PI quality and achieves comparable accuracy, while providing interpretability.
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