发表机构
Ferdowsi University of Mashhad（马什哈德菲尔多西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对三角形隶属函数不可微导致梯度优化受限的问题，提出SoftTri可微三角形隶属函数，保留几何结构并实现C∞光滑，集成于TS模糊神经网络，在多个基准上提升优化稳定性和逼近精度。
AI 中文摘要
三角形隶属函数因其可解释性、低参数化复杂性和强局部性而被广泛应用于模糊系统。然而，其固有的在结点处不可微性限制了基于梯度的优化在自适应神经模糊架构中的有效性，通常需要次梯度近似或启发式平滑技术。本文提出了SoftTri，一种通过受Swish型激活启发的平滑软铰链机制构造的可微三角形隶属函数。所提出的公式保留了经典三角形隶属函数的几何结构和局部行为，同时对输入变量和隶属参数(a,b,c)提供C∞光滑性，且对于任何有限的尖锐度参数β>0均成立。推导了闭式解析梯度，以实现高效且完全可微的基于反向传播的学习。SoftTri被集成到一个具有网格划分规则的Takagi-Sugeno模糊神经网络中，并在多个一维和二维非线性逼近基准以及使用Airfoil Self-Noise数据集的真实回归任务上进行了评估。实验结果表明，与经典三角形隶属函数相比，SoftTri在相同规则结构和训练设置下，持续提高了优化稳定性和逼近精度，同时达到与高斯隶属函数相当或更好的性能。所提出的方法在现代神经模糊学习系统中提供了可解释性与可微优化之间的有效折衷。
英文摘要
Triangular membership functions (MFs) are widely used in fuzzy systems because of their interpretability, low parameterization complexity, and strong locality properties. However, their inherent nondifferentiability at knot points limits the effectiveness of gradient-based optimization in adaptive neuro-fuzzy architectures, often necessitating subgradient approximations or heuristic smoothing techniques. In this paper, we propose \emph{SoftTri}, a differentiable triangular membership function constructed using a smooth soft-hinge mechanism inspired by Swish-type activations. The proposed formulation preserves the geometric structure and localized behavior of classical triangular MFs while providing $C^\infty$ smoothness with respect to both the input variable and the membership parameters $(a,b,c)$ for any finite sharpness parameter $β>0$. Closed-form analytical gradients are derived to enable efficient and fully differentiable backpropagation-based learning. SoftTri is integrated into a Takagi--Sugeno fuzzy neural network with grid-partitioned rules and evaluated on multiple one-dimensional and two-dimensional nonlinear approximation benchmarks as well as a real-world regression task using the Airfoil Self-Noise dataset. Experimental results demonstrate that SoftTri consistently improves optimization stability and approximation accuracy compared with classical triangular membership functions, while achieving performance comparable to or better than Gaussian MFs under identical rule structures and training settings. The proposed approach provides an effective compromise between interpretability and differentiable optimization in modern neuro-fuzzy learning systems.
