SoftTri: Smooth Triangular Membership Functions for Adaptive Fuzzy Inference Systems
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.