Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator

• URL: http://arxiv.org/abs/2511.04804v1
• Date: Thu, 06 Nov 2025 20:49:13 GMT
• Title: Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator
• Authors: Chaymae Yahyati, Ismail Lamaakal, Khalid El Makkaoui, Ibrahim Ouahbi, Yassine Maleh,
• Abstract summary: SiFEN is a learned piecewise-polynomial predictor that represents f: Rd -> Rk as a globally Cr finite-element field on a learned simplicial mesh.<n>Each query activates exactly one simplex and at most d+1 basis functions via bary coordinates.
• Score: 1.565870461096057
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce Simplex-FEM Networks (SiFEN), a learned piecewise-polynomial predictor that represents f: R^d -> R^k as a globally C^r finite-element field on a learned simplicial mesh in an optionally warped input space. Each query activates exactly one simplex and at most d+1 basis functions via barycentric coordinates, yielding explicit locality, controllable smoothness, and cache-friendly sparsity. SiFEN pairs degree-m Bernstein-Bezier polynomials with a light invertible warp and trains end-to-end with shape regularization, semi-discrete OT coverage, and differentiable edge flips. Under standard shape-regularity and bi-Lipschitz warp assumptions, SiFEN achieves the classic FEM approximation rate M^(-m/d) with M mesh vertices. Empirically, on synthetic approximation tasks, tabular regression/classification, and as a drop-in head on compact CNNs, SiFEN matches or surpasses MLPs and KANs at matched parameter budgets, improves calibration (lower ECE/Brier), and reduces inference latency due to geometric locality. These properties make SiFEN a compact, interpretable, and theoretically grounded alternative to dense MLPs and edge-spline networks.

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