{"ID":2845525,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.04804","arxiv_id":"2511.04804","title":"Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator","abstract":"We introduce Simplex-FEM Networks (SiFEN), a learned piecewise-polynomial predictor that represents f: R^d -\u003e 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.","short_abstract":"We introduce Simplex-FEM Networks (SiFEN), a learned piecewise-polynomial predictor that represents f: R^d -\u003e 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,...","url_abs":"https://arxiv.org/abs/2511.04804","url_pdf":"https://arxiv.org/pdf/2511.04804v2","authors":"[\"Chaymae Yahyati\",\"Ismail Lamaakal\",\"Khalid El Makkaoui\",\"Ibrahim Ouahbi\",\"Yassine Maleh\"]","published":"2025-11-06T20:49:13Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Convolutional Neural Network\"]","has_code":false}