Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917190422233088 |
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| author | Yahyati, Chaymae Lamaakal, Ismail Makkaoui, Khalid El Ouahbi, Ibrahim Maleh, Yassine |
| author_facet | Yahyati, Chaymae Lamaakal, Ismail Makkaoui, Khalid El Ouahbi, Ibrahim Maleh, Yassine |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_04804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator Yahyati, Chaymae Lamaakal, Ismail Makkaoui, Khalid El Ouahbi, Ibrahim Maleh, Yassine Machine Learning 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. |
| title | Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.04804 |