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.

翻译：本文提出单纯形有限元网络（SiFEN），这是一种学习得到的分片多项式预测器，它将函数 f: R^d -> R^k 表示为一个定义在可选的扭曲输入空间中、基于学习得到的单纯形网格上的全局 C^r 有限元场。每个查询通过重心坐标精确激活一个单纯形和至多 d+1 个基函数，从而具有显式的局部性、可控的光滑性和缓存友好的稀疏性。SiFEN 将 m 次 Bernstein-Bezier 多项式与一个轻量可逆扭曲函数配对，并通过形状正则化、半离散最优输运覆盖和可微分边翻转进行端到端训练。在标准的形状正则性和双 Lipschitz 扭曲假设下，SiFEN 在网格顶点数为 M 时达到经典的有限元逼近速率 M^(-m/d)。实验表明，在合成逼近任务、表格数据回归/分类任务以及作为紧凑 CNN 的即插即用头部时，SiFEN 在参数量相当的情况下，性能匹配或超越了多层感知机（MLP）和 Kolmogorov-Arnold 网络（KAN），改善了校准性（更低的预期校准误差/ Brier 分数），并因几何局部性而降低了推理延迟。这些特性使 SiFEN 成为密集 MLP 和边缘样条网络的一种紧凑、可解释且具有理论依据的替代方案。