SiFEN用可学习的三角网格逼近函数，兼具精度、速度与可解释性。
Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator
- 基于学习的单纯形网格与伯恩斯坦-贝齐多项式构建分段多项式预测器
- 在相同参数量下性能媲美甚至超越MLP和KAN，且推理更快
- 适合追求高效、可解释模型的场景，如嵌入紧凑CNN
我们提出简单形有限元网络（SiFEN），一种可学习的分段多项式预测器，将函数 f: R^d → R^k 表示为在可学习单纯形网格上的全局 C^r 有限元场，输入空间可选非线性扭曲。每次查询仅激活一个单纯形及最多 d+1 个基函数，通过重心坐标实现显式局部性、可控平滑度与缓存友好稀疏性。SiFEN结合 m 次伯恩斯坦-贝齐多项式与轻量可逆扭曲，端到端训练并引入形状正则化、半离散最优传输覆盖及可微边翻转。在标准形状正则性与双李普希茨扭曲假设下，达到经典有限元近似率 M^(-m/d)，其中 M 为网格顶点数。实验显示，在合成逼近任务、表格回归/分类以及作为紧凑CNN的替换头时，SiFEN在相同参数量下表现优于或等同于MLP和KAN，校准性更优（更低ECE/Brier），推理延迟更低，因几何局部性显著。该特性使其成为紧凑、可解释且理论可靠的密集MLP与边样条网络替代方案。
原文摘要 · Abstract (English)
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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