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

TL;DR

SiFEN introduces a learned finite-element network that efficiently approximates functions with explicit locality, controllable smoothness, and strong empirical performance, serving as a compact and interpretable alternative to traditional neural networks.

Contribution

The paper presents SiFEN, a novel learned piecewise-polynomial predictor using a learned simplicial mesh, combining finite element theory with end-to-end training for improved approximation and efficiency.

Findings

Matches or surpasses MLPs and KANs at similar parameter budgets.

Improves calibration metrics like ECE and Brier score.

Reduces inference latency through geometric locality.

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.