Simplex-FEM Networks (SiFEN): Learning A Triangulated Function Approximator
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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Simplicial Neural Networks:First Steps and Future Applications
It is well-known that artificial neural networks are universal approximators. The Universal Approximation Theorem proves that, given a continuous function on a compact set embedded in an $n$-dimensional space, there exis…
Convergence of the Deep Galerkin Method for Mean Field Control Problems
We establish the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean fiel…
On Approximate Dynamic Programming with Multivariate Splines for Adaptive Control
We define a SDP framework based on the RLSTD algorithm and multivariate simplex B-splines. We introduce a local forget factor capable of preserving the continuity of the simplex splines. This local forget factor is integ…
rho-POMDPs have Lipschitz-Continuous epsilon-Optimal Value Functions
Many state-of-the-art algorithms for solving Partially Observable Markov Decision Processes (POMDPs) rely on turning the problem into a “fully observable” problem—a belief MDP—and exploiting the piece-wise linearity and …
Parameterized Convex Minorant for Objective Function Approximation in Amortized Optimization
Parameterized convex minorant (PCM) method is proposed for the approximation of the objective function in amortized optimization. In the proposed method, the objective function approximator is expressed by the sum of a P…
Model Predictive Control