Divergence-free Linearized Neural Networks: Integral Representation and Optimal Approximation Rates

TL;DR

This paper develops a new integral representation for divergence-free vector fields using linearized ReLU$^k$ neural networks, achieving optimal approximation rates under divergence-free constraints, with numerical validation.

Contribution

It introduces a novel integral representation for divergence-free fields via antisymmetric potentials parameterized by linearized ReLU$^k$ networks, enabling optimal approximation.

Findings

Numerical experiments confirm theoretical approximation rates.

Exactly divergence-free neural networks improve computational effectiveness.

The approach applies to 2D and 3D steady Stokes problems.

Abstract

This paper studies the numerical approximation of divergence-free vector fields by linearized shallow neural networks, also referred to as random feature models or finite neuron spaces. Combining the stable potential lifting for divergence-free fields with the scalar Sobolev integral representation theory via ReLU$^k$ networks, we derive a core integral representation of divergence-free Sobolev vector fields through antisymmetric potentials parameterized by linearized ReLU$^k$ neural networks. This representation, together with a quasi-uniform distribution argument for the inner parameters, yields optimal approximation rates for such linearized ReLU$^k$ neural networks under an exact divergence-free constraint. Numerical experiments in two and three spatial dimensions, including $L^2$ projection and steady Stokes problems, confirm the theoretical rates and illustrate the effectiveness of exactly divergence-free conditions in computation.