The Finite Neuron Method and Convergence Analysis

TL;DR

This paper introduces the finite neuron method (FNM), a neural network-based approach for solving high-order PDEs, providing convergence analysis and error estimates, and discusses its relation to traditional finite element methods.

Contribution

The paper constructs an $H^m$-conforming neural network-based finite element space with proven convergence and error bounds for solving elliptic PDEs of any order and dimension.

Findings

Error estimate: $ orm{u-u_N}_{H^m(ody)}=O(N^{-rac{1}{2}-rac{1}{d}})$

FNM constructs conforming spaces using neural networks for arbitrary order PDEs

Discussion on differences between FNM and classical FEM

Abstract

We study a family of $H^m$-conforming piecewise polynomials based on artificial neural network, named as the finite neuron method (FNM), for numerical solution of $2m$-th order partial differential equations in $\mathbb{R}^d$ for any $m,d \geq 1$ and then provide convergence analysis for this method. Given a general domain $\Omega\subset\mathbb R^d$ and a partition $\mathcal T_h$ of $\Omega$, it is still an open problem in general how to construct conforming finite element subspace of $H^m(\Omega)$ that have adequate approximation properties. By using techniques from artificial neural networks, we construct a family of $H^m$-conforming set of functions consisting of piecewise polynomials of degree $k$ for any $k\ge m$ and we further obtain the error estimate when they are applied to solve elliptic boundary value problem of any order in any dimension. For example, the following error estimates between the exact solution $u$ and finite neuron approximation $u_N$ are obtained. $$ \|u-u_N\|_{H^m(\Omega)}=\mathcal O(N^{-{1\over 2}-{1\over d}}). $$ Discussions will also be given on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can only be obtained by solving a non-linear and non-convex optimization problem. Despite of many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value is a subject of further investigation since the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and also convenience to readers, some basic known results and their proofs are also included in this manuscript.