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