Finite Neuron Method and Convergence Analysis (Communications in Computational Physics, 2020)
We study a family of $H^m$-conforming piecewise polynomials based on theartificial neural network, referred to as the finite neuron method (FNM), for numericalsolution of $2m$-th-order partial differential equations in $\mathbb{R}^d$ for any $m,d≥1$ and thenprovide convergence analysis for this method. Given a general domain Ω$⊂\mathbb{R}^d$ and apartition $\mathcal{T}_h$ of Ω, it is still an open problem in general how to construct a conforming finite element subspace of $H^m$(Ω) that has adequate approximation properties. By usingtechniques from artificial neural networks, we construct a family of $H^m$-conformingfunctions 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 boundaryvalue problem of any order in any dimension. For example, the error estimates that $‖u−u_N‖_{H^m(\rm{Ω})}=\mathcal{O}(N^{−\frac{1}{2}−\frac{1}{d}})$ is obtained for the error between the exact solution $u$ andthe finite neuron approximation $u_N$. A discussion is also provided on the differenceand relationship between the finite neuron method and finite element methods (FEM).For example, for the finite neuron method, the underlying finite element grids are notgiven a priori and the discrete solution can be obtained by only solving a non-linearand non-convex optimization problem. Despite the many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value requiresfurther investigation as the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness andthe convenience of the reader, some basic known results and their proofs are introduced.