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We present in this talk a family of $H^m$-conforming piecewise polynomials based on

 artificial neural network, named as the finite neuron method,

 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 finite element 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.  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.  

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Cantrell test

When \(a \ne 0\), there are two solutions to \(ax^2 + bx + c = 0\) and they are \[x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\]


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