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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.

## 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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