Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is $\epsilon$-close with respect to the $H^1$ norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension $d$ of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the $H^1$ norm with a dimension-explicit convergence rate.
翻译:以神经网络为基础的高维部分偏差方程(PDEs)的数值解决方案出现了令人振奋的发展。本文对Barron空间中以美元为维维的二阶椭圆形 PDE(PDEs)解决方案进行了复杂估计,这是一套功能,根据参数的概率计量,承认某些参数的分数脊函数的有机体。我们根据一些适当的假设证明,如果利滑式PDE的系数和源术语存在于Barron空间,那么PDE的解决方案在Barron函数的$H1美元规范方面是$\epsilon-cloose。此外,我们证明这一近似解决方案的Barron规范的维度界限,主要取决于PDE的维度。作为复杂估计的一个直接结果,PDE的解决方案可以通过两层神经网络在任何受约束的域中近似于$H1美元标准,并具有维异的趋同率。