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.

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关于巴伦空间中椭圆偏微分方程解的表示

基于神经网络的高维偏微分方程 (PDE) 的数值解已经取得了令人兴奋的进展。本文推导了 Barron 空间中 $d$ 维二阶椭圆偏微分方程的解的复杂度估计，这是一组函数，允许某些参数脊函数对参数的概率测度进行积分。我们在一些适当的假设下证明，如果椭圆 PDE 的系数和源项位于 Barron 空间中，那么 PDE 的解是 $\epsilon$-close 相对于 Barron 函数的 $H^1$ 范数. 此外，我们证明了这个近似解的 Barron 范数的维度显式边界，最多取决于 PDE 的维度 $d$。作为复杂性估计的直接结果，