We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE solutions under appropriate structural assumptions.

翻译：我们在Barron空间框架下研究单位超立方体上具有齐次边界条件的高维二阶椭圆偏微分方程的逼近复杂度。在系数属于适当定义的Barron空间的假设下，我们证明该方程的解可以通过两层神经网络高效逼近，从而规避维度灾难。我们的结果表明，在适当的结构假设下，浅层网络具有捕捉高维偏微分方程解的强大表达能力。