This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.
翻译:本文涉及深Ritz方法[W. E和B. Yu, 2017年] 的先验概括分析,这是解决高维部分偏差方程的一种流行的神经网络方法。我们从DRM框架内的两层神经网络的概括误差中推断出解决两个原型椭圆形PDE的两层神经网络:Poisson方程式和静态Schr\'odinger方程式在$d$d$-syu 超立方体上。具体地说,我们证明,在先验假设PDE的确切解决办法存在于一个叫做光谱Barron空间的适当的低兼容度空间中的情况下,一般误差率独立于维度$d$d。此外,我们对强制术语和潜在功能提供了充分的条件,保证解决方案是光谱Barron功能。我们通过为光谱Barron空间的PDEs开发新的解决方案理论来实现这一目标,这可以被视为PDES古典Sobolev规律理论的类比。