In certain polytopal domains \(\varOmega \), in space dimension \(d=2,3\), we prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in \(H^1(\varOmega )\) for weighted analytic function classes. These classes comprise in particular solution sets of source and eigenvalue problems for elliptic PDEs with analytic data. Functions in these classes are locally analytic on open subdomains \(D\subset \varOmega \), but may exhibit isolated point singularities in the interior of \(\varOmega \) or corner and edge singularities at the boundary \(\partial \varOmega \). The exponential approximation rates are shown to hold in space dimension \(d = 2\) on Lipschitz polygons with straight sides, and in space dimension \(d=3\) on Fichera-type polyhedral domains with plane faces. The constructive proofs indicate that NN depth and size increase poly-logarithmically with respect to the target NN approximation accuracy \(\varepsilon >0\) in \(H^1(\varOmega )\). The results cover solution sets of linear, second-order elliptic PDEs with analytic data and certain nonlinear elliptic eigenvalue problems with analytic nonlinearities and singular, weighted analytic potentials as arise in electron structure models. Here, the functions correspond to electron densities that exhibit isolated point singularities at the nuclei.

在某些多面域\(\varOmega \)中，在空间维度\(d=2,3\)中，我们用\(H^1(\varOmega )\)中的稳定 ReLU 神经网络 (ReLU NNs) 证明了指数表达性加权分析函数类。这些类特别包括具有分析数据的椭圆 PDE 的源和特征值问题的特定解决方案集。这些类中的函数在开放子域\(D\subset \varOmega \)上是局部分析的，但可能在\(\varOmega \)内部表现出孤立的点奇异性，或者在边界\(\partial \ varOmega \) 处表现出角和边奇异性\)。指数近似率显示在空间维度\(d = 2\)中保持不变在具有直边的 Lipschitz 多边形上，以及在具有平面的 Fichera 型多面体域上的空间维度\(d=3\)上。建设性证明表明，NN 深度和大小相对于\(H^1(\varOmega )\)中的目标 NN 近似精度\(\varepsilon >0\)呈多对数增加。结果涵盖了具有解析数据的线性二阶椭圆 PDE 的解集，以及在电子结构模型中出现的具有解析非线性和奇异加权解析势的某些非线性椭圆特征值问题。在这里，函数对应于在原子核处表现出孤立点奇点的电子密度。