We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physics-informed neural networks based on strong and variational formulations.

翻译：我们提出了一个抽象框架，用于分析当可行解为神经网络时基于残差最小化的最小二乘方法的收敛性。利用范数关系和紧性论证，我们推导了强形式和弱形式下残差最小化的连续和离散公式的误差估计。这些公式涵盖了最近基于强形式和变分公式发展起来的物理信息神经网络。