We introduce a conceptual framework for numerically solving linear elliptic, parabolic, and hyperbolic PDEs on bounded, polytopal domains in euclidean spaces by deep neural networks. The PDEs are recast as minimization of a least-squares (LSQ for short) residual of an equivalent, well-posed first-order system, over parametric families of deep neural networks. The associated LSQ residual is a) equal or proportional to a weak residual of the PDE, b) additive in terms of contributions from localized subnetworks, indicating locally ``out-of-equilibrium'' of neural networks with respect to the PDE residual, c) serves as numerical loss function for neural network training, and d) constitutes, even with incomplete training, a computable, (quasi-)optimal numerical error estimator in the context of adaptive LSQ finite element methods. In addition, an adaptive neural network growth strategy is proposed which, assuming exact numerical minimization of the LSQ loss functional, yields sequences of neural networks with realizations that converge rate-optimally to the exact solution of the first order system LSQ formulation.

本研究提出了一种利用深度神经网络数值解 bounded polymorphic domains 中线性椭圆、抛物线和双曲型偏微分方程（PDEs）的概念框架。通过将PDE转化为一阶系统的最小二乘残差的优化，提出的策略能够在适应性最小二乘有限元方法中实现高效的误差估计，在网络训练过程中还有助于局部的平衡调整。

一阶系统最小二乘神经网络