用神经网络求解各类偏微分方程，通过最小化残差实现高精度计算。
First Order System Least Squares Neural Networks
- 将偏微分方程转化为一阶系统，用神经网络最小化残差进行求解。
- 训练中残差可作为误差估计器，支持自适应优化。
- 自适应增长策略保证解收敛速度最优，适合复杂方程求解。
我们提出一种基于深度神经网络的数值方法，用于在欧氏空间中有界多面体域上求解线性椭圆、抛物和双曲型偏微分方程。将原方程重写为等价的适定一阶系统，并最小化其最小二乘（LSQ）残差。该残差：a）等于或正比于弱形式下的残差；b）可分解为局部子网络贡献之和，反映神经网络局部“失衡”状态；c）作为神经网络训练的数值损失函数；d）即使训练不完全，仍可作为自适应最小二乘有限元方法中的可计算（准）最优误差估计器。此外，提出一种自适应神经网络增长策略，假设能精确最小化LSQ损失泛函，则生成的神经网络序列实现实例收敛到一阶系统最小二乘解的最优速率。
原文摘要 · Abstract (English)
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.
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