Partitioned neural network functions are used to approximate the solution of partial differential equations. The problem domain is partitioned into non-overlapping subdomains and the partitioned neural network functions are defined on the given non-overlapping subdomains. Each neural network function then approximates the solution in each subdomain. To obtain the convergent neural network solution, certain continuity conditions on the partitioned neural network functions across the subdomain interface need to be included in the loss function, that is used to train the parameters in the neural network functions. In our work, by introducing suitable interface values, the loss function is reformulated into a sum of localized loss functions and each localized loss function is used to train the corresponding local neural network parameters. In addition, to accelerate the neural network solution convergence, the localized loss function is enriched with an augmented Lagrangian term, where the interface condition and the boundary condition are enforced as constraints on the local solutions by using Lagrange multipliers. The local neural network parameters and Lagrange multipliers are then found by optimizing the localized loss function. To take the advantage of the localized loss function for the parallel computation, an iterative algorithm is also proposed. For the proposed algorithms, their training performance and convergence are numerically studied for various test examples.

翻译：采用分区神经网络函数来逼近偏微分方程的解。将问题域划分为不重叠的子域，并在给定的不重叠子域上定义分区神经网络函数。每个神经网络函数分别逼近各子域内的解。为获得收敛的神经网络解，需在损失函数（用于训练神经网络参数）中纳入跨子域界面的分区神经网络函数的连续性条件。本文通过引入合适的界面值，将损失函数重构为局部损失函数之和，每个局部损失函数用于训练对应的局部神经网络参数。此外，为加速神经网络解的收敛，在局部损失函数中引入增广拉格朗日项，利用拉格朗日乘子将界面条件和边界条件作为约束施加于局部解。随后通过优化局部损失函数求解局部神经网络参数和拉格朗日乘子。为利用局部损失函数进行并行计算的优势，还提出了迭代算法。通过多个数值算例对所提算法的训练性能和收敛性进行了数值研究。