Neural networks (NNs) have gained significant attention across various engineering disciplines, particularly in design optimization, where they are used to build surrogate models for high-dimensional regression problems. Despite their power as global approximators, NNs often fail to accurately capture local nonlinearities without relying on a large number of training parameters. To address these limitations, in this paper we propose domain decomposition methods (DDM), which divide the input feature space into multiple local subdomains, each modeled by a simpler NN, trained in parallel. To recover the accuracy of a global approximation, interface constraints are introduced in the local loss functions to enforce continuity between subdomains. The interface constraints are enforced with two different approaches, by utilizing Lagrange multiplier or augmented Lagrange multiplier methods. Both approaches are validated using synthetic data from 2D and 3D linear compression problems, numerically solved using the finite element method. The study investigates computational time and accuracy across varying numbers of subdomains to identify optimal partitioning strategies. Compared to unconstrained approximations, both methods significantly improve continuity across subdomain interfaces. Also, the use of DDMs improves approximation accuracy in nonlinear regions when compared to standard global NN training. The augmented Lagrange method outperforms the standard Lagrange formulation by converging faster due to lower convergence requirements, albeit with a slightly lower accuracy. Its scalability makes it the preferred choice for large-scale problems, as the faster convergence outweighs the minor loss in accuracy. Overall, these results highlight the augmented Lagrange method as a promising DDM approach for training efficient and scalable NN surrogate models.

翻译：神经网络（NNs）已在各工程领域获得广泛关注，特别是在设计优化中用于构建高维回归问题的代理模型。尽管作为全局逼近器具有强大能力，但神经网络常因依赖大量训练参数而难以准确捕捉局部非线性特征。为解决此问题，本文提出区域分解方法（DDM），该方法将输入特征空间划分为多个局部子域，每个子域由更简单的神经网络建模并并行训练。为恢复全局逼近精度，在局部损失函数中引入界面约束以强制子域间连续性。通过拉格朗日乘子法和增广拉格朗日乘子法两种方式施加界面约束。采用有限元法数值求解的二维与三维线性压缩问题中的合成数据验证两种方法，通过研究不同子域数量下的计算时间与精度，确定最优划分策略。相较于无约束逼近，两种方法均显著改善了子域界面的连续性。此外，与标准全局神经网络训练相比，区域分解方法提高了非线性区域的逼近精度。增广拉格朗日法因收敛要求更低而收敛速度更快，但精度略逊于标准拉格朗日法，其可扩展性使其成为大规模问题的优选方案——更快的收敛性足以弥补精度轻微损失。总体而言，这些结果凸显了增广拉格朗日法作为训练高效可扩展神经网络代理模型的一种有前景的区域分解方法。