发表机构
Rice University; Sandia National Laboratories（莱斯大学; 桑迪亚国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种混合建模框架，利用重叠 Schwarz 交替方法耦合预训练数值信息神经网络与经典全阶模型，求解高 Peclet 数对流扩散方程，两种训练方法均取得与全 FOM 求解相当的精度。
AI 中文摘要
我们开发了一个混合建模框架，用于使用重叠 Schwarz 交替方法将预训练的数值信息神经网络（NINNs）与经典全阶模型（FOMs）耦合。我们考虑在对流主导、Peclet 数为 10^6 的区域中的二维对流扩散方程。我们首先证明，与相应的物理信息神经网络（PINN）不同，一个整体的 NINN 可以在我们的模型问题上被精确训练，而无需域分解。然后，我们采用重叠乘法 Schwarz 作为部署机制，用于将预训练的、子域局部的 NINN 与相邻的 FOM 耦合，且在 Schwarz 迭代过程中保持 NINN 权重固定。我们考虑了子域局部 NINN 的两种训练方法：自上而下的方法，其中边界数据从全域上的耦合 Schwarz 求解中获得，每个子域上使用 FOM（FOM-FOM Schwarz）；以及自下而上的方法，其中边界迹线在 NINN 子域上合成生成，无需任何全域求解。所得的混合 NINN-FOM 解与相应的 FOM-FOM Schwarz 解非常吻合，自上而下和自下而上的训练方法产生了相当的精度。
英文摘要
We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.
