Title: Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs): learning neural networks for designing neutral inclusions Show Chinese title

Title: 共形映射坐标物理信息神经网络（CoCo-PINNs）：用于设计中性夹杂物的神经网络学习

Abstract: We focus on designing and solving the neutral inclusion problem via neural networks. The neutral inclusion problem has a long history in the theory of composite materials, and it is exceedingly challenging to identify the precise condition that precipitates a general-shaped inclusion into a neutral inclusion. Physics-informed neural networks (PINNs) have recently become a highly successful approach to addressing both forward and inverse problems associated with partial differential equations. We found that traditional PINNs perform inadequately when applied to the inverse problem of designing neutral inclusions with arbitrary shapes. In this study, we introduce a novel approach, Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs), which integrates complex analysis techniques into PINNs. This method exhibits strong performance in solving forward-inverse problems to construct neutral inclusions of arbitrary shapes in two dimensions, where the imperfect interface condition on the inclusion's boundary is modeled by training neural networks. Notably, we mathematically prove that training with a single linear field is sufficient to achieve neutrality for untrained linear fields in arbitrary directions, given a minor assumption. We demonstrate that CoCo-PINNs offer enhanced performances in terms of credibility, consistency, and stability. Show Chinese abstract

Abstract: 我们专注于通过神经网络设计和求解中性夹杂物问题。 中性夹杂物问题在复合材料理论中有着悠久的历史，确定使任意形状的夹杂物成为中性夹杂物的确切条件极具挑战性。 物理信息神经网络（PINNs）最近已成为解决与偏微分方程相关的正向和反向问题的一种非常成功的方法。 我们发现，传统PINNs在应用于设计任意形状中性夹杂物的反问题时表现不佳。 在这项研究中，我们引入了一种新颖的方法，即共形映射坐标物理信息神经网络（CoCo-PINNs），它将复分析技术集成到PINNs中。 该方法在解决正逆问题以构造二维中任意形状的中性夹杂物方面表现出色，在这种方法中，夹杂物边界的非理想界面条件通过训练神经网络来建模。 值得注意的是，我们从数学上证明了，只要满足一个较小的假设条件，在单一线性场下进行训练就足以实现任意方向未训练线性场的中性化。 我们证明了CoCo-PINNs在可信度、一致性和稳定性方面表现出增强的性能。