In this paper, we make the first attempt to apply the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. BINNs combine artificial neural networks with the well-established boundary integral equations (BIEs) to effectively solve partial differential equations (PDEs). The BIEs are utilized to map all the unknowns onto the boundary, after which these unknowns are approximated using artificial neural networks and resolved via a training process. In contrast to traditional neural network-based methods, the current BINNs offer several distinct advantages. First, by embedding BIEs into the learning procedure, BINNs only need to discretize the boundary of the solution domain, which can lead to a faster and more stable learning process (only the boundary conditions need to be fitted during the training). Second, the differential operator with respect to the PDEs is substituted by an integral operator, which effectively eliminates the need for additional differentiation of the neural networks (high-order derivatives of neural networks may lead to instability in learning). Third, the loss function of the BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighing functions, which are commonly used in traditional methods to balance the gradients among different objective functions. Moreover, BINNs possess the ability to tackle PDEs in unbounded domains since the integral representation remains valid for both bounded and unbounded domains. Extensive numerical experiments show that BINNs are much easier to train and usually give more accurate learning solutions as compared to traditional neural network-based methods.

翻译：本文首次尝试应用边界集成神经网络（BINNs）数值求解二维弹性静力学与压电问题。BINNs将人工神经网络与成熟的边界积分方程相结合，有效求解偏微分方程。通过边界积分方程将全部未知量映射至边界，随后利用人工神经网络逼近这些未知量，并通过训练过程求解。与传统基于神经网络的方法相比，当前BINNs具有若干显著优势：第一，通过将边界积分方程嵌入学习过程，BINNs仅需离散求解域边界，从而可实现更快速、更稳定的学习过程（训练时仅需拟合边界条件）；第二，将偏微分方程中的微分算子替换为积分算子，有效消除了对神经网络额外求导的需求（神经网络高阶导数可能导致学习不稳定）；第三，由于所有边界条件已天然融入公式体系，BINNs的损失函数仅包含边界积分方程的残差，因此无需采用传统方法中用于平衡不同目标函数梯度的权重函数。此外，由于积分表示在有界域与无界域中均有效，BINNs具备求解无界域偏微分方程的能力。大量数值实验表明，与传统基于神经网络的方法相比，BINNs更易训练且通常能获得更精确的学习解。