Continuum solid mechanics form the foundation of numerous theoretical studies and engineering applications. Distinguished from traditional mesh-based numerical solutions, the rapidly developing field of scientific machine learning, exemplified by methods such as physics-informed neural networks (PINNs), shows great promise for the study of forward and inverse problems in mechanics. However, accurately imposing boundary conditions (BCs) in the training and prediction of neural networks (NNs) has long been a significant challenge in the application and research of PINNs. This paper integrates the concept of isogeometric analysis (IGA) by parameterizing the physical model of the structure with spline basis functions to form analytical distance functions (DFs) for arbitrary structural boundaries. Meanwhile, by means of the energy approach to circumvent the solution of boundary stress components, the accurate imposition of both Dirichlet and Neumann BCs is ultimately achieved in PINNs. Additionally, to accommodate the complex mixed BCs often encountered in engineering applications, where Dirichlet and Neumann BCs simultaneously appear on adjacent irregular boundary segments, structural subdomain decomposition and multi-subdomain stitching strategies are introduced. The effectiveness and accuracy of the proposed method are verified through two numerical experiments with various cases.

中文翻译：

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用于具有复杂混合边界条件的固体力学的参数化扩展物理信息神经网络


连续体固体力学构成了许多理论研究和工程应用的基础。与传统的基于网格的数值解不同，以物理信息神经网络 （PINN） 等方法为代表的快速发展的科学机器学习领域在力学中正向和逆向问题的研究方面显示出巨大的前景。然而，在神经网络 （NN） 的训练和预测中准确施加边界条件 （BC） 一直是 PINN 应用和研究的重大挑战。本文集成了等几何分析 （IGA） 的概念，方法是使用样条基函数将结构的物理模型参数化，以形成任意结构边界的解析距离函数 （DF）。同时，通过利用能量方法来规避边界应力分量的解，最终在 PINN 中实现了狄利克雷和诺依曼 BC 的精确施加。此外，为了适应工程应用中经常遇到的复杂混合 BC，其中狄利克雷和诺依曼 BC 同时出现在相邻的不规则边界段上，引入了结构子域分解和多子域拼接策略。通过两次不同情况下的数值实验验证了所提方法的有效性和准确性。