We present the application of a class of deep learning, known as Physics Informed Neural Networks (PINN), to learning and discovery in solid mechanics. We explain how to incorporate the momentum balance and constitutive relations into PINN, and explore in detail the application to linear elasticity, although the framework is rather general and can be extended to other solid-mechanics problems. While common PINN algorithms are based on training one deep neural network (DNN), we propose a multi-network model that results in more accurate representation of the field variables. To validate the model, we test the framework on synthetic data generated from analytical and numerical reference solutions. We study convergence of the PINN model, and show that Isogeometric Analysis (IGA) results in superior accuracy and convergence characteristics compared with classic low-order Finite Element Method (FEM). We also show the applicability of the framework for transfer learning, and find vastly accelerated convergence during network re-training. Finally, we find that honoring the physics leads to improved robustness: when trained only on a few parameters, we find that the PINN model can accurately predict the solution for a wide range of parameters new to the network---thus pointing to an important application of this framework to sensitivity analysis and surrogate modeling.

本文介绍了物理信息神经网络在固体力学中应用的方法，展示了通过使用多网络模型，结合动量平衡和本构关系，可以更准确地呈现一些场量变量。同时，通过测试合成数据并和解析解和数值解进行比较，验证了模型的有效性和精度，并指出了等几何分析在准确性和收敛性方面的优于有限元法的特点。我们还探索了该框架在机器学习中的应用，并发现物理信息对于提高模型的鲁棒性有很大作用。

固体力学领域的解决方案和发现的深度学习框架