一个用于固体力学求解和发现的深度学习框架
A deep learning framework for solution and discovery in solid mechanics
摘要：
我们提出了一类深度学习的应用，称为物理信息神经网络（PINN），学习和发现固体力学。我们解释了如何将动量平衡和本构关系纳入PINN，并详细探讨了线性弹性的应用，并通过一个展示von~Mises弹塑性的例子说明其扩展到非线性问题。虽然常见的PINN算法是基于训练一个深度神经网络（DNN），但我们提出了一个多网络模型，可以更准确地表示字段变量。为了验证模型，我们测试的框架上生成的合成数据的分析和数值参考解决方案。我们研究了PINN模型的收敛性，并表明等几何分析（IGA）的结果优越的精度和收敛特性相比，经典的低阶有限元法（FEM）。我们还展示了该框架对迁移学习的适用性，并发现在网络重新训练过程中收敛速度大大加快。最后，我们发现，尊重物理学可以提高鲁棒性：当只在少数参数上训练时，我们发现PINN模型可以准确地预测网络新参数范围的解决方案-从而指出该框架在灵敏度分析和代理建模中的重要应用。
英文摘要：
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, and illustrate its extension to nonlinear problems through an example that showcases von~Mises elastoplasticity. 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.
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