[论文解读] Physics informed deep learning for computational elastodynamics without labeled data

Chengping Rao, Hao Sun+1 more|arXiv (Cornell University)|Jun 10, 2020

Model Reduction and Neural Networks参考文献 39被引用 33

一句话总结

本文开发了一种用于弹性动力学的物理信息神经网络（PINN），在没有带标签数据的情况下进行训练，使用混合位移-应力输出，以及通过由三个人工神经网络组成的组合来硬性执行初始/边界条件，并展示静态、动态和波传播问题。

ABSTRACT

Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning is a novel approach recently developed for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data. The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neural network (DNN) and embed the physical law to regularize the network. To this end, training the network is equivalent to minimization of a well-designed loss function that contains the PDE residuals and initial/boundary conditions (I/BCs). In this paper, we present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data, in which the I/BCs are hardly imposed. In particular, both the displacement and stress components are taken as the DNN output, inspired by the hybrid finite element analysis, which largely improves the accuracy and trainability of the network. Since the conventional PINN framework augments all the residual loss components in a "soft" manner with Lagrange multipliers, the weakly imposed I/BCs cannot not be well satisfied especially when complex I/BCs are present. To overcome this issue, a composite scheme of DNNs is established based on multiple single DNNs such that the I/BCs can be satisfied forcibly in a "hard" manner. The propose PINN framework is demonstrated on several numerical elasticity examples with different I/BCs, including both static and dynamic problems as well as wave propagation in truncated domains. Results show the promise of PINN in the context of computational mechanics applications.

研究动机与目标

• 在不使用带标签数据的情况下，使用PINNs来激发和建模弹性动力学问题。
• 将位移场和应力场都作为DNN输出纳入，以提升准确性和可训练性。
• 开发一个对初始/边界条件进行硬性执行的方案，以确保解的唯一性。
• 在有界/截断域中的静态与动态弹性动力学问题及波传播上演示该框架。

提出的方法

• 使用混合变量PINN输出（u, sigma）来近似位移场和应力场。
• 通过强形式，结合控制方程和本构关系，对弹性动力学进行表述。
• 引入一个由三个相互连接的网络（particular、distance、general）组成的复合硬边界条件执行方案，以精确满足I/BCs。
• 先对distance和particular解网络进行预训练，然后训练general解网络以使PDE残量满足。
• 在共点处评估残差，并利用自动微分获得精确导数。

实验结果

研究问题

• RQ1 在不使用带标签数据的情况下，PINN模型是否能解决弹性动力学问题，同时准确捕捉位移和应力场？
• RQ2 通过复合DNN对初始/边界条件进行硬性执行，是否能在弹性动力学问题中比软性执行获得更高的准确性？
• RQ3 混合位移-应力输出如何影响弹性动力学PINN的训练稳定性与预测准确性？
• RQ4 该框架能否在有界和截断域内处理静态、动态和波传播情景？

主要发现

• 混合的u、sigma PINN输出相较于仅位移的形式在训练和准确性上有改进。
• 通过复合DNN方案对I/BCs进行硬性执行比软性执行在边界与初始条件满足方面获得更高的准确性。
• 当网络容量增加时，该方法显示应力和位移场向有限元解收敛。
• 动态及循环加载情况表明PINN能够在有界域内捕捉随时间演化的应力分布和波传播。
• 在所给出的示例中，预测的应力场和位移场与参考FE解高度一致。