[论文解读] Physics-aware deep learning framework for linear elasticity

Arunabha M. Roy, Rikhi Bose|arXiv (Cornell University)|Feb 19, 2023

Model Reduction and Neural Networks被引用 9

一句话总结

论文为求解线性弹性问题提出一个物理信息神经网络（PINN）框架，通过使用一个多目标损失来强制 PDE 残差、边界条件、本构关系以及数据驱动项，每个场变量有独立的神经网络。

ABSTRACT

The paper presents an efficient and robust data-driven deep learning (DL) computational framework developed for linear continuum elasticity problems. The methodology is based on the fundamentals of the Physics Informed Neural Networks (PINNs). For an accurate representation of the field variables, a multi-objective loss function is proposed. It consists of terms corresponding to the residual of the governing partial differential equations (PDE), constitutive relations derived from the governing physics, various boundary conditions, and data-driven physical knowledge fitting terms across randomly selected collocation points in the problem domain. To this end, multiple densely connected independent artificial neural networks (ANNs), each approximating a field variable, are trained to obtain accurate solutions. Several benchmark problems including the Airy solution to elasticity and the Kirchhoff-Love plate problem are solved. Performance in terms of accuracy and robustness illustrates the superiority of the current framework showing excellent agreement with analytical solutions. The present work combines the benefits of the classical methods depending on the physical information available in analytical relations with the superior capabilities of the DL techniques in the data-driven construction of lightweight, yet accurate and robust neural networks. The models developed herein can significantly boost computational speed using minimal network parameters with easy adaptability in different computational platforms.

研究动机与目标

• 将线性弹性物理定律融合到深度学习框架中，以实现鲁棒、数据高效的解。
• 开发一个多目标损失，强制 PDE 残差、边界条件和本构关系。
• 通过求解经典弹性基准问题，展示轻量级、精确的神经网络。
• 证明智能初始化和解析解可以提升训练速度与准确性。
• 说明该框架在不同弹性问题和网络结构中的适应性。

提出的方法

• 使用独立的密集连接神经网络在二维弹性中近似每个场变量（位移 u、应力 σ、应变 ε）。
• 构造一个多目标损失 ΔL，包含 PDE 残差、边界条件惩罚和数据拟合项。
• 通过 ΔΩ、Δe、ΔΓu、ΔΓt 及数据项将兼容性、平衡与本构关系嵌入损失中。
• 应用自动微分来计算 PDE 残差和本构方程所需的导数。
• 研究不同的激活函数与结构，并使用智能初始化以减少训练时间。
• 求解平面应力端加载悬臂梁的 Airy 解与 Kirchhoff–Love 薄板等基准问题，以验证精度。

Figure 1 : PINNs network architecture for solving linear elasticity problem consisting of multi-ANN ( $\mathsf{NN}_{i}\,\forall\,i=1,k$ ) for each output variables $\tilde{\mathsf{u}}^{\mathsf{NN}}_{x}(\mbox{\boldsymbol{$x$}})$ , $\tilde{\mathsf{u}}^{\mathsf{NN}}_{y}(\mbox{\boldsymbol{$x$}})$ , $\ti

实验结果

研究问题

• RQ1 如何在多目标损失中构建 PINN，使线性弹性支配方程（兼容性、平衡、本构关系）得以实现？
• RQ2 在弹性问题中加入数据驱动的物理知识项以及边界条件惩罚对准确性和鲁棒性有何影响？
• RQ3 在二维弹性中，独立的位移、应力与应变神经网络是否可以以较少的网络参数获得准确解？
• RQ4 网络架构与激活函数的选择如何影响弹性 PINN 的性能？
• RQ5 智能初始化和利用解析解来加速训练有哪些好处？

主要发现

• 多目标 PINN 框架在所测试的弹性问题中与解析解具有极佳的一致性。
• 位移 u、应力 σ、应变 ε 的独立神经网络可以在同一个 PINN 框架中准确近似各自的场。
• 在损失中包含 PDE 残差、本构关系和边界条件相较于标准数据驱动方法提升了鲁棒性。
• 智能初始化和解析洞见可以在降低训练时间的同时提高精度。
• 与传统线性弹性解法相比，该方法显示出计算速度提升和参数效率。

Figure 2 : (a) Elastic plane-stress problem for an end-loaded cantilever beam of length $L$ , height $2a$ and out-of-plane thickness $b$ which has been clamped at $x=L$ ; (b) distributions of total collocations points $N_{c}=5,000$ on the problem domain and various boundaries during PINNs training.