AI 中文总结
该研究提出一种基于图神经网络和弱形式的无数据神经偏微分方程求解器，可推广至未知载荷工况与几何，残差误差低于1%，避免昂贵的合成数据获取过程，克服物理信息神经网络的困难。
AI 中文摘要
我们提出了一种基于物理信息、无数据的偏微分方程神经求解器，该求解器构建于利用消息传递的图神经网络架构之上。通过依赖问题的弱形式，我们使用有限元形函数的梯度（这些梯度因此是多项式），而非自动微分算子，来从网络自身预测的位移中计算方程的残差。我们的方法可推广至先前未见的载荷工况和几何结构，残差收敛误差轻易达到1%以下，且能够扩展到规模相当大、几何形状任意的模型。为确保符合物理定律并为推理提供保证，可将残差本身用作推理的误差指标，因此若未达到用户预先设定的残差容差，可在测试阶段进行细化。本文提供了示例以展示所提方法的性能。该方法避免了获取、整理和存储用于训练神经网络的高保真合成数据这一昂贵过程。尽管这并非我们方法所独有，但这是首次将其与几何机器学习技术相结合，该技术能够提供必要的几何偏置，以克服物理信息神经网络的众所周知的困难。
英文摘要
We present a physics-informed, data-free neural solver for partial differential equations, built on a graph neural network architecture that utilises message passing. By relying on the weak form of the problem, we use gradients of finite-element shape functions (which are therefore polynomials) rather than automatic differentiation operators to compute the residuals of the equation from the displacements predicted by the network itself. Our approach generalises to previously unseen load cases and geometries, achieving easily convergence errors in the residuals of less than 1% and being capable of scaling up to models of considerable size and arbitrary geometries. To ensure compliance with the laws of physics and provide guarantees regarding the inference, it is possible to use the residual itself as an error indicator for the inference, and thus perform a refinement at the testing stage if the residual tolerance set in advance by the user is not met. Examples are provided to demonstrate the performance of the proposed method. This results in a method that avoids the costly process of obtaining, curating and storing high-fidelity synthetic data for training the neural network. Whilst this is not unique to our method, it is the first time it has been combined with a geometric machine learning technique capable of providing the necessary geometric bias to overcome the well-known difficulties of physics-informed neural networks.
