发表机构
University of Stuttgart（斯图加特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用深度学习方法（物理信息神经网络和Deep Ritz）替代传统数值方法求解随机域椭圆PDE的随机伽辽金系统，并在二维随机几何上验证了有效性。
AI 中文摘要
本文研究了椭圆随机偏微分方程（PDE）的随机域问题的强和弱随机伽辽金近似。随机域问题通过足够正则的随机域映射实现，允许在确定性参考域上进行变换。求解由此产生的高维耦合随机伽辽金系统的传统数值方法被深度学习技术所取代。我们比较了一种基于强随机伽辽金残差的物理信息神经网络方法与一种基于弱随机伽辽金系统（重新表述为Ritz能量最小化）的Deep Ritz方法。神经网络作为相应随机伽辽金解的确定性谱系数的替代模型。随机参数不作为神经网络的输入参数，且在训练过程中不对随机域进行采样。该方法的效率在二维空间中的随机拉伸区间和随机变形环上得到了验证。
英文摘要
This work considers strong and weak stochastic Galerkin approximations of random domain problems for the elliptic random partial differential equation (PDE). The random domain problems are realized by sufficiently regular random domain mapping, allowing a transformation on deterministic reference domain. A traditional numerical method for solving the resulting high-dimensional coupled stochastic Galerkin systems is replaced by deep learning techniques. We compare a physics-informed neural network approach, based on the strong stochastic Galerkin residual, with a Deep Ritz approach, based on the weak stochastic Galerkin system, reformulated as Ritz energy minimization. The neural networks serve as surrogates for the deterministic spectral coefficients of the respective stochastic Galerkin solution. The stochastic parameters are not used as neural network input parameters and the stochastic domain is not sampled during training. The efficiency of the methods is demonstrated on a randomly stretched interval and a randomly deformed annulus in two spatial dimensions.
