神经网络引导的对偶加权残差估计伴随计算
Neural network guided adjoint computations in dual weighted residual error estimation
摘要：
在这项工作中，我们关注的是神经网络引导的面向目标的后验误差估计和使用双加权残差方法的自适应性。用经典的Galerkin有限元求解原问题。伴随问题的解决，在强的形式与前馈神经网络使用两个或三个隐藏层。我们的方法的主要目标是探索解决伴随问题的替代方案，具有更大的数值成本降低的潜力。该算法是基于一般的面向目标的误差估计定理，包括线性和非线性静态偏微分方程和目标泛函。我们的发展证实了一些数值实验，包括神经网络计算的伴随和经典的有限元解决方案的伴随比较。在编程软件中，开源库deal.II成功地与PyTorch C++应用编程接口LibTorch耦合。
英文摘要：
In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
