In this letter, we propose and demonstrate a data-driven machine learning-based approach to accelerate the finite element method (FEM), method of moments (MoM), finite difference (FD) method, and related variational methods, while maintaining the attractive properties that have allowed such methods to dominate computational science and engineering fields like computational electromagnetics. We use a neural network to predict a set of macro basis functions for a given problem, using only the solution to an extremely coarse description of the problem as input. We then solve the problem using the predicted macro basis. Unlike some existing methods, ours does not rely on the direct prediction of the solution. We show that our macro basis function approach corrects errors in the raw prediction of the network, achieving a far more accurate solution. Results are presented for a class of finite element scattering problems, with error statistics presented from 1000 validation examples and compared to standard and naïve approaches. These results suggest the described macro basis function approach is superior to machine learning approaches that directly predict the solution. Meanwhile, our method achieves comparable accuracy to the full solution while requiring only a fraction of the degrees of freedom.

中文翻译：

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工程计算的数据支持进步：一种稳健的机器学习方法，可加速电磁学和其他学科的变分方法

在这封信中，我们提出并展示了一种基于数据驱动的机器学习的方法来加速有限元法 (FEM)、矩量法 (MoM)、有限差分 (FD) 方法和相关变分方法，同时保持吸引力允许此类方法在计算科学和工程领域（如计算电磁学）中占据主导地位的特性。我们使用神经网络来预测给定问题的一组宏基函数，仅使用对问题极其粗略描述的解决方案作为输入。然后我们使用预测的宏观基础解决问题。与一些现有方法不同，我们的方法不依赖于对解决方案的直接预测。我们展示了我们的宏基函数方法可以纠正网络原始预测中的错误，从而实现更准确的解决方案。给出了一类有限元散射问题的结果，给出了来自 1000 个验证示例的错误统计数据，并与标准和朴素的方法进行了比较。这些结果表明，所描述的宏基函数方法优于直接预测解的机器学习方法。同时，我们的方法实现了与完整解决方案相当的精度，而只需要一小部分自由度。