论文标题
神经网络增强方法的多尺度框架，用于解决部分微分方程的解决方案
A multi-scale framework for neural network enhanced methods to the solution of partial differential equations
论文作者
论文摘要
在目前的工作中，提出了一个用于神经网络增强方法的多尺度框架，以近似和偏微分方程（PDE）的函数和解决方案。通过引入多尺度概念，可以将目标问题的总解决方案分解为两个部分，即粗尺度解决方案和精细的比例解决方案。在粗刻度中，应用了常规数值方法（例如有限元方法），并可以获得粗略的解决方案。在细节中，引入了神经网络以制定解决方案。自定义损失函数是通过考虑PDE的管理方程和边界条件，约束和从粗尺度的相互作用来开发的。通过各种测试案例对所提出的方法进行了说明和检查。
In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of the target problem could be decomposed into two parts, i.e. the coarse scale solution and the fine scale solution. In the coarse scale, the conventional numerical methods (e.g. finite element methods) are applied and the coarse scale solution could be obtained. In the fine scale, the neural networks is introduced to formulate the solution. The custom loss functions are developed by taking into account the governing equations and boundary conditions of PDEs, the constraints and the interaction from coarse scale. The proposed methods are illustrated and examined by various of testing cases.
