论文标题
在深层混合剩余方法中执行精确边界和初始条件
Enforcing exact boundary and initial conditions in the deep mixed residual method
论文作者
论文摘要
从理论上讲，边界和初始条件对于部分微分方程（PDE）的良好性很重要。从数值上讲，这些条件可以通过经典的数值方法（例如有限差异方法和有限元方法）来执行。近年来，深层神经网络（DNN）尤其是在高维情况下，对解决PDE的兴趣日益增强。但是，在一般情况下，仔细的文献综述表明，边界条件不能完全针对DNN执行，这不可避免地导致建模误差。在这项工作中，基于最近开发的深层残留方法（MIM），我们演示了如何以系统的方式自动满足边界和初始条件。结果，MIM中的损失函数不含罚款项，并且没有任何建模误差。使用许多示例，包括Dirichlet，Neumann，Mixed，Robin和椭圆方程的周期性边界条件，以及抛物线和双曲线方程的初始条件，我们表明，执行精确的边界和初始条件不仅提供了更好的近似解决方案，还提供了训练过程。
In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method and finite element method. Recent years have witnessed growing interests in solving PDEs by deep neural networks (DNNs), especially in the high-dimensional case. However, in the generic situation, a careful literature review shows that boundary conditions cannot be enforced exactly for DNNs, which inevitably leads to a modeling error. In this work, based on the recently developed deep mixed residual method (MIM), we demonstrate how to make DNNs satisfy boundary and initial conditions automatically in a systematic manner. As a consequence, the loss function in MIM is free of the penalty term and does not have any modeling error. Using numerous examples, including Dirichlet, Neumann, mixed, Robin, and periodic boundary conditions for elliptic equations, and initial conditions for parabolic and hyperbolic equations, we show that enforcing exact boundary and initial conditions not only provides a better approximate solution but also facilitates the training process.
