arXiv 2008.014912020-08-04math.NAcs.NA

深层混合残量法中精确边界条件和初始条件的强化

Enforcing exact boundary and initial conditions in the deep mixed residual method

Liyao Lyu, Keke Wu, Rui Du, Jingrun Chen

摘要：

在理论上，边界条件和初始条件对于偏微分方程（PDE）的适定性是非常重要的。在数值上，这些条件可以在经典的数值方法，如有限差分法和有限元法中精确地实现。近年来，人们对深度神经网络（DNN）解决偏微分方程的兴趣越来越大，特别是在高维情况下。然而，在一般情况下，仔细的文献回顾表明，边界条件不能完全强制DNN，这不可避免地导致建模错误。在这项工作中，基于最近开发的深度混合残差方法（MIM），我们演示了如何使DNN系统地自动满足边界和初始条件。因此，MIM中的损失函数没有惩罚项，并且没有任何建模误差。使用大量的例子，包括狄利克雷，诺依曼，混合，罗宾，和周期边界条件的椭圆型方程，抛物和双曲型方程的初始条件，我们表明，强制执行精确的边界和初始条件不仅提供了一个更好的近似解，但也有利于训练过程。

英文摘要：

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.

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。

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