论文标题
Friedrichs学习：通过深度学习的部分微分方程的薄弱解决方案
Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning
论文作者
论文摘要
本文提出，弗里德里奇（Friedrichs）学习是一种新颖的深度学习方法，可以通过Minmax公式学习PDE的弱解决方案，这将PDE问题转化为最小值优化问题以识别弱解决方案。 “ Friedrichs Learning”这个名称是为了强调我们的学习策略与PDES对称系统的Friedrichs理论之间的紧密关系。弱的解决方案和弱公式中的测试功能以无网格的方式被参数化为深神经网络，该方式分别更新以接近近似弱解决方案和最佳测试功能的最佳解决方案网络。广泛的数值结果表明，我们的无网格方法可以为在各个维度上定义在常规和不规则域上定义的广泛的PDE提供合理的解决方案，在各个维度上，经典的数值方法（例如有限差异方法和有限元方法）可能是乏味的或难以应用的。
This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minmax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name "Friedrichs learning" is for highlighting the close relationship between our learning strategy and Friedrichs theory on symmetric systems of PDEs. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free method can provide reasonably good solutions to a wide range of PDEs defined on regular and irregular domains in various dimensions, where classical numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied.
