Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning
Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning
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DOI：
10.2139/ssrn.3964424
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发表时间：
2020-12
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影响因子：
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通讯作者：
Fan Chen;J. Huang;Chunmei Wang;Haizhao Yang
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文献类型：
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作者：
Fan Chen;J. Huang;Chunmei Wang;Haizhao Yang
本文提出弗里德里希学习作为一种新颖的深度学习方法，可以通过弗里德里希开创性的极小极大公式来学习偏微分方程的弱解，将偏微分方程问题转化为极小极大优化问题来识别弱解。 “弗里德里希学习”这个名字是为了纪念弗里德里希对弱形式偏微分方程极小极大框架的贡献。弱公式中的弱解和测试函数以无网格方式参数化为深度神经网络，交替更新以分别逼近弱解和最优测试函数的最优解网络。大量的数值结果表明，我们的无网格方法可以为各种维度的规则和不规则域上定义的各种偏微分方程提供相当好的解，而经典数值方法（例如有限差分法和有限元法）可能很繁琐或难以应用。
This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via Friedrichs' seminal minimax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name "Friedrichs learning" is for Friedrichs' contribution to the minimax framework for PDEs in a weak form. 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.