Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs
CoRR(2023)
摘要
In this paper, we provide a theoretical analysis of the recently introduced
weakly adversarial networks (WAN) method, used to approximate partial
differential equations in high dimensions. We address the existence and
stability of the solution, as well as approximation bounds. More precisely, we
prove the existence of discrete solutions, intended in a suitable weak sense,
for which we prove a quasi-best approximation estimate similar to Cea's lemma,
a result commonly found in finite element methods. We also propose two new
stabilized WAN-based formulas that avoid the need for direct normalization.
Furthermore, we analyze the method's effectiveness for the Dirichlet boundary
problem that employs the implicit representation of the geometry. The key
requirement for achieving the best approximation outcome is to ensure that the
space for the test network satisfies a specific condition, known as the inf-sup
condition, essentially requiring that the test network set is sufficiently
large when compared to the trial space. The method's accuracy, however, is only
determined by the space of the trial network. We also devise a pseudo-time
XNODE neural network class for static PDE problems, yielding significantly
faster convergence results than the classical DNN network.
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关键词
weak adversarial network discretizations,essential boundary conditions,boundary conditions,best approximation results
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