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2023

Convergence Analysis of the Deep Galerkin Method for Weak Solutions

Jiao, Yuling, Lai, Yanming, Wang, Yang et al.

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This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on $\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions, respectively.

In DGMW, a deep neural network is applied to parametrize the PDE solution, and a second neural network is adopted to parametrize the test function in the traditional Galerkin formulation.
By properly choosing the depth and width of these two networks in terms of the number of training samples $n$, it is shown that the convergence rate of DGMW is $\mathcal{O}(n^{-1/d})$, which is the first convergence result for weak solutions.
The main idea of the proof is to divide the error of the DGMW into an approximation error and a statistical error.

Convergence Analysis of the Deep Galerkin Method for Weak Solutions · Around

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