A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs (Communications in Computational Physics, 2023)
In this paper, we give the first rigorous error estimation of the Weak Adversarial Neural Networks (WAN) in solving the second order parabolic PDEs. Bydecomposing the error into approximation error and statistical error, we first show theweak solution can be approximated by the $ReLU^2$ with arbitrary accuracy, then provethat the statistical error can also be efficiently bounded by the Rademacher complexityof the network functions, which can be further bounded by some integral related withthe covering numbers and pseudo-dimension of $ReLU^2$ space. Finally, by combiningthe two bounds, we prove that the error of the WAN method can be well controlled ifthe depth and width of the neural network as well as the sample numbers have beenproperly selected. Our result also reveals some kind of freedom in choosing samplenumbers on $∂Ω$ and in the time axis.