Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs. (1st May 2022) - Liu, Min; Cai, Zhiqiang
Highlights: Proposing a self-adaptive algorithm (ANE) for designing a nearly optimal two-layer NN for solving PDEs. A method of continuation by the ANE method for providing a good initialization in training the neural network. Analyzing effect of numerical integration. Introducing a posteriori error estimators of recovery type for the ANE method. Demonstrating superior performance of the ANE method for problems with interface singularities and sharp interior layers. Abstract: In the companion paper [1], we introduce adaptive network enhancement (ANE) method for the best least-squares approximation to a target function by using two-layer ReLU neural networks (NNs). In this paper, we apply the ANE method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented.