Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs

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