  In this paper, we study adaptive neuron enhancement (ANE) method for solving
self-adjoint second-order elliptic partial differential equations (PDEs). The
ANE method is a self-adaptive method generating a two-layer spline NN and a
numerical integration mesh such that the approximation accuracy is within the
prescribed tolerance. Moreover, the ANE method provides a natural process for
obtaining a good initialization which is crucial for training nonlinear
optimization problem.
  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.
