Deep Learning for Singular PDEs: A Weighted Neural Network Approach

Abstract

We propose a deep learning framework for the numerical approximation of weakly singular semilinear elliptic equations subject to homogeneous Dirichlet boundary conditions and positivity constraints. These problems are challenging because the nonlinear source term becomes unbounded as the positive solution approaches zero near the boundary. The proposed approach combines three components. First, the Dirichlet boundary conditions are embedded directly into the neural representation through a hard-constraint formulation. Second, a Softplus transformation preserves positivity in the interior of the computational domain, ensuring that the singular nonlinear term remains well-defined during training. Third, a singularity-aware weighted residual loss emphasizes regions where the predicted solution becomes small and the singular behavior is more pronounced. The method is evaluated in the weak singularity regime through numerical reference solutions and controlled verification tests. Comparisons with the standard residual formulation are used to assess approximation accuracy and convergence behavior. The numerical experiments illustrate the potential of combining hard boundary enforcement, positivity preservation, and singularity-aware residual weighting for weakly singular elliptic problems.