Deep learning-complex variable meshless method for inverse Poisson problems

This study proposes a deep learning-complex variable meshless method (DL-CVMM) framework that integrates deep neural networks (DNNs) with an improved complex variable element-free Galerkin (ICVEFG) method for solving inverse Poisson problem. The framework takes 2D coordinates as input and predicts source terms via forward propagation in DNNs. These predicted source terms are then incorporated into the ICVEFG discretization scheme to reconstruct the physical field. The inverse problem is formulated as an optimization problem by minimizing the empirical risk function in the problem domain between the reconstructed and observed values. This framework leverages DNNs for source term prediction, harnessing their generalization and learning capabilities, while employing the ICVEFG method for efficient physical field reconstruction. A key advantage of ICVEFG compared to the element-free Galerkin (EFG) method is its reduction in the number of unknown coefficients, reducing the matrix order in the shape function computations. Rigorous validations on three 2D Poisson inverse problem benchmarks demonstrate that the DL-CVMM framework achieves good convergence and computational accuracy.