A residual-based adaptive deep learning hybrid block algorithm for partial differential equations with singularities

Abstract

This paper presents an algorithm composed of a class of hybrid block methods blended with a class of neural network optimisation algorithms. The neural network was introduced to enhance the accuracy and stability of numerical hybrid solutions. The hybrid framework synergises the precision of advanced block numerical methods with the function approximation and generalisation capabilities of a radial basis function neural network (RBFNN). By leveraging the strengths of both approaches, the proposed method yields solutions that are computationally efficient and robust. The numerical results obtained from block methods serve as inputs. At the same time, the exact solutions are used as targets to train the RBFNN, enabling the network to refine the solution and effectively handle complex boundary conditions and singularities. This extension significantly improves upon the limitations of existing approaches, particularly in dealing with singular behaviours and achieving superior accuracy. The proposed method is applied to solve a range of challenging partial differential equations, demonstrating its robustness and effectiveness. Comparative analysis highlights its advantages over traditional numerical methods and standalone neural network approaches in terms of accuracy, convergence, and computational efficiency. This hybrid framework establishes a promising direction for integrating numerical methods and machine learning to solve complex mathematical and engineering problems.