An Intelligent SPH Framework Based on Machine-Learned Residual Correction for Elliptic PDEs

We present an intelligent, non-intrusive framework to enhance the performance of Symmetric Smoothed Particle Hydrodynamics (SSPH) for elliptic partial differential equations, focusing on the linear and nonlinear Poisson equations. Classical Smoothed Particle Hydrodynamics methods, while meshfree, suffer from discretization errors due to kernel truncation and irregular particle distributions. To address this, we employ a machine-learning-based residual correction, where a neural network learns the difference between the SSPH solution and a reference solution. The predicted residuals are added to the SSPH solution, yielding a corrected approximation with significantly reduced errors. The method preserves numerical stability and consistency while systematically reducing errors. Numerical results demonstrate that the proposed approach outperforms standard SSPH.