The search for numerical methods that combine accuracy, stability, and adaptability is essential for modeling complex physical phenomena governed by diffusion processes, such as heat conduction, viscous flow, and mass transport. In this context, the Smoothed Particle Hydrodynamics (SPH) method stands out for its mesh-free Lagrangian formulation, making it a promising tool for thermal and diffusive applications in computational mechanics and fluid dynamics. This work proposes a comprehensive approach to evaluate, improve, and automate SPH for solving parabolic partial differential equations. Robust implementations were developed to solve the Poisson equation and heat conduction problems in steady-state and transient regimes, in one- and two-dimensional domains, under Dirichlet and Neumann boundary conditions, using different kernel functions. To identify the ideal kernel function for each configuration, several machine learning algorithms were incorporated, including Extreme Learning Machine (ELM), Multilayer Perceptron (MLP), Random Forest (RF), and Extreme Gradient Boosting (XGBoost). These models enable the automatic selection of the most suitable configuration according to variations in the compact support scaling factor. Based on thousands of SPH simulations, the predictive models achieved high accuracy in choosing optimal kernel configurations, eliminating the need for exhaustive parameter sweeps. The integration of SPH and machine learning reduced computational time by up to five orders of magnitude in transient two-dimensional cases while maintaining accuracy above 97%. This data-driven strategy demonstrates that supervised learning can effectively replace empirical calibration without compromising solution quality, establishing a solid methodological foundation for predictive numerical modeling in Physics-Informed Machine Learning (PIML). The proposed framework is directly extensible to other SPH applications including multiphase flows, fluid–structure interaction, and coupled thermo-mechanical problems.
Augusto et al. (Sun,) studied this question.