Integrating Machine Learning with Statistical Analysis and Mathematical Modeling for Advanced Population Balance Equation Solutions

This research presents a novel approach to solving complex nonlinear Population Balance Equations (PBEs) by integrating machine learning with statistical analysis and mathematical modeling. We enhance the homotopy analysis technique using neural networks to optimize sub-problem selection and weighting in PBE solutions. This hybrid method transforms PBEs into manageable linear sub-problems, with machine learning dynamically adjusting convergence parameters. Results show remarkable agreement with analytical solutions, demonstrating accuracy improvements up to 95% compared to initial approximations. Statistical analysis confirms the method's robustness across diverse parameters, highlighting its potential for previously intractable particulate process modeling problems. This innovative approach not only advances PBE solving but also showcases the power of interdisciplinary methods in computational science, opening avenues for application in complex particulate systems and real-time process control.