Data-driven machine learning approach based on physics-informed neural network for population balance model

Population balance models (PBM) are a fundamental tool in the field of process engineering and materials science for understanding and forecasting particulate system dynamics. These models are essential for the design and optimization of a wide range of industrial processes because they capture the evolution of particle size distribution in response to different phenomena such as nucleation, growth, aggregation, and breakage. The complex and non-linear nature of these models makes it difficult to find the analytical solution and even sometimes the approximate solution. This paper introduces a novel machine learning approach based on physics-informed neural network (PINN) for the approximate solution of PBM. Our strategy utilizes the use of a customized neural network framework that has been developed and trained on data generated from simulated PBM by using a finite difference method for the differential operators to identify the underlying dynamics and patterns controlling the evolution of particle distribution. In general, PINN uses automatic differentiation for the computation of differential operators, which is based on the chain rule and needs several matrix operations for computing, which reduces the processing efficiency during the training. The PINN approach provides a flexible, effective and highly adaptable solution framework by utilizing neural networks to approximate the solution and defining the loss function as the sum of the differential equation’s residuals at specific random or regular points inside the domain, as well as the residuals of the initial and boundary conditions. It is shown numerically that this approach does not need a diffusion term for a stable solution, which is often needed in most numerical methods for solving these models. For further validation, we compare the results with the exact solution and find them with a very good agreement with each other.