Application of machine learning technique for a fast forecast of aggregation kinetics in space-inhomogeneous systems

Modelling of aggregation processes in space-inhomogeneous systems is extremely numerically challenging since complicated aggregation equations, Smoluchowski equations, are to be solved at each space point along with the computation of particle propagation. Low rank approximation for the aggregation kernels can significantly speed up the solution of Smoluchowski equations, while the particle propagation could be done in parallel. Yet the numerics with many aggregate sizes remains quite resource-demanding. Here, we explore the way to reduce the amount of direct computations by replacing the actual numerical solution of the Smoluchowski equations with the respective density transformations learned with the application of one of machine learning (ML) methods, the conditional normalising flow. We demonstrate that the ML predictions for the space distribution of aggregates and their size distribution require drastically shorter computation time and agree fairly well with the results of direct numerical simulations. Such an opportunity of a quick forecast of space-dependent particle size distribution could be important in practice, especially for the fast (on the timescale of data reading) prediction and visualisation of pollution processes, providing a tool with a reasonable trade off between the prediction accuracy and the computational time.