Modeling 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 particle
propagation could be done in parallel. Yet the simulations with many aggregate
sizes remain quite resource-demanding. Here, we explore the way to reduce the
amount of direct computations with the use of modern machine learning (ML)
techniques. Namely, we propose to replace the actual numerical solution of the
Smoluchowki equations with the respective density transformations learned with
the application of the conditional normalising flow. We demonstrate that the ML
predictions for the space distribution of aggregates and their size
distribution requires drastically less computation time and agrees 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 online prediction and visualisation of
pollution processes, providing a tool with a reasonable tradeoff between the
prediction accuracy and the computational time.

使用现代机器学习技术通过条件归一化流学习取代 Smoluchowski 方程的数值解，可以显著减少计算时间并与直接数值模拟结果较好地吻合。