The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects caused by the formation of wakes and by boundary layer effects. As a result, the force acting on a particle depends on its past trajectory, which complicates the numerical solution of the equations. The Basset force is therefore often neglected, despite substantial evidence that it affects the movement patterns of modelled particles both quantitatively and qualitatively.
Using the concept of universal differential equations (UDE), the authors approximate the history term with neural networks. This turns the MaRGE into a system of ordinary differential equations that can be solved with standard numerical solvers such as Runge-Kutta methods, without a bespoke implementation. Two network types were compared, a feedforward network and a long short-term memory network, each with around 13,000 trainable parameters.
The approach was tested on an analytical vortex field in three dimensions and on a flow field interpolated from experimental data of a stirred tank reactor on laboratory scale. In the vortex field, both networks reproduced the particle trajectories with an average relative deviation of about 0.5 percent, two orders of magnitude more accurate than neglecting the Basset force. In the experimental flow field, the accuracy improved by one order of magnitude compared with neglecting the history term.
Especially for long simulations of trajectories with high resolution, where a large number of time steps is required, the UDE approach can reduce the time to solution.
The study was carried out at the Chair of Computational Mathematics at Hamburg University of Technology within the Collaborative Research Centre CRC 1615 (SMART Reactors), with the flow data of the stirred tank reactor provided by colleagues at HAW Hamburg.
Finn Sommer, Vamika Rathi, Sebastian Götschel, Daniel Ruprecht (2026). Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations. arXiv:2604.08194.
