Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations

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 due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods. In particular, for long-term and high-resolution simulations of trajectories, where the number of required time-steps is large, the UDE approach can reduce time-to-solution.

Subjects

cs.LG

math.NA

DDC Class

510: Mathematics

Funding(s)

SFB 1615 - SMARTe Reaktoren für die Verfahrenstechnik der Zukunft

SFB 1615 - Teilprojekt A08: Lagrangesche Bauelemente mit einem validierten Modell zur Vielteilchen-Positionsbestimmung

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https://creativecommons.org/licenses/by/4.0/

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acceptedVersion

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Sommer+et+al_2026_Mach._Learn.__Sci._Technol._10.1088_2632-2153_aeba36.pdf

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1.11 MB

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Adobe PDF

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Approximation of the Basset force in the Maxey-Riley-Gatignol equations via universal differential equations