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

2026-04-09

AI Summary

This work addresses the computational expense of the Basset history force in the Maxey–Riley–Gatignol equation, which is often neglected despite its significant influence on particle dynamics. To overcome this challenge, the authors propose, for the first time, embedding a neural network within the universal differential equations (UDE) framework to data-drivenly approximate the Basset history term. This approach effectively transforms the original integro-differential equation into a standard system of ordinary differential equations, eliminating the need for explicit evaluation of the costly history integral. Consequently, classical ODE solvers such as Runge–Kutta can be employed directly, yielding high-fidelity simulations of particle motion with substantially improved computational efficiency while preserving physical accuracy.

Abstract

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.

Problem

• Basset force
• Maxey-Riley-Gatignol equations
• history effects
• inertial particles
• numerical solution

Innovation

• Basset force
• Maxey-Riley-Gatignol equations
• universal differential equations
• neural networks
• history effects