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

TL;DR

This paper introduces a neural network-based approximation of the Basset force in the Maxey-Riley-Gatignol equations, simplifying their numerical solution while preserving important history effects.

Contribution

It proposes a novel method using universal differential equations to approximate the history-dependent Basset force with neural networks, enabling standard ODE solvers.

Findings

Neural network approximation effectively models the Basset force.

The approach simplifies numerical simulation of MaRGE.

Preserves key qualitative and quantitative effects of history terms.

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.