Data-driven modeling of a settling sphere in a quiescent medium

We present two data-driven models for predicting the dynamics of a freely settling sphere in a quiescent Newtonian fluid, trained on experimentally measured trajectories obtained via particle tracking velocimetry (PTV). Unlike traditional methods that explicitly resolve fluid-structure interactions, our models learn the sphere's motion directly from data. We implement deterministic neural ordinary differential equations (NODEs) and stochastic neural stochastic differential equations (NSDEs) to reconstruct 3D trajectories and analyze key statistical features of the settling process. Model performance is evaluated using short- and long-time dynamics, including ensemble-averaged velocity evolution, settling time distributions, and final position statistics. We also assess the correlations between displacements and streamwise velocity, as well as the effect of training dataset size on predictive accuracy. Results show that NODEs excel in trajectory reconstructions and generalization across varying initial conditions, while NSDEs effectively capture statistical trends in the long-time behavior but are more sensitive to data availability. Acceleration estimates derived from second-order finite differences confirm both model's ability to accurately represent long-time dynamics, though capturing short-time transients poses challenges.