Machine learning wall effects of eccentric spheres for convenient computation

In confined systems, such as the inside of a biological cell, the outer boundary or wall can affect the dynamics of internal particles. In many cases of interest both the internal particle and outer wall are approximately spherical. Therefore, quantifying the wall effects from an outer spherical boundary on the motion of an internal eccentric sphere is very useful. However, when the two spheres are not concentric, the problem becomes nontrivial. In this paper we improve existing analytical methods to evaluate these wall effects and then train a feed-forward artificial neural network within a broader model. The final model generally performed with $\sim 0.001\%$ error within the training domain and $\sim 0.05\%$ when the outer spherical wall was extrapolated to an infinite plane. Through this model, the wall effects of an outer spherical boundary on the arbitrary motion of an internal sphere for all experimentally achievable configurations can now be conveniently and efficiently determined.

Figure 1

The inner sphere, with radius $a$ is offset from the outer spherical boundary by $\chi$ along the $z$ axis. The minimum clearance distance between boundaries can be related to the radii and vertical offset by $d=b-a-\chi$.

Figure 2

The four distinct motions of the inner sphere. The top and bottom rows distinguish rotation and translation. The left and right columns distinguish axisymmetry and asymmetry.

Figure 3

Histograms of the relative error between model outputs and testing data. The coupling wall effect $f_x^c$ is kept separate because of its larger relative errors. The solid red lines represent the cumulative densities showing the proportion of points less than the given relative error.

Figure 4

The ratio of the asymmetric rotation-translation coupling force and the asymmetric translation is very small. It approaches zero in the concentric limit and decreases as $\lambda$ decreases.

Figure 5

A Q–Q plot comparing the performance of the model on the testing data when using the network and when using interpolation. The blue lines represent distributions of errors from linear interpolations over $11\times 10$ grids of $\frac{d}{b-a}\times \lambda$. The red lines are corresponding results from cubic interpolations. The network outperforms the interpolation methods in all cases.

Figure 6

The relative error of the model when extrapolating from the training region to calculate infinite plane wall effects ($\lambda =0$).

Figure 7

A comparison between approximations for the axisymmetric rotational wall effect in the zero clearance limit. The black dots are the true values of the wall effects [calculated using many terms in sum Eq. (A14)], while the solid lines are the $\lambda \to 0$ and $\lambda \to 1$ approximations given by Eqs. (A16) and (A17), respectively. The relative errors of these approximations are plotted using the dashed lines and correspond to the right vertical axis.

Figure 8

The axisymmetric translational wall effects [Eq. (B10)] subtract the conjectured singular terms [Eq. (B14)] appear to converge towards finite values in the zero clearance limit.