  The secondary Bjerknes force plays a significant role in the evolution of
bubble clusters. However, due to the complex dependence of the force on
multiple parameters, it is highly non-trivial to include the effects of this
force in the simulations of bubble clusters. In this paper, machine learning is
used to develop a data-driven model for the secondary Bjerknes force between
two insonated bubbles as a function of the equilibrium radii of the bubbles,
the distance between the bubbles, the amplitude and the frequency of the
pressure. The force varies over several orders of magnitude, which poses a
serious challenge for the usual machine learning models. To overcome this
difficulty, the magnitudes and the signs of the force are separated and
modelled separately. A nonlinear regression is obtained with a feed-forward
network model for the logarithm of the magnitude, whereas the sign is modelled
by a support-vector machine model. The principle, the practical aspects related
to the training and validation of the machine models are introduced. The
predictions from the models are checked against the values computed from the
Keller-Miksis equations. The results show that the models are extremely
efficient while providing accurate estimate of the force. The models make it
computationally feasible for the future simulations of the bubble clusters to
include the effects of the secondary Bjerknes force.
