  Recently, physics-informed neural networks (PINNs) have emerged as a
promising method for solving partial differential equations (PDEs). In this
study, we establish a deep learning computational framework, HL-nets, for
computing the flow field of hydrodynamic lubrication involving cavitation
effects. Two classical cavitation conditions, i.e., the Swift-Stieber (SS)
condition and the Jakobsson-Floberg-Olsson (JFO) condition, are implemented in
the PINNs to solve the Reynolds equation. For the non-negativity constraint of
the SS cavitation condition, a penalizing scheme with a residual of the
non-negativity and an imposing scheme with a continuous differentiable
non-negative function are proposed. For the complementarity constraint of the
JFO cavitation condition, the pressure and cavitation fraction are taken as the
neural network outputs, and the residual of the Fischer-Burmeister (FB)
equation constrains their complementary relationships. Multi-task learning
(MTL) methods are applied to balance the newly introduced loss terms described
above. To estimate the accuracy of HL-nets, we present a numerical solution of
the Reynolds equation for oil-lubricated bearings involving cavitation. The
results indicate that the proposed HL-nets can highly accurately simulate
hydrodynamic lubrication involving cavitation phenomena. The imposing scheme
can effectively improve the accuracy of the training results of PINNs, and it
is expected to have great potential to be applied to different fields where the
non-negativity constraint is needed.
