  In this paper, we leverage the recent advances in physics-informed neural
network (PINN) and develop a generic PINN-based framework to assess the
reliability of multi-state systems (MSSs). The proposed methodology consists of
two major steps. In the first step, we recast the reliability assessment of MSS
as a machine learning problem using the framework of PINN. A feedforward neural
network with two individual loss groups are constructed to encode the initial
condition and state transitions governed by ordinary differential equations
(ODEs) in MSS. Next, we tackle the problem of high imbalance in the magnitude
of the back-propagated gradients in PINN from a multi-task learning
perspective. Particularly, we treat each element in the loss function as an
individual task, and adopt a gradient surgery approach named projecting
conflicting gradients (PCGrad), where a task's gradient is projected onto the
norm plane of any other task that has a conflicting gradient. The gradient
projection operation significantly mitigates the detrimental effects caused by
the gradient interference when training PINN, thus accelerating the convergence
speed of PINN to high-precision solutions to MSS reliability assessment. With
the proposed PINN-based framework, we investigate its applications for MSS
reliability assessment in several different contexts in terms of
time-independent or dependent state transitions and system scales varying from
small to medium. The results demonstrate that the proposed PINN-based framework
shows generic and remarkable performance in MSS reliability assessment, and the
incorporation of PCGrad in PINN leads to substantial improvement in solution
quality and convergence speed.
