The Hard-Constraint PINNs for Interface Optimal Control Problems
Ming-Chih Lai Yongcun Song Xiaoming Yuan Hangrui Yue Tianyou Zeng
We show that the physics-informed neural networks (PINNs), in combination
with some recently developed discontinuity capturing neural networks, can be
applied to solve optimal control problems subject to partial differential
equations (PDEs) with interfaces and some control constraints. The resulting
algorithm is mesh-free and scalable to different PDEs, and it ensures the
control constraints rigorously. Since the boundary and interface conditions, as
well as the PDEs, are all treated as soft constraints by lumping them into a
weighted loss function, it is necessary to learn them simultaneously and there
is no guarantee that the boundary and interface conditions can be satisfied
exactly. This immediately causes difficulties in tuning the weights in the
corresponding loss function and training the neural networks. To tackle these
difficulties and guarantee the numerical accuracy, we propose to impose the
boundary and interface conditions as hard constraints in PINNs by developing a
novel neural network architecture. The resulting hard-constraint PINNs approach
guarantees that both the boundary and interface conditions can be satisfied
exactly and they are decoupled from the learning of the PDEs. Its efficiency is
promisingly validated by some elliptic and parabolic interface optimal control
problems.