The Hard-Constraint PINNs for Interface Optimal Control Problems
Ming-Chih Lai, Yongcun Song, Xiaoming Yuan, Hangrui Yue, Tianyou Zeng
TL;DR
This work tackles optimal control problems constrained by PDEs with interfaces, where states exhibit jumps across an interface and controls are bounded. It advances a hard-constraint physics-informed neural network (PINN) framework that enforces boundary and interface conditions exactly (or nearly so) via a novel architecture that augments the neural solution with an auxiliary function $\phi$, effectively decoupling constraint satisfaction from PDE learning. The method extends to both elliptic and parabolic interface OCIPs and to scenarios with interface controls, delivering mesh-free solutions that satisfy the constraints rigorously and exhibit improved accuracy over soft-constraint PINNs. The results demonstrate strong numerical performance, including comparisons to immersed FEM and extensions to time-dependent problems, highlighting the approach’s potential for scalable, high-accuracy interface problems in engineering and physics.
Abstract
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 or with a high degree of accuracy, and they are decoupled from the learning of the PDEs. Its efficiency is promisingly validated by some elliptic and parabolic interface optimal control problems.