The Hard-Constraint PINNs for Interface Optimal Control Problems

TL;DR

This paper introduces a novel hard-constraint PINNs approach for solving interface optimal control problems governed by PDEs, ensuring exact satisfaction of boundary and interface conditions and improving training accuracy.

Contribution

The paper develops a new neural network architecture that enforces boundary and interface conditions as hard constraints within PINNs, enhancing solution accuracy and robustness.

Findings

Guarantees exact satisfaction of boundary and interface conditions.

Demonstrates effectiveness on elliptic and parabolic interface control problems.

Improves training stability and accuracy over soft-constraint PINNs.

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.