Physics-Informed Neural Networks for PDE-Constrained Optimization and Control

• Published in: Communications on Applied Mathematics and Computation (Online)
• Main Authors: Barry-Straume, Jostein, Sarshar, Arash, Popov, Andrey A., Sandu, Adrian
• Format: Journal Article
• Language: English
• Published: 18.08.2025
• ISSN: 2096-6385; 2661-8893; 2661-8893
• DOI: 10.1007/s42967-025-00499-x

The goal of optimal control is to determine a sequence of inputs for maximizing or minimizing a given performance criterion subject to the dynamics and constraints of the system under observation. This work introduces Control Physics-Informed Neural Networks (PINNs), which simultaneously learn both the system states and the optimal control signal in a single-stage framework that leverages the system’s underlying physical laws. While prior approaches often follow a two-stage process-modeling, the system first and then devising its control—the presented novel framework embeds the necessary optimality conditions directly into the network architecture and loss function. We demonstrate the effectiveness of the novel methodology by solving various open-loop optimal control problems governed by analytical, one-dimensional, and two-dimensional partial differential equations (PDEs).