Pontryagin Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

Abstract

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This work introduces Pontryagin Neural Networks (PoNNs), a specialised subset of Physics-Informed Neural Networks (PINNs) that aim to learn optimal control actions for optimal control problems (OCPs) characterised by integral quadratic cost functions. PoNNs employ the Pontryagin Minimum Principle (PMP) to establish necessary conditions for optimality, resulting in a two-point boundary value problem (TPBVP) that involves both state and costate variables within a system of ordinary differential equations (ODEs). By modelling the unknown solutions of the TPBVP with neural networks, PoNNs effectively learn the optimal control strategies. We also derive upper bounds on the generalisation error of PoNNs in solving these OCPs, taking into account the selection and quantity of training points along with the training error. To validate our theoretical analysis, we perform numerical experiments on benchmark linear and nonlinear OCPs. The results indicate that PoNNs can successfully learn open-loop control actions for the considered class of OCPs, outperforming the commercial software GPOPS-II in terms of both accuracy and computational efficiency. The reduced computational time suggests that PoNNs hold promise for real-time applications.

Keywords

• optimal control problem
• Pontryagin minimum principle
• two points boundary value problem
• physics-informed neural networks
• scientific machine learning