Is Pontryagin's Maximum Principle All You Need? Solving optimal control problems with PMP-inspired neural networks

Kawisorn Kamtue, Jose M F Moura, Orathai Sangpetch

Keywords: prior knowledge, Pontryagin's Maximum Principle, optimal control

TL;DR: Incorporating Pontryagin's Maximum Principle into neural networks to learn optimal control

Abstract: Calculus of Variations is the mathematics of functional optimization, i.e., when the solutions are functions over a time interval. This is particularly important when the time interval, or support, is unknown like in minimum-time control problems, so that forward-in-time solutions are not possible. Calculus of Variations also offers a robust framework for learning optimal control and inference with moving boundaries. How can this framework be leveraged to design neural networks to solve challenges in control and inference? We propose the Pontryagin's Maximum Principle Neural Network (PMP-Net) that is tailored to estimate control and inference solutions, in accordance with the necessary conditions outlined by Pontryagin’s Maximum Principle. We assess PMP-Net on two classic optimal control and inference problems: optimal linear filtering and minimum-time control. Our findings indicate that PMP-Net can be effectively trained in an unsupervised manner to solve these problems without the need for ground-truth data, successfully deriving the classical ''Kalman filter'' and "bang-bang'' control solution. This establishes a new approach for addressing general, possibly yet unsolved, inference and optimal control problems.

Primary Area: learning theory

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Submission Number: 11251