Is Pontryagin's Maximum Principle all you need? Solving optimal control problems with PMP-inspired neural networks

2024-10-08

arXiv.org

AI Summary

This work addresses function optimization and optimal control problems characterized by the absence of ground-truth labels and unknown time domains. We propose PMP-Net, a neural architecture that intrinsically encodes the variational necessary conditions of Pontryagin’s Maximum Principle (PMP) into its structure. Trained end-to-end via an unsupervised loss, PMP-Net requires neither optimal solution labels nor predefined temporal support intervals. The method integrates optimal control theory with differential constraint embedding and automatic differentiation–based optimization. Experiments demonstrate that PMP-Net exactly recovers analytical solutions—including the Kalman filter and bang-bang control—in zero-shot settings. Moreover, it exhibits strong generalization on linear filtering and minimum-time control tasks. Overall, PMP-Net establishes a theoretically grounded, interpretable deep learning framework for solving previously intractable optimal control problems.

Technology Category

• Machine Learning: Optimization
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 is unknown like in minimum-time control problems, so that forward in time solutions are not possible. Calculus of Variations offers a robust framework for learning optimal control and inference. 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, optimal control problems.

Problem

• Solve functional optimization without training data
• Use deep networks and variational principles
• Address problems lacking analytical solutions

Innovation

• Deep networks without training data
• Variational principles in network design
• Unsupervised learning for functional optimization