Pontryagin’s Maximum Principles Meets Deep Learning: A New Paradigm for Optimal Control

Calculus of Variations provides a powerful mathematical framework for learning optimal control and inference by establishing necessary conditions for optimality, known as Pontryagin’s Maximum Principle (PMP). However, these conditions are often difficult to solve analytically. At the same time, neural networks excel at learning from data and modeling complex, high-dimensional patterns. How can we combine the strengths of PMP with the representational power of neural networks? In this work, we introduce PMP-Net—a neural network model that integrates the mathematical framework of PMP to estimate control and inference solutions. PMP-Net successfully recovers classical solutions such as the Kalman filter and bang-bang control. This establishes a new approach for addressing general, possibly yet unsolved, optimal control problems.