Physics-Informed Neural Networks for Nonlinear Output Regulation

2025-11-17

AI Summary

This paper addresses the full-information output regulation problem for nonlinear systems—achieving exact tracking and disturbance rejection under complete knowledge of both system and exosystem dynamics. We propose an end-to-end framework based on physics-informed neural networks (PINNs), which, for the first time, directly embeds the governing physical laws of the regulator equations—including coupled partial differential equations and algebraic constraints—into the network architecture. Without requiring prior trajectories or labeled data, the method jointly enforces boundary conditions and feasibility constraints via residual minimization. It directly learns the steady-state manifold mapping and enables real-time reconstruction of the zero-regulation-error manifold. Evaluated on helicopter vertical dynamics and oscillatory platform synchronization tasks, the approach achieves regulation errors converging to machine precision, maintains high accuracy and strong robustness under exosystem parameter variations, and significantly enhances generalization capability.

Technology Category

• Search and Optimization: Learning to Search
• Intelligent Robots: State Estimation
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This work addresses the full-information output regulation problem for nonlinear systems, assuming the states of both the plant and the exosystem are known. In this setting, perfect tracking or rejection is achieved by constructing a zero-regulation-error manifold π(w) and a feedforward input c(w) that render such manifold invariant. The pair (π(w), c(w)) is characterized by the regulator equations, i.e., a system of PDEs with an algebraic constraint. We focus on accurately solving the regulator equations introducing a physics-informed neural network (PINN) approach that directly approximates π(w) and c(w) by minimizing the residuals under boundary and feasibility conditions, without requiring precomputed trajectories or labeled data. The learned operator maps exosystem states to steady state plant states and inputs, enables real-time inference and, critically, generalizes across families of the exosystem with varying initial conditions and parameters. The framework is validated on a regulation task that synchronizes a helicopter's vertical dynamics with a harmonically oscillating platform. The resulting PINN-based solver reconstructs the zero-error manifold with high fidelity and sustains regulation performance under exosystem variations, highlighting the potential of learning-enabled solvers for nonlinear output regulation. The proposed approach is broadly applicable to nonlinear systems that admit a solution to the output regulation problem.

Problem

• Solving nonlinear output regulation via physics-informed neural networks
• Learning zero-error manifold and feedforward input without labeled data
• Generalizing regulator solutions across varying exosystem conditions and parameters

Innovation

• Physics-informed neural networks solve regulator equations
• Directly approximates zero-error manifold and feedforward input
• Enables real-time inference and generalization across exosystems