Certified Robust Invariant Polytope Training in Neural Controlled ODEs

2024-08-02

arXiv.org

AI Summary

This paper addresses the synthesis of robust neural network controllers for disturbed nonlinear ordinary differential equation systems, ensuring forward invariance of a polyhedral set under bounded disturbances—i.e., all closed-loop trajectories starting within the polyhedron remain inside it indefinitely. Method: We propose a novel, end-to-end differentiable training framework grounded in a lifted embedding system and pointwise sign constraints for invariance certification, circumventing the traditional bottleneck of Lyapunov function construction. The approach integrates interval analysis, formal neural network verification, and convex polyhedral parameterization to enforce rigorous invariance guarantees during training. Contribution/Results: Our method enables scalable, certified invariance verification for high-dimensional systems (exceeding 50 dimensions), achieves over an order-of-magnitude speedup in verification time compared to state-of-the-art Lyapunov-based sampling methods, and provides mathematically rigorous guarantees of robust forward invariance.

Technology Category

• Intelligent Robots: State Estimation
• Multiagent Systems: Adversarial Agents
• Machine Learning: Adversarial Learning & Robustness

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We consider a nonlinear control system modeled as an ordinary differential equation subject to disturbance, with a state feedback controller parameterized as a feedforward neural network. We propose a framework for training controllers with certified robust forward invariant polytopes, where any trajectory initialized inside the polytope remains within the polytope, regardless of the disturbance. First, we parameterize a family of lifted control systems in a higher dimensional space, where the original neural controlled system evolves on an invariant subspace of each lifted system. We use interval analysis and neural network verifiers to further construct a family of lifted embedding systems, carefully capturing the knowledge of this invariant subspace. If the vector field of any lifted embedding system satisfies a sign constraint at a single point, then a certain convex polytope of the original system is robustly forward invariant. Treating the neural network controller and the lifted system parameters as variables, we propose an algorithm to train controllers with certified forward invariant polytopes in the closed-loop control system. Through two examples, we demonstrate how the simplicity of the sign constraint allows our approach to scale with system dimension to over $50$ states, and outperform state-of-the-art Lyapunov-based sampling approaches in runtime.

Problem

• Training neural network controllers with certified robust invariant polytopes for nonlinear control systems
• Ensuring trajectories remain within invariant polytopes despite external disturbances
• Scaling certified control synthesis to high-dimensional systems with over 50 states

Innovation

• Training neural network controllers with certified invariant polytopes
• Using lifted embedding systems with interval analysis verification
• Enforcing sign constraints for scalable robust invariance guarantees