# Certified Robust Invariant Polytope Training in Neural Controlled ODEs

arXiv: https://arxiv.org/abs/2408.01273 · HTML: https://mlsanity.com/paper/2408.01273/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 41 references that are in the map.
- First posted on arXiv 2 Aug 2024.

## Builds on (most cited first, 24 of 24)

- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [On the Effectiveness of Interval Bound Propagation for Training Verifiably Robust Models](https://mlsanity.com/paper/1810.12715/) (2018)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systems](https://mlsanity.com/paper/2004.05519/) (2020)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [First Three Years of the International Verification of Neural Networks Competition (VNN-COMP)](https://mlsanity.com/paper/2301.05815/) (2023)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Adversarial Robustness of Deep Neural Networks: A Survey from a Formal Verification Perspective](https://mlsanity.com/paper/2206.12227/) (2022)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems](https://mlsanity.com/paper/2106.13867/) (2021)
- [Verification of Neural-Network Control Systems by Integrating Taylor Models and Zonotopes](https://mlsanity.com/paper/2112.09197/) (2021)
- [Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopes](https://mlsanity.com/paper/2207.02715/) (2022)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [Interval Reachability of Nonlinear Dynamical Systems with Neural Network Controllers](https://mlsanity.com/paper/2301.07912/) (2023)
- [Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loops](https://mlsanity.com/paper/2307.14938/) (2023)
- [Safety verification of Neural-Network-based controllers: a set invariance approach](https://mlsanity.com/paper/2312.11352/) (2023)
- [textttimmrax: A Parallelizable and Differentiable Toolbox for Interval Analysis and Mixed Monotone Reachability in JAX](https://mlsanity.com/paper/2401.11608/) (2024)
- [FI-ODE: Certifiably Robust Forward Invariance in Neural ODEs](https://mlsanity.com/paper/2210.16940/) (2022)
- [Forward Invariance in Neural Network Controlled Systems](https://mlsanity.com/paper/2309.09043/) (2023)

## Built on by (most cited first, 3 of 3)

- [Certified Training with Branch-and-Bound for Lyapunov-stable Neural Control](https://mlsanity.com/paper/2411.18235/) (2024)
- [Provably-Safe Neural Network Training Using Hybrid Zonotope Reachability Analysis](https://mlsanity.com/paper/2501.13023/) (2025)
- [ShardNet: Training Neural Controllers with Hard, Non-Convex Constraints](https://mlsanity.com/paper/2606.30935/) (2026)

## Abstract (arXiv)

We propose a framework for training neural network controllers with certified robust forward invariant polytopes. 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.