# Backward Reachability Analysis of Neural Feedback Loops: Techniques for Linear and Nonlinear Systems

arXiv: https://arxiv.org/abs/2209.14076 · HTML: https://mlsanity.com/paper/2209.14076/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 56 references that are in the map.
- First posted on arXiv 28 Sep 2022.

## Builds on (most cited first, 30 of 37)

- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [QT-Opt: Scalable Deep Reinforcement Learning for Vision-Based Robotic Manipulation](https://mlsanity.com/paper/1806.10293/) (2018)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Deep Reinforcement Learning for Robotic Manipulation with Asynchronous Off-Policy Updates](https://mlsanity.com/paper/1610.00633/) (2016)
- [Adversarial Examples: Attacks and Defenses for Deep Learning](https://mlsanity.com/paper/1712.07107/) (2017)
- [Sim-to-Real: Learning Agile Locomotion For Quadruped Robots](https://mlsanity.com/paper/1804.10332/) (2018)
- [Learning to Drive in a Day](https://mlsanity.com/paper/1807.00412/) (2018)
- [HG-DAgger: Interactive Imitation Learning with Human Experts](https://mlsanity.com/paper/1810.02890/) (2018)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [DeepDriving: Learning Affordance for Direct Perception in Autonomous Driving](https://mlsanity.com/paper/1505.00256/) (2015)
- [Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances](https://mlsanity.com/paper/1709.07523/) (2017)
- [Semidefinite relaxations for certifying robustness to adversarial examples](https://mlsanity.com/paper/1811.01057/) (2018)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Neural Lyapunov Control](https://mlsanity.com/paper/2005.00611/) (2020)
- [Control of a Quadrotor with Reinforcement Learning](https://mlsanity.com/paper/1707.05110/) (2017)
- [Verisig: verifying safety properties of hybrid systems with neural network controllers](https://mlsanity.com/paper/1811.01828/) (2018)
- [Learning Safe Multi-Agent Control with Decentralized Neural Barrier Certificates](https://mlsanity.com/paper/2101.05436/) (2021)
- [Lyapunov-stable neural-network control](https://mlsanity.com/paper/2109.14152/) (2021)
- [Deep Neural Network Compression for Aircraft Collision Avoidance Systems](https://mlsanity.com/paper/1810.04240/) (2018)
- [Learning Certified Control using Contraction Metric](https://mlsanity.com/paper/2011.12569/) (2020)
- [Actor-Critic Reinforcement Learning for Control with Stability Guarantee](https://mlsanity.com/paper/2004.14288/) (2020)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [OVERT: An Algorithm for Safety Verification of Neural Network Control Policies for Nonlinear Systems](https://mlsanity.com/paper/2108.01220/) (2021)
- [Reachable Set Estimation for Neural Network Control Systems: A Simulation-Guided Approach](https://mlsanity.com/paper/2004.12273/) (2020)

## Built on by (most cited first, 18 of 18)

- [Provably Bounding Neural Network Preimages](https://mlsanity.com/paper/2302.01404/) (2023)
- [Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation](https://mlsanity.com/paper/2404.07956/) (2024)
- [How Generalizable Is My Behavior Cloning Policy? A Statistical Approach to Trustworthy Performance Evaluation](https://mlsanity.com/paper/2405.05439/) (2024)
- [PREMAP: A Unifying PREiMage APproximation Framework for Neural Networks](https://mlsanity.com/paper/2408.09262/) (2024)
- [A Hybrid Partitioning Strategy for Backward Reachability of Neural Feedback Loops](https://mlsanity.com/paper/2210.07918/) (2022)
- [Reachable Polyhedral Marching (RPM): An Exact Analysis Tool for Deep-Learned Control Systems](https://mlsanity.com/paper/2210.08339/) (2022)
- [DRIP: Domain Refinement Iteration with Polytopes for Backward Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2212.04646/) (2022)
- [Constraint-Aware Refinement for Safety Verification of Neural Feedback Loops](https://mlsanity.com/paper/2410.00145/) (2024)
- [TTT: A Temporal Refinement Heuristic for Tenuously Tractable Discrete Time Reachability Problems](https://mlsanity.com/paper/2407.14394/) (2024)
- [Automated Reachability Analysis of Neural Network-Controlled Systems via Adaptive Polytopes](https://mlsanity.com/paper/2212.07553/) (2022)
- [BURNS: Backward Underapproximate Reachability for Neural-Feedback-Loop Systems](https://mlsanity.com/paper/2505.03643/) (2025)
- [Efficient Preimage Approximation for Neural Network Certification](https://mlsanity.com/paper/2505.22798/) (2025)
- [Learning Verifiable Control Policies Using Relaxed Verification](https://mlsanity.com/paper/2504.16879/) (2025)
- [The FABRIC Strategy for Verifying Neural Feedback Systems](https://mlsanity.com/paper/2603.08964/) (2026)
- [Collision Avoidance Verification of Multiagent Systems with Learned Policies](https://mlsanity.com/paper/2403.03314/) (2024)
- [Efficient Reachability Analysis for Convolutional Neural Networks Using Hybrid Zonotopes](https://mlsanity.com/paper/2503.10840/) (2025)
- [A New Strategy for Verifying Reach-Avoid Specifications in Neural Feedback Systems](https://mlsanity.com/paper/2601.08065/) (2026)
- [Stochastic World Models for Verifying Vision-Based Neural Feedback Systems](https://mlsanity.com/paper/2609.38120/) (2026)

## Abstract (arXiv)

As neural networks (NNs) become more prevalent in safety-critical applications such as control of vehicles, there is a growing need to certify that systems with NN components are safe. This paper presents a set of backward reachability approaches for safety certification of neural feedback loops (NFLs), i.e., closed-loop systems with NN control policies. While backward reachability strategies have been developed for systems without NN components, the nonlinearities in NN activation functions and general noninvertibility of NN weight matrices make backward reachability for NFLs a challenging problem. To avoid the difficulties associated with propagating sets backward through NNs, we introduce a framework that leverages standard forward NN analysis tools to efficiently find over-approximations to backprojection (BP) sets, i.e., sets of states for which an NN policy will lead a system to a given target set. We present frameworks for calculating BP over approximations for both linear and nonlinear systems with control policies represented by feedforward NNs and propose computationally efficient strategies. We use numerical results from a variety of models to showcase the proposed algorithms, including a demonstration of safety certification for a 6D system.