mlsanityData to 7 Oct 2026

POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems

arXiv 2106.13867 ↗First posted 25 Jun 2021eess.SY cs.LG cs.SYcited by 38

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Abstract (arXiv)

We present POLAR, a polynomial arithmetic-based framework for efficient bounded-time reachability analysis of neural-network controlled systems (NNCSs). Existing approaches that leverage the standard Taylor Model (TM) arithmetic for approximating the neural-network controller cannot deal with non-differentiable activation functions and suffer from rapid explosion of the remainder when propagating the TMs. POLAR overcomes these shortcomings by integrating TM arithmetic with textbfBernstein B'ezier Form and symbolic remainder. The former enables TM propagation across non-differentiable activation functions and local refinement of TMs, and the latter reduces error accumulation in the TM remainder for linear mappings in the network. Experimental results show that POLAR significantly outperforms the current state-of-the-art tools in terms of both efficiency and tightness of the reachable set overapproximation. The source code can be found in https://github.com/ChaoHuang2018/POLARTool

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). Builds on 10 of its 45 references that are in the map. First posted on arXiv 25 Jun 2021.

Builds on

10 of 45 references.

References · 10

  1. 2015Continuous control with deep reinforcement learningcited by 3,279
  2. 2015End-to-End Training of Deep Visuomotor Policiescited by 847
  3. 2017Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networkscited by 739
  4. 2018Efficient Neural Network Robustness Certification with General Activation Functionscited by 439
  5. 2016Safety Verification of Deep Neural Networkscited by 262
  6. 2018Towards Fast Computation of Certified Robustness for ReLU Networkscited by 254
  7. 2018Formal Security Analysis of Neural Networks using Symbolic Intervalscited by 185
  8. 2020NNV: The Neural Network Verification Tool for Deep Neural Networks and Learning-Enabled Cyber-Physical Systemscited by 128
  9. 2018Verisig: verifying safety properties of hybrid systems with neural network controllerscited by 83
  10. 2019ReachNN: Reachability Analysis of Neural-Network Controlled Systemscited by 26

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 11; 2023: 12; 2024: 8; 2025: 4; 2026: 3.

38 papers.

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  1. 2022Enforcing Hard Constraints with Soft Barriers: Safe Reinforcement Learning in Unknown Stochastic Environmentscited by 54
  2. 2023Empowering Autonomous Driving with Large Language Models: A Safety Perspectivecited by 34
  3. 2022Open- and Closed-Loop Neural Network Verification using Polynomial Zonotopescited by 24
  4. 2022Efficient Global Robustness Certification of Neural Networks via Interleaving Twin-Network Encodingcited by 15
  5. 2024Vehicle: Bridging the Embedding Gap in the Verification of Neuro-Symbolic Programscited by 14
  6. 2023POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled Systemscited by 12
  7. 2025Neural Network Verification is a Programming Language Challengecited by 12
  8. 2022Joint Differentiable Optimization and Verification for Certified Reinforcement Learningcited by 11
  9. 2023Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loopscited by 9
  10. 2023Data-Driven Reachability Analysis of Stochastic Dynamical Systems with Conformal Inferencecited by 9
  11. 2024Formal Verification of Graph Convolutional Networks with Uncertain Node Features and Uncertain Graph Structurecited by 8
  12. 2024Provably Safe Neural Network Controllers via Differential Dynamic Logiccited by 7
26 more
  1. 2022Risk Verification of Stochastic Systems with Neural Network Controllerscited by 6
  2. 2023Fully Automatic Neural Network Reduction for Formal Verificationcited by 6
  3. 2023A Toolbox for Fast Interval Arithmetic in numpy with an Application to Formal Verification of Neural Network Controlled Systemscited by 6
  4. 2024Unifying Qualitative and Quantitative Safety Verification of DNN-Controlled Systemscited by 6
  5. 2022BERN-NN: Tight Bound Propagation For Neural Networks Using Bernstein Polynomial Interval Arithmeticcited by 5
  6. 2023Bridging Dimensions: Confident Reachability for High-Dimensional Controllerscited by 5
  7. 2024Real-Time Safe Control of Neural Network Dynamic Models with Sound Approximationcited by 5
  8. 2024Statistical Reachability Analysis of Stochastic Cyber-Physical Systems under Distribution Shiftcited by 4
  9. 2023A Neurosymbolic Approach to the Verification of Temporal Logic Properties of Learning enabled Control Systemscited by 3
  10. 2023Contraction-Guided Adaptive Partitioning for Reachability Analysis of Neural Network Controlled Systemscited by 3
  11. 2024Certified Robust Invariant Polytope Training in Neural Controlled ODEscited by 3
  12. 2024Certified Training with Branch-and-Bound for Lyapunov-stable Neural Controlcited by 3
  13. 2022Reachability Verification Based Reliability Assessment for Deep Reinforcement Learning Controlled Robotics and Autonomous Systemscited by 2
  14. 2023Neural Abstraction-Based Controller Synthesis and Deploymentcited by 2
  15. 2023Forward Invariance in Neural Network Controlled Systemscited by 2
  16. 2026Parallel Differentiable Reachability for Learning and Planning with Certified Neural Dynamics and Controllerscited by 2
  17. 2025Certified Neural Approximations of Nonlinear Dynamicscited by 1
  18. 2026The FABRIC Strategy for Verifying Neural Feedback Systemscited by 1
  19. 2022Polynomial-Time Reachability for LTI Systems with Two-Level Lattice Neural Network Controllerscited by 0
  20. 2022Towards Global Neural Network Abstractions with Locally-Exact Reconstructioncited by 0
  21. 2022Taming Reachability Analysis of DNN-Controlled Systems via Abstraction-Based Trainingcited by 0
  22. 2022Safety Analysis of Autonomous Driving Systems Based on Model Learningcited by 0
  23. 2023Compositional Inductive Invariant Based Verification of Neural Network Controlled Systemscited by 0
  24. 2025Neural Network-assisted Interval Reachability for Systems with Control Barrier Function-Based Safe Controllerscited by 0
  25. 2025Of Good Demons and Bad Angels: Guaranteeing Safe Control under Finite Precisioncited by 0
  26. 2026Probabilistic Verification of Neural Networks via Efficient Probabilistic Hull Generationcited by 0