research-article

Synthesizing ReLU neural networks with two hidden layers as barrier certificates for hybrid systems

Authors:

Zhengfeng Yang,

Xuandong LiAuthors Info & Claims

HSCC '21: Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control

Article No.: 17, Pages 1 - 11

https://doi.org/10.1145/3447928.3456638

Published: 19 May 2021 Publication History

Abstract

Barrier certificates provide safety guarantees for hybrid systems. In this paper, we propose a novel approach to synthesizing neural networks as barrier certificates. Candidate networks are trained from a special structure: ReLU neural networks consisting of two hidden layers. Then, the problem of identifying real barrier certificates from candidates is transformed into a group of mixed integer linear programming problems and a mixed integer quadratically constrained problem. Taking full advantage of the recent advance in optimization, barrier certificates validation can be performed effectively. We implement the tool SyntheBC and evaluate its performance over 3 hybrid systems and 8 continuous systems up to 12-dimensional state space. The experimental results show that our method is more scalable and effective than the classical polynomial barrier certificate method and the existing neural network based method.

References

[1]

Hirokazu Anai and Volker Weispfenning. 2001. Reach Set Computations Using Real Quantifier Elimination. In Proceedings of the 4th International Workshop on Hybrid Systems: Computation and Control (HSCC '01, Vol. 14). Springer, London, UK, 63--76.

[2]

Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe. 2004. Convex optimization. Cambridge university press.

Digital Library

[3]

George Cybenko. 1989. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems 2, 4 (1989), 303--314.

[4]

Liyun Dai, Ting Gan, Bican Xia, and Naijun Zhan. 2017. Barrier Certificates Revisited. Journal of Symbolic Computation 80 (2017), 62--86.

Digital Library

[5]

Souradeep Dutta, Susmit Jha, Sriram Sankaranarayanan, and Ashish Tiwari. 2018. Output range analysis for deep feedforward neural networks. In NASA Formal Methods Symposium. Springer, 121--138.

[6]

Chuchu Fan and Sayan Mitra. 2015. Bounded Verification with On-the-Fly Discrepancy Computation. In Automated Technology for Verification and Analysis, Bernd Finkbeiner, Geguang Pu, and Lijun Zhang (Eds.). Springer International Publishing, Cham, 446--463.

[7]

Sicun Gao. [n.d.]. Quadcopter Model. ([n. d.]). https://github.com/dreal/benchmarks/blob/master/inv/quadcopter_nonlinear.inv.

[8]

Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. 2016. Deep learning. Vol. 1. MIT press Cambridge.

Digital Library

[9]

Eric Goubault, J-H Jourdan, Sylvie Putot, and Sriram Sankaranarayanan. 2014. Finding non-polynomial positive invariants and Lyapunov functions for polynomial systems through Darboux polynomials. In Proceedings of the 2014 American Control Conference (ACC). IEEE, 3571--3578.

[10]

Sumit Gulwani and Ashish Tiwari. 2008. Constraint-Based Approach for Analysis of Hybrid Systems. In Proc. of the 20th International Conference on Computer Aided Verification (CAV). 190--203.

Digital Library

[11]

Incorporate Gurobi Optimization. 2020. Gurobi optimizer reference manual. URL https://www.gurobi.com (2020).

[12]

Kurt Hornik, Maxwell B. Stinchcombe, and Halbert White. 1989. Multilayer feedforward networks are universal approximators. Neural Networks 2, 5 (1989), 359--366.

[13]

James Kapinski and Jyotirmoy Deshmukh. 2015. Discovering forward invariant sets for nonlinear dynamical systems. In Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. Springer, 259--264.

[14]

James Kapinski, Jyotirmoy V Deshmukh, Sriram Sankaranarayanan, and Nikos Aréchiga. 2014. Simulation-guided lyapunov analysis for hybrid dynamical systems. In Proc. of the Hybrid Systems: Computation and Control (HSCC). ACM, 133--142.

Digital Library

[15]

Guy Katz, Clark Barrett, David L Dill, Kyle Julian, and Mykel J Kochenderfer. 2017. Reluplex: An efficient SMT solver for verifying deep neural networks. In International Conference on Computer Aided Verification. Springer, 97--117.

[16]

R Baker Kearfott and Vladik Kreinovich. 2013. Applications of interval computations. Vol. 3. Springer Science & Business Media.

[17]

Diederik P Kingma and Jimmy Ba. 2014. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014).

[18]

Edda Klipp, Ralf Herwig, Axel Kowald, Christoph Wierling, and Hans Lehrach. 2005. Systems Biology in Practice: Concepts, Implementation and Application. Wiley-Blackwell.

[19]

Michal Kocvara and Michael Stingl. 2005. PENBMI user's guide. Avaiable from http://www.penopt.com (2005).

[20]

Hui Kong, Fei He, Xiaoyu Song, William NN Hung, and Ming Gu. 2013. Exponential-condition-based barrier certificate generation for safety verification of hybrid systems. In Proceedings of the 25th International Conference on Computer Aided Verification (CAV). Springer, 242--257.

[21]

Hui Kong, Xiaoyu Song, Dong Han, Ming Gu, and Jiaguang Sun. 2014. A new barrier certificate for safety verification of hybrid systems. Comput. J. 57, 7 (2014), 1033--1045.

[22]

Jiang Liu, Naijun Zhan, and Zhao Hengjun. 2011. Computing semi-algebraic invariants for polynomial dynamical systems. In Proceedings of the International Conference on Embedded Software (EMSOFT). ACM, 97--106.

Digital Library

[23]

Jiang Liu, Naijun Zhan, Hengjun Zhao, and Liang Zou. 2015. Abstraction of Elementary Hybrid Systems by Variable Transformation. In Proceedings of the 20th International Symposium on Formal Methods. 360--377.

[24]

Zhou Lu, Hongming Pu, Feicheng Wang, Zhiqiang Hu, and Liwei Wang. 2017. The Expressive Power of Neural Networks: A View from the Width. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4--9 December 2017, Long Beach, CA, USA. 6231--6239.

Digital Library

[25]

Nadir Matringe, Arnaldo Vieira Moura, and Rachid Rebiha. 2010. Generating invariants for non-linear hybrid systems by linear algebraic methods. In Proc. of the Static Analysis. Springer, 373--389.

[26]

Andrea Peruffo, Daniele Ahmed, and Alessandro Abate. 2020. Automated Formal Synthesis of Neural Barrier Certificates for Dynamical Models. arXiv preprint arXiv:2007.03251 (2020).

[27]

André Platzer and Edmund M. Clarke. 2009. Computing differential invariants of hybrid systems as fixedpoints. Formal Methods in System Design 35, 1 (2009), 98--120.

Digital Library

[28]

Stephen Prajna and Ali Jadbabaie. 2004. Safety verification of hybrid systems using barrier certificates. In Proceedings of the 7th International Workshop on Hybrid Systems: Computation and Control (HSCC). Springer, 477--492.

[29]

S. Prajna, A. Jadbabaie, and G.J. Pappas. 2007. A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Automat. Control 52, 8 (2007), 1415--1429.

[30]

Maithra Raghu, Ben Poole, Jon M. Kleinberg, Surya Ganguli, and Jascha Sohl-Dickstein. 2017. On the Expressive Power of Deep Neural Networks. In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6--11 August 2017. 2847--2854.

[31]

Enric Rodríguez-Carbonell and Ashish Tiwari. 2005. Generating Polynomial Invariants for Hybrid Systems. In Proc. of the 8th ACM International Conference on Hybrid Systems: Computation and Control. 590--605.

Digital Library

[32]

Sriram Sankaranarayanan. 2010. Automatic invariant generation for hybrid systems using ideal fixed points. In Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control. ACM, 221--230.

Digital Library

[33]

Sriram Sankaranarayanan. 2020. Reachability Analysis Using Message Passing over Tree Decompositions. In Computer Aided Verification, Shuvendu K. Lahiri and Chao Wang (Eds.). Springer International Publishing, Cham, 604--628.

[34]

Sriram Sankaranarayanan, Henny Sipma, and Zohar Manna. 2008. Constructing invariants for hybrid systems. Formal Methods in System Design 32, 1 (2008), 25--55.

Digital Library

[35]

Gagandeep Singh, Timon Gehr, Matthew Mirman, Markus Püschel, and Martin Vechev. 2018. Fast and effective robustness certification. In Advances in Neural Information Processing Systems. 10802--10813.

[36]

Gagandeep Singh, Timon Gehr, Markus Püschel, and Martin Vechev. 2019. An abstract domain for certifying neural networks. Proceedings of the ACM on Programming Languages 3, POPL (2019), 1--30.

Digital Library

[37]

Christoffer Sloth, George J Pappas, and Rafael Wisniewski. 2012. Compositional safety analysis using barrier certificates. In Proceedings of the 15th ACM International Conference on Hybrid Systems: Computation and Control. ACM, 15--24.

Digital Library

[38]

Jasper Snoek, Hugo Larochelle, and Ryan P Adams. 2012. Practical bayesian optimization of machine learning algorithms. In Advances in neural information processing systems. 2951--2959.

[39]

Andrew Sogokon, Khalil Ghorbal, Paul B Jackson, and André Platzer. 2016. A Method for Invariant Generation for Polynomial Continuous Systems. In Proceedings of the International Conference on Verification, Model Checking, and Abstract Interpretation (VMCAI). Springer, 268--288.

Digital Library

[40]

Thomas Sturm and Ashish Tiwari. 2011. Verification and synthesis using real quantifier elimination. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC). ACM Press, 329--336.

Digital Library

[41]

Vincent Tjeng and Russ Tedrake. 2017. Verifying neural networks with mixed integer programming. arXiv preprint arXiv:1711.07356 (2017), 945--950.

[42]

Vincent Tjeng, Kai Xiao, and Russ Tedrake. 2017. Evaluating robustness of neural networks with mixed integer programming. arXiv preprint arXiv:1711.07356 (2017).

[43]

Cumhur Erkan Tuncali, James Kapinski, Hisahiro Ito, and Jyotirmoy V. Deshmukh. 2018. Reasoning about safety of learning-enabled components in autonomous cyber-physical systems. In Proceedings of the 55th Annual Design Automation Conference, DAC 2018, San Francisco, CA, USA, June 24--29, 2018. ACM, 30:1--30:6.

[44]

Shiqi Wang, Kexin Pei, Justin Whitehouse, Junfeng Yang, and Suman Jana. 2018. Formal security analysis of neural networks using symbolic intervals. In 27th {USENIX} Security Symposium ({USENIX} Security 18). 1599--1614.

[45]

Felix Winterer. 2017. isat3. URL https://projects.informatik.unifreiburg.de/projects/isat3 (2017).

[46]

Eric Wong and Zico Kolter. 2018. Provable defenses against adversarial examples via the convex outer adversarial polytope. In International Conference on Machine Learning. PMLR, 5286--5295.

[47]

Bai Xue, Martin Fränzle, Hengjun Zhao, Naijun Zhan, and Arvind Easwaran. 2019. Probably Approximate Safety Verification of Hybrid Dynamical Systems. In International Conference on Formal Engineering Methods. Springer, 236--252.

[48]

Zhengfeng Yang, Wang Lin, and Min Wu. 2015. Exact Verification of Hybrid Systems Based on Bilinear SOS Representation. ACM Transactions on Embedded Computing Systems 14, 1 (2015), 1--19.

Digital Library

[49]

Zhengfeng Yang, Min Wu, and Wang Lin. 2020. An efficient framework for barrier certificate generation of uncertain nonlinear hybrid systems. Nonlinear Analysis: Hybrid Systems 36 (2020), 100837.

[50]

Xia Zeng, Wang Lin, Zhengfeng Yang, Xin Chen, and Lilei Wang. 2016. Darbouxtype barrier certificates for safety verification of nonlinear hybrid systems. In Proc. of 2016 International Conference on Embedded Software, EMSOFT 2016, Pittsburgh, Pennsylvania, USA, October 1--7, 2016, Petru Eles and Rahul Mangharam (Eds.). ACM, 11:1--11:10.

[51]

Hengjun Zhao, Xia Zeng, Taolue Chen, and Zhiming Liu. 2020. Synthesizing barrier certificates using neural networks. In Proceedings of the 23rd International Conference on Hybrid Systems: Computation and Control. 1--11.

Digital Library

Cited By

Chen DLin WDing Z(2025)A Learner-Refiner Framework for Barrier Certificate GenerationMathematics10.3390/math1305084813:5(848)Online publication date: 4-Mar-2025
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Samanipour PPoonawala H(2024)Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier functionEuropean Journal of Control10.1016/j.ejcon.2024.10111580(101115)Online publication date: Nov-2024
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Synthesizing ReLU neural networks with two hidden layers as barrier certificates for hybrid systems

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cover image ACM Conferences

HSCC '21: Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control

May 2021

300 pages

ISBN:9781450383394

DOI:10.1145/3447928

Program Chairs:
Sergiy Bogomolov
Newcastle University, UK
,
Raphaël Jungers
UCLouvain, Belgium

Copyright © 2021 ACM.

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Published: 19 May 2021

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National Natural Science Foundation of China
Zhejiang Provincial Natural Science Foundation of China
Jiangsu Natural Science Foundation

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HSCC '21

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SIGBED

HSCC '21: 24th ACM International Conference on Hybrid Systems: Computation and Control

May 19 - 21, 2021

Tennessee, Nashville

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HSCC '21 Paper Acceptance Rate 27 of 77 submissions, 35%;

Overall Acceptance Rate 153 of 373 submissions, 41%

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Cited By

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Samanipour PPoonawala H(2024)Invariant set estimation for piecewise affine dynamical systems using piecewise affine barrier functionEuropean Journal of Control10.1016/j.ejcon.2024.10111580(101115)Online publication date: Nov-2024
https://doi.org/10.1016/j.ejcon.2024.101115
Zhang PZhou YSmolka SStoller SWang XZhao RLing TXing YRoy SDamare A(2024)AI‐Grid: AI‐Enabled, Smart Programmable MicrogridsMicrogrids10.1002/9781119890881.ch2(7-58)Online publication date: 15-Mar-2024
https://doi.org/10.1002/9781119890881.ch2
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Silva Neto ASilva HCamargo JAlmeida JCugnasca P(2023)Design and Assurance of Safety-Critical Systems with Artificial Intelligence in FPGAs: The Safety ArtISt Method and a Case Study of an FPGA-Based Autonomous Vehicle Braking Control SystemElectronics10.3390/electronics1224490312:24(4903)Online publication date: 6-Dec-2023
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