Abstract
Recurrent neural networks are increasingly employed in safety-critical applications, such as control in cyber-physical systems, and therefore their verification is crucial for guaranteeing reliability and correctness. We present a novel approach for verifying the dynamic behavior of Long short-term memory networks (LSTMs), a popular type of recurrent neural network (RNN). Our approach employs the satisfiability modulo theories (SMT) solver iSAT solving complex Boolean combinations of linear and non-linear constraint formulas (including transcendental functions), and it therefore is able to verify safety properties of these networks.
This work is supported by the German Research Foundation DFG through the Research Training Group “SCARE: System Correctness under Adverse Conditions” (DFG-GRK 1765/2) and project grant FR 2715/5-1.
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References
Economic commission for Europe, inland transport committee, “vienna convention on road traffic”. http://www.unece.org/fileadmin/DAM/trans/conventn/crt1968e.pdf
European space agency, “collision detection dataset”. https://www.kelvins.esa.int
Federal highway administration research and technology, “US highway 101 dataset”. https://www.fhwa.dot.gov/publications/research/operations/07030
space track. www.space-track.org
iSAT Quick Start Guide. AVACS H1/2 iSAT Developer Team (2010)
Agency, E.S.: Space debris by the numbers. https://www.esa.int
Akintunde, M.E., Kevorchian, A., Lomuscio, A., Pirovano, E.: Verification of RNN-based neural agent-environment systems. In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, The Thirty-First Innovative Applications of Artificial Intelligence Conference, IAAI 2019, The Ninth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2019, Honolulu, Hawaii, USA, 27 January - 1 February 2019, pp. 6006–6013. AAAI Press (2019). https://doi.org/10.1609/aaai.v33i01.33016006
Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-49059-0_14
Ehlers, R.: Formal verification of piece-wise linear feed-forward neural networks. In: D’Souza, D., Narayan Kumar, K. (eds.) ATVA 2017. LNCS, vol. 10482, pp. 269–286. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68167-2_19
Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. J. Satisf. Boolean Model. Comput. 1(3–4), 209–236 (2007). https://doi.org/10.3233/sat190012
Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Comput. 9(8), 1735–1780 (1997). https://doi.org/10.1162/neco.1997.9.8.1735
Jacoby, Y., Barrett, C., Katz, G.: Verifying recurrent neural networks using invariant inference. In: Hung, D.V., Sokolsky, O. (eds.) ATVA 2020. LNCS, vol. 12302, pp. 57–74. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-59152-6_3
Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.: Reluplex: an efficient SMT solver for verifying deep neural networks. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 97–117. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63387-9_5
Mohammadinejad, S., Paulsen, B., Deshmukh, J.V., Wang, C.: DiffRNN: differential verification of recurrent neural networks. In: Dima, C., Shirmohammadi, M. (eds.) FORMATS 2021. LNCS, vol. 12860, pp. 117–134. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-85037-1_8
Pulina, L., Tacchella, A.: Never: a tool for artificial neural networks verification. Ann. Math. Artif. Intell. 62, 403–425 (2011)
Rizaldi, A., Immler, F.: A formally verified checker of the safe distance traffic rules for autonomous vehicles. Arch. Formal Proofs 2020 (2020). https://www.isa-afp.org/entries/Safe_Distance.html
Scheibler, K.: isat3 manual. isat3 0.04-20170301 (2017)
Sighencea, B.I., Stanciu, R.I., Caleanu, C.: A review of deep learning-based methods for pedestrian trajectory prediction. Sensors 21(22), 7543 (2021). https://doi.org/10.3390/s21227543
Uriot, T., et al.: Spacecraft collision avoidance challenge: design and results of a machine learning competition. Astrodynamics 6(2), 121–140 (2022)
Yu, Y., Si, X., Hu, C., Zhang, J.: A review of recurrent neural networks: LSTM cells and network architectures. Neural Comput. 31(7), 1235–1270 (2019). https://doi.org/10.1162/neco_a_01199
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Moradkhani, F., Fibich, C., Fränzle, M. (2023). Verification of LSTM Neural Networks with Non-linear Activation Functions. In: Rozier, K.Y., Chaudhuri, S. (eds) NASA Formal Methods. NFM 2023. Lecture Notes in Computer Science, vol 13903. Springer, Cham. https://doi.org/10.1007/978-3-031-33170-1_1
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