{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T16:04:33Z","timestamp":1781193873555,"version":"3.54.1"},"publisher-location":"Cham","reference-count":35,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783032280787","type":"print"},{"value":"9783032280794","type":"electronic"}],"license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"DOI":"10.1007\/978-3-032-28079-4_15","type":"book-chapter","created":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T15:33:28Z","timestamp":1781192008000},"page":"334-355","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Safer Policies via\u00a0Affine Representations Using Koopman Dynamics"],"prefix":"10.1007","author":[{"given":"Tanmay","family":"Ambadkar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Darshan","family":"Chudiwal","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Greg","family":"Anderson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abhinav","family":"Verma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,12]]},"reference":[{"key":"15_CR1","unstructured":"Achiam, J., Held, D., Tamar, A., Abbeel, P.: Constrained policy optimization. In: Proceedings of the 34th International Conference on Machine Learning - Volume 70, pp. 22\u201331. ICML2017, JMLR.org (2017)"},{"key":"15_CR2","unstructured":"Alshiekh, M., Bloem, R., Ehlers, R., K\u00f6nighofer, B., Niekum, S., Topcu, U.: Safe reinforcement learning via shielding. In: McIlraith, S.A., Weinberger, K.Q. (eds.) Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018, pp. 2669\u20132678. AAAI Press (2018). https:\/\/www.aaai.org\/ocs\/index.php\/AAAI\/AAAI18\/paper\/view\/17211"},{"key":"15_CR3","unstructured":"Anderson, G., Chaudhuri, S., Dillig, I.: Guiding safe exploration with weakest preconditions. In: The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net (2023). https:\/\/iclr.cc\/virtual\/2023\/poster\/12258"},{"key":"15_CR4","unstructured":"Anderson, G., Verma, A., Dillig, I., Chaudhuri, S.: Neurosymbolic reinforcement learning with formally verified exploration. In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., Lin, H. (eds.) Advances in Neural Information Processing Systems, vol.\u00a033, pp. 6172\u20136183. Curran Associates, Inc. (2020). https:\/\/proceedings.neurips.cc\/paper\/2020\/file\/448d5eda79895153938a8431919f4c9f-Paper.pdf"},{"key":"15_CR5","doi-asserted-by":"publisher","unstructured":"Bacci, E., Giacobbe, M., Parker, D.: Verifying reinforcement learning up to infinity. In: Zhou, Z.H. (ed.) Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21, pp. 2154\u20132160. International Joint Conferences on Artificial Intelligence Organization (2021). https:\/\/doi.org\/10.24963\/ijcai.2021\/297","DOI":"10.24963\/ijcai.2021\/297"},{"key":"15_CR6","doi-asserted-by":"crossref","unstructured":"Banerjee, A., Rahmani, K., Biswas, J., Dillig, I.: Dynamic model predictive shielding for provably safe reinforcement learning. In: The Thirty-eighth Annual Conference on Neural Information Processing Systems (2024). https:\/\/openreview.net\/forum?id=x2zY4hZcmg","DOI":"10.52202\/079017-3177"},{"key":"15_CR7","doi-asserted-by":"publisher","unstructured":"Bastani, O.: Safe reinforcement learning with nonlinear dynamics via model predictive shielding. In: 2021 American Control Conference (ACC), pp. 3488\u20133494 (2021). https:\/\/doi.org\/10.23919\/ACC50511.2021.9483182","DOI":"10.23919\/ACC50511.2021.9483182"},{"key":"15_CR8","unstructured":"Berkenkamp, F., Turchetta, M., Schoellig, A., Krause, A.: Safe model-based reinforcement learning with stability guarantees. In: Advances in Neural Information Processing Systems, vol. 30 (2017)"},{"key":"15_CR9","unstructured":"Bharadhwaj, H., Kumar, A., Rhinehart, N., Levine, S., Shkurti, F., Garg, A.: Conservative safety critics for exploration. In: 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net (2021). https:\/\/openreview.net\/forum?id=iaO86DUuKi"},{"key":"15_CR10","unstructured":"Brunke, L., et al.: Safe learning in robotics: from learning-based control to safe reinforcement learning. ArXiv abs\/2108.06266 (2021). https:\/\/arxiv.org\/pdf\/2108.06266.pdf"},{"key":"15_CR11","doi-asserted-by":"crossref","unstructured":"Brunton, S.L., Budi\u0161i\u0107, M., Kaiser, E., Kutz, J.N.: Modern Koopman theory for dynamical systems. arXiv preprint arXiv:2102.12086 (2021)","DOI":"10.1137\/21M1401243"},{"key":"15_CR12","unstructured":"Dalal, G., Dvijotham, K., Vecer\u00edk, M., Hester, T., Paduraru, C., Tassa, Y.: Safe exploration in continuous action spaces. CoRR abs\/1801.08757 (2018). http:\/\/arxiv.org\/abs\/1801.08757"},{"key":"15_CR13","doi-asserted-by":"publisher","unstructured":"Fulton, N., Platzer, A.: Verifiably safe off-model reinforcement learning. In: International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp. 413\u2013430. Springer (2019). https:\/\/doi.org\/10.1007\/978-3-030-17462-0_28","DOI":"10.1007\/978-3-030-17462-0_28"},{"key":"15_CR14","doi-asserted-by":"publisher","unstructured":"Gillula, J.H., Tomlin, C.J.: Guaranteed safe online learning via reachability: tracking a ground target using a quadrotor. In: IEEE International Conference on Robotics and Automation, ICRA 2012, 14-18 May, 2012, St. Paul, Minnesota, USA, pp. 2723\u20132730 (2012). https:\/\/doi.org\/10.1109\/ICRA.2012.6225136","DOI":"10.1109\/ICRA.2012.6225136"},{"key":"15_CR15","doi-asserted-by":"publisher","unstructured":"Goodall, A.W., Belardinelli, F.: Approximate model-based shielding for safe reinforcement learning. In: Gal, K., Now\u00e9, A., Nalepa, G.J., Fairstein, R., Radulescu, R. (eds.) ECAI 2023 - 26th European Conference on Artificial Intelligence, September 30 - October 4, 2023, Krak\u00f3w, Poland - Including 12th Conference on Prestigious Applications of Intelligent Systems (PAIS 2023). Frontiers in Artificial Intelligence and Applications, vol.\u00a0372, pp. 883\u2013890. IOS Press (2023). https:\/\/doi.org\/10.3233\/FAIA230357","DOI":"10.3233\/FAIA230357"},{"key":"15_CR16","doi-asserted-by":"crossref","unstructured":"Gu, S., et al.: Balance reward and safety optimization for safe reinforcement learning: a perspective of gradient manipulation. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol. 38, pp. 21099\u201321106 (2024)","DOI":"10.1609\/aaai.v38i19.30102"},{"key":"15_CR17","unstructured":"Gu, S., et al.: A review of safe reinforcement learning: Methods, theory and applications. ArXiv abs\/2205.10330 (2022). https:\/\/api.semanticscholar.org\/CorpusId:248965265"},{"key":"15_CR18","unstructured":"Ji, J., et al.: Safety gymnasium: a unified safe reinforcement learning benchmark. In: Thirty-seventh Conference on Neural Information Processing Systems Datasets and Benchmarks Track (2023). https:\/\/openreview.net\/forum?id=WZmlxIuIGR"},{"key":"15_CR19","unstructured":"Ji, J., et al.: OmniSafe: an infrastructure for accelerating safe reinforcement learning research. J. Mach. Learn. Res. 25(285), \u00a01\u20136 (2024). http:\/\/jmlr.org\/papers\/v25\/23-0681.html"},{"key":"15_CR20","unstructured":"Jothimurugan, K., Alur, R., Bastani, O.: A composable specification language for reinforcement learning tasks. In: Wallach, H., Larochelle, H., Beygelzimer, A., d\u2019Alch\u00e9-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol.\u00a032. Curran Associates, Inc. (2019). https:\/\/proceedings.neurips.cc\/paper_files\/paper\/2019\/file\/f5aa4bd09c07d8b2f65bad6c7cd3358f-Paper.pdf"},{"key":"15_CR21","unstructured":"Jovanovi\u2019c, M.R., Ding, D., Wei, X., Yang, Z., Wang, Z.: Provably efficient safe exploration via primal-dual policy optimization. In: International Conference on Artificial Intelligence and Statistics (2020). https:\/\/api.semanticscholar.org\/CorpusId:211677570"},{"key":"15_CR22","unstructured":"Ladosz, P., Weng, L., Kim, M., Oh, H.: Exploration in deep reinforcement learning: a survey. ArXiv abs\/2205.00824 (2022). https:\/\/api.semanticscholar.org\/CorpusId:247900285"},{"key":"15_CR23","unstructured":"Liu, Z., Zhou, H., Chen, B., Zhong, S., Hebert, M., Zhao, D.: Constrained model based reinforcement learning with robust cross-entropy method (2020)"},{"key":"15_CR24","doi-asserted-by":"publisher","unstructured":"Ma, Y.J., Shen, A., Bastani, O., Jayaraman, D.: Conservative and adaptive penalty for model-based safe reinforcement learning (2021). https:\/\/doi.org\/10.48550\/ARXIV.2112.07701","DOI":"10.48550\/ARXIV.2112.07701"},{"key":"15_CR25","unstructured":"Satija, H., Amortila, P., Pineau, J.: Constrained Markov decision processes via backward value functions. In: Proceedings of the 37th International Conference on Machine Learning. ICML 2020, JMLR.org (2020)"},{"key":"15_CR26","unstructured":"Shi, H., Meng, M.Q.H.: Deep Koopman operator with control for nonlinear systems (2022). https:\/\/arxiv.org\/abs\/2202.08004"},{"key":"15_CR27","unstructured":"Sootla, A., et al.: Saut\u00e9 rl: almost surely safe reinforcement learning using state augmentation. In: International Conference on Machine Learning, pp. 20423\u201320443. PMLR (2022)"},{"key":"15_CR28","unstructured":"Sootla, A., Cowen-Rivers, A.I., Wang, J., Ammar, H.B.: Effects of safety state augmentation on safe exploration. arXiv preprint arXiv:2206.02675 (2022)"},{"key":"15_CR29","doi-asserted-by":"publisher","unstructured":"Wang, Y., Zhu, H.: Safe exploration in reinforcement learning by reachability analysis over learned models, pp. 232\u2013255 (2024). https:\/\/doi.org\/10.1007\/978-3-031-65633-0_11","DOI":"10.1007\/978-3-031-65633-0_11"},{"key":"15_CR30","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.1007\/s00332-015-9258-5","volume":"25","author":"MO Williams","year":"2015","unstructured":"Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25, 1307\u20131346 (2015)","journal-title":"J. Nonlinear Sci."},{"key":"15_CR31","unstructured":"Yang, L., et al.: Constrained update projection approach to safe policy optimization. In: Koyejo, S., Mohamed, S., Agarwal, A., Belgrave, D., Cho, K., Oh, A. (eds.) Advances in Neural Information Processing Systems, vol.\u00a035, pp. 9111\u20139124. Curran Associates, Inc. (2022). https:\/\/proceedings.neurips.cc\/paper_files\/paper\/2022\/file\/3ba7560b4c3e66d760fbdd472cf4a5a9-Paper-Conference.pdf"},{"key":"15_CR32","unstructured":"Yang, T.Y., Rosca, J., Narasimhan, K., Ramadge, P.J.: Projection-based constrained policy optimization. arXiv preprint arXiv:2010.03152 (2020)"},{"key":"15_CR33","doi-asserted-by":"publisher","unstructured":"Zhang, L., et al.: Penalized proximal policy optimization for safe reinforcement learning. In: Raedt, L.D. (ed.) Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI-22, pp. 3744\u20133750. International Joint Conferences on Artificial Intelligence Organization (2022). https:\/\/doi.org\/10.24963\/ijcai.2022\/520","DOI":"10.24963\/ijcai.2022\/520"},{"key":"15_CR34","unstructured":"Zhang, Y., Vuong, Q., Ross, K.: First order constrained optimization in policy space. In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., Lin, H. (eds.) Advances in Neural Information Processing Systems, vol.\u00a033, pp. 15338\u201315349. Curran Associates, Inc. (2020). https:\/\/proceedings.neurips.cc\/paper\/2020\/file\/af5d5ef24881f3c3049a7b9bfe74d58b-Paper.pdf"},{"key":"15_CR35","doi-asserted-by":"crossref","unstructured":"Zhu, H., Xiong, Z., Magill, S., Jagannathan, S.: An inductive synthesis framework for verifiable reinforcement learning. In: ACM Conference on Programming Language Design and Implementation (SIGPLAN) (2019)","DOI":"10.1145\/3314221.3314638"}],"container-title":["Lecture Notes in Computer Science","NASA Formal Methods"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-28079-4_15","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T15:33:37Z","timestamp":1781192017000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-28079-4_15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"ISBN":["9783032280787","9783032280794"],"references-count":35,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-28079-4_15","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026]]},"assertion":[{"value":"12 June 2026","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"NFM","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"NASA Formal Methods Symposium","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Los Angeles, CA","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"USA","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2026","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"5 May 2026","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"7 May 2026","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"18","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"nfm2026","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/nfm2026.github.io\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}