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Despite that, accurately estimating domains of attraction for nonlinear systems remains a challenging task, where existing methods are conservative or limited to low-dimensional systems. The estimation becomes even more challenging when accounting for state constraints. In this work, we propose a framework to accurately estimate safe (state-constrained) domains of attraction for discrete-time autonomous nonlinear systems. In establishing this framework, we first derive a new Zubov equation, whose solution corresponds to the exact safe domain of attraction. The solution to the aforementioned Zubov equation is shown to be unique and continuous over the whole state space. We then present a physics-informed approach to approximating the solution of the Zubov equation using neural networks. To obtain certifiable estimates of the domain of attraction from the neural network approximate solutions, we propose a verification framework that can be implemented using standard verification tools (e.g., $\\alpha,\\!\\beta$-CROWN and dReal). To illustrate its effectiveness, we demonstrate our approach through numerical examples concerning nonlinear systems with state constraints.","url_abs":"https://arxiv.org/abs/2506.13961v1","url_pdf":"https://arxiv.org/pdf/2506.13961v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"safe-domains-of-attraction-for-discrete-time","repo_url":"https://github.com/ruikunzhou/safe-roa-dt-system","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"safe-domains-of-attraction-for-discrete-time","repo_url":"https://github.com/RuikunZhou/Safe_ROA_dt_LyZNet","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2506.13961","atlas_url":"https://app.syntology.ai/?focus=2506.13961","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2506.13961"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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