{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/markov-neural-operators-for-learning-chaotic","title":"Learning Dissipative Dynamics in Chaotic Systems","arxiv_id":"2106.06898","date":"2021-06-13","proceeding":null,"authors":["Zongyi Li","Miguel Liu-Schiaffini","Nikola Kovachki","Burigede Liu","Kamyar Azizzadenesheli","Kaushik Bhattacharya","Andrew Stuart","Anima Anandkumar"],"abstract":"Chaotic systems are notoriously challenging to predict because of their sensitivity to perturbations and errors due to time stepping. Despite this unpredictable behavior, for many dissipative systems the statistics of the long term trajectories are governed by an invariant measure supported on a set, known as the global attractor; for many problems this set is finite dimensional, even if the state space is infinite dimensional. For Markovian systems, the statistical properties of long-term trajectories are uniquely determined by the solution operator that maps the evolution of the system over arbitrary positive time increments. In this work, we propose a machine learning framework to learn the underlying solution operator for dissipative chaotic systems, showing that the resulting learned operator accurately captures short-time trajectories and long-time statistical behavior. Using this framework, we are able to predict various statistics of the invariant measure for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to 5000.","url_abs":"https://arxiv.org/abs/2106.06898v2","url_pdf":"https://arxiv.org/pdf/2106.06898v2.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":"markov-neural-operators-for-learning-chaotic","repo_url":"https://github.com/NeuralOperator/markov_neural_operator","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"markov-neural-operators-for-learning-chaotic","repo_url":"https://github.com/sciml/neuraloperators.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2106.06898","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2106.06898"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+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. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/NeuralOperator/markov_neural_operator","reach":null},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/sciml/neuraloperators.jl","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran_fixture":1},"by_repo_kind":{"official":{"samples":1,"ran":1,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":0,"samples":[{"code_sha256_prefix":"ad9e509694013de8","entry":"compl_mul2d","repo":"NeuralOperator/markov_neural_operator","repo_kind":"official","path":"models/fno_2d.py","file_url":"https://github.com/NeuralOperator/markov_neural_operator/blob/HEAD/models/fno_2d.py","link_basis":"first_harvest_node","language":"python","status":"ran_fixture","verification_level":1,"contract_check":"RAISES","metamorphic_tier":"invariant","behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"ad9e509694013de8"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}