{"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/enabling-local-neural-operators-to-perform","title":"Enabling Local Neural Operators to perform Equation-Free System-Level Analysis","arxiv_id":"2505.02308","date":"2025-05-05","proceeding":null,"authors":["Gianluca Fabiani","Hannes Vandecasteele","Somdatta Goswami","Constantinos Siettos","Ioannis G. Kevrekidis"],"abstract":"Neural Operators (NOs) provide a powerful framework for computations involving physical laws that can be modelled by (integro-) partial differential equations (PDEs), directly learning maps between infinite-dimensional function spaces that bypass both the explicit equation identification and their subsequent numerical solving. Still, NOs have so far primarily been employed to explore the dynamical behavior as surrogates of brute-force temporal simulations/predictions. Their potential for systematic rigorous numerical system-level tasks, such as fixed-point, stability, and bifurcation analysis - crucial for predicting irreversible transitions in real-world phenomena - remains largely unexplored. Toward this aim, inspired by the Equation-Free multiscale framework, we propose and implement a framework that integrates (local) NOs with advanced iterative numerical methods in the Krylov subspace, so as to perform efficient system-level stability and bifurcation analysis of large-scale dynamical systems. Beyond fixed point, stability, and bifurcation analysis enabled by local in time NOs, we also demonstrate the usefulness of local in space as well as in space-time (\"patch\") NOs in accelerating the computer-aided analysis of spatiotemporal dynamics. We illustrate our framework via three nonlinear PDE benchmarks: the 1D Allen-Cahn equation, which undergoes multiple concatenated pitchfork bifurcations; the Liouville-Bratu-Gelfand PDE, which features a saddle-node tipping point; and the FitzHugh-Nagumo (FHN) model, consisting of two coupled PDEs that exhibit both Hopf and saddle-node bifurcations.","url_abs":"https://arxiv.org/abs/2505.02308v1","url_pdf":"https://arxiv.org/pdf/2505.02308v1.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":"enabling-local-neural-operators-to-perform","repo_url":"https://github.com/centrum-intelliphysics/time-stepper","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2505.02308","atlas_url":"https://app.syntology.ai/?focus=2505.02308","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2505.02308"}},"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. 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/centrum-intelliphysics/time-stepper","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran":3},"by_repo_kind":{"official":{"samples":3,"ran":3,"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":"1b4c2406b8506bf4","entry":"psi","repo":"centrum-intelliphysics/time-stepper","repo_kind":"official","path":"Bratu/EulerTimestepper.py","file_url":"https://github.com/centrum-intelliphysics/time-stepper/blob/HEAD/Bratu/EulerTimestepper.py","link_basis":"harvester_set","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"1b4c2406b8506bf4"}},{"code_sha256_prefix":"180ca1666c4fe375","entry":"rhs","repo":"centrum-intelliphysics/time-stepper","repo_kind":"official","path":"Bratu/EulerTimestepper.py","file_url":"https://github.com/centrum-intelliphysics/time-stepper/blob/HEAD/Bratu/EulerTimestepper.py","link_basis":"harvester_set","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"180ca1666c4fe375"}},{"code_sha256_prefix":"bc2ca895051669cb","entry":"timestepper","repo":"centrum-intelliphysics/time-stepper","repo_kind":"official","path":"Bratu/EulerTimestepper.py","file_url":"https://github.com/centrum-intelliphysics/time-stepper/blob/HEAD/Bratu/EulerTimestepper.py","link_basis":"harvester_set","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"bc2ca895051669cb"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}