{"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/message-passing-neural-pde-solvers-1","title":"Message Passing Neural PDE Solvers","arxiv_id":"2202.03376","date":"2022-02-07","proceeding":"ICLR 2022 4","authors":["Johannes Brandstetter","Daniel Worrall","Max Welling"],"abstract":"The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.","url_abs":"https://arxiv.org/abs/2202.03376v3","url_pdf":"https://arxiv.org/pdf/2202.03376v3.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":"message-passing-neural-pde-solvers-1","repo_url":"https://github.com/brandstetter-johannes/mp-neural-pde-solvers","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"domain-adaptation","task_name":"Domain Adaptation"}],"methods":[{"method_slug":"speed","method_name":"SPEED"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2202.03376","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2202.03376"}},"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":"deterministic:regex_extraction","url":"https://github.com/brandstetter-johannes/MP-Neural-PDE-Solvers","reach":{"status":"ok","spdx":"MIT"}},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/brandstetter-johannes/mp-neural-pde-solvers","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran_honours":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":"bb7d0ae780033c78","entry":"cheb_grid","repo":"brandstetter-johannes/mp-neural-pde-solvers","repo_kind":"official","path":"generate/generate_data.py","file_url":"https://github.com/brandstetter-johannes/mp-neural-pde-solvers/blob/HEAD/generate/generate_data.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"well_formed","behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"bb7d0ae780033c78"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}