{"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/neural-conservation-laws-a-divergence-free","title":"Neural Conservation Laws: A Divergence-Free Perspective","arxiv_id":"2210.01741","date":"2022-10-04","proceeding":null,"authors":["Jack Richter-Powell","Yaron Lipman","Ricky T. Q. Chen"],"abstract":"We investigate the parameterization of deep neural networks that by design satisfy the continuity equation, a fundamental conservation law. This is enabled by the observation that any solution of the continuity equation can be represented as a divergence-free vector field. We hence propose building divergence-free neural networks through the concept of differential forms, and with the aid of automatic differentiation, realize two practical constructions. As a result, we can parameterize pairs of densities and vector fields that always exactly satisfy the continuity equation, foregoing the need for extra penalty methods or expensive numerical simulation. Furthermore, we prove these models are universal and so can be used to represent any divergence-free vector field. Finally, we experimentally validate our approaches by computing neural network-based solutions to fluid equations, solving for the Hodge decomposition, and learning dynamical optimal transport maps.","url_abs":"https://arxiv.org/abs/2210.01741v3","url_pdf":"https://arxiv.org/pdf/2210.01741v3.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":"neural-conservation-laws-a-divergence-free","repo_url":"https://github.com/facebookresearch/neural-conservation-law","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2210.01741","atlas_url":"https://app.syntology.ai/?focus=2210.01741","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2210.01741"}},"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/facebookresearch/neural-conservation-law","reach":null}],"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":1,"samples":[{"code_sha256_prefix":"37fb022c60d70c5c","entry":"normal_logprob","repo":"facebookresearch/neural-conservation-law","repo_kind":"official","path":"pytorch/main_ot.py","file_url":"https://github.com/facebookresearch/neural-conservation-law/blob/HEAD/pytorch/main_ot.py","link_basis":"first_harvest_node","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"NOASSERTION","inline_ok":false,"mcp_get_code":{"code_sha256":"37fb022c60d70c5c"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}