{"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/a-multiscale-neural-network-based-on","title":"A multiscale neural network based on hierarchical nested bases","arxiv_id":"1808.02376","date":"2018-08-04","proceeding":null,"authors":["Yuwei Fan","Jordi Feliu-Faba","Lin Lin","Lexing Ying","Leonardo Zepeda-Nunez"],"abstract":"In recent years, deep learning has led to impressive results in many fields. In this paper, we introduce a multi-scale artificial neural network for high-dimensional non-linear maps based on the idea of hierarchical nested bases in the fast multipole method and the $\\mathcal{H}^2$-matrices. This approach allows us to efficiently approximate discretized nonlinear maps arising from partial differential equations or integral equations. It also naturally extends our recent work based on the generalization of hierarchical matrices [Fan et al. arXiv:1807.01883] but with a reduced number of parameters. In particular, the number of parameters of the neural network grows linearly with the dimension of the parameter space of the discretized PDE. We demonstrate the properties of the architecture by approximating the solution maps of non-linear Schr{\\\"o}dinger equation, the radiative transfer equation, and the Kohn-Sham map.","url_abs":"http://arxiv.org/abs/1808.02376v1","url_pdf":"http://arxiv.org/pdf/1808.02376v1.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-multiscale-neural-network-based-on","repo_url":"https://github.com/ywfan/mnn-H2","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1808.02376","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1808.02376"}},"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/ywfan/mnn-H2","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"unverified":6},"by_repo_kind":{"listed":{"samples":6,"ran":0,"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":"9ea5cc2aa8a485cb","entry":"CNNR","repo":"ywfan/mnn-H2","repo_kind":"listed","path":"NLSE/testH2matrix.py","file_url":"https://github.com/ywfan/mnn-H2/blob/HEAD/NLSE/testH2matrix.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"9ea5cc2aa8a485cb"}},{"code_sha256_prefix":"6703a86d3fc73bff","entry":"matrix2tensor","repo":"ywfan/mnn-H2","repo_kind":"listed","path":"NLSE/testH2matrix2d.py","file_url":"https://github.com/ywfan/mnn-H2/blob/HEAD/NLSE/testH2matrix2d.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"6703a86d3fc73bff"}},{"code_sha256_prefix":"d1c1b561a0454059","entry":"padding","repo":"ywfan/mnn-H2","repo_kind":"listed","path":"NLSE/testH2matrix.py","file_url":"https://github.com/ywfan/mnn-H2/blob/HEAD/NLSE/testH2matrix.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"d1c1b561a0454059"}},{"code_sha256_prefix":"5141bff929400d6c","entry":"padding2d","repo":"ywfan/mnn-H2","repo_kind":"listed","path":"NLSE/testH2matrix2d.py","file_url":"https://github.com/ywfan/mnn-H2/blob/HEAD/NLSE/testH2matrix2d.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"5141bff929400d6c"}},{"code_sha256_prefix":"a3111a464f03d264","entry":"reshape_interpolation","repo":"ywfan/mnn-H2","repo_kind":"listed","path":"RTE/testH2Mix2d.py","file_url":"https://github.com/ywfan/mnn-H2/blob/HEAD/RTE/testH2Mix2d.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"a3111a464f03d264"}},{"code_sha256_prefix":"2329f2fb74e28bb7","entry":"tensor2matrix","repo":"ywfan/mnn-H2","repo_kind":"listed","path":"NLSE/testH2matrix2d.py","file_url":"https://github.com/ywfan/mnn-H2/blob/HEAD/NLSE/testH2matrix2d.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"2329f2fb74e28bb7"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}