{"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/oden-a-framework-to-solve-ordinary","title":"ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks","arxiv_id":"2005.14090","date":"2020-05-28","proceeding":null,"authors":["Liam L. H. Lau","Denis Werth"],"abstract":"We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to evaluate neural networks' performance. Neural networks are shown to be proficient at approximating continuous solutions within their training domains. We illustrate neural networks' ability to outperform traditional standard numerical techniques. Training is thoroughly examined and three universal phases are found: (i) a prior tangent adjustment, (ii) a curvature fitting, and (iii) a fine-tuning stage. The main limitation of the method is the nontrivial task of finding the appropriate neural network architecture and the choice of neural network hyperparameters for efficient optimization. However, we observe an optimal architecture that matches the complexity of the differential equation. A user-friendly and adaptable open-source code (ODE$\\mathcal{N}$) is provided on GitHub.","url_abs":"https://arxiv.org/abs/2005.14090v2","url_pdf":"https://arxiv.org/pdf/2005.14090v2.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":"oden-a-framework-to-solve-ordinary","repo_url":"https://github.com/deniswerth/ODEN","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2005.14090"}},"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/deniswerth/ODEN","reach":null}],"summary":{"ran":1,"unverified":1},"by_repo_kind":{"official":{"samples":2,"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":2,"samples":[{"code_sha256_prefix":"bed676817f300cde","entry":"DiffEq","repo":"deniswerth/ODEN","repo_kind":"official","path":"ODEsolver.py","file_url":"https://github.com/deniswerth/ODEN/blob/HEAD/ODEsolver.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"bed676817f300cde"}},{"code_sha256_prefix":"b20b837ce3d11555","entry":"ODEsolver","repo":"deniswerth/ODEN","repo_kind":"official","path":"ODEsolver.py","file_url":"https://github.com/deniswerth/ODEN/blob/HEAD/ODEsolver.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"b20b837ce3d11555"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}