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The trial space is the space of neural network, which is defined globally over the whole computational domain, while the test space contains the piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in accuracy and training cost for several numerical examples of function approximation and solving differential equations.","url_abs":"https://arxiv.org/abs/2003.05385v1","url_pdf":"https://arxiv.org/pdf/2003.05385v1.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":"hp-vpinns-variational-physics-informed-neural","repo_url":"https://github.com/cmgcds/fastvpinns","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"hp-vpinns-variational-physics-informed-neural","repo_url":"https://github.com/ehsankharazmi/hp-VPINNs","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2003.05385","atlas_url":"https://app.syntology.ai/?focus=2003.05385","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2003.05385"}},"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. 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