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A proposed step is accepted if its error, \\emph{relative to some norm}, is sufficiently small; else it is rejected, the step is shrunk, and the process is repeated. Here, we demonstrate that the particular structure of the adjoint equations makes the usual choices of norm (such as $L^2$) unnecessarily stringent. By replacing it with a more appropriate (semi)norm, fewer steps are unnecessarily rejected and the backpropagation is made faster. This requires only minor code modifications. Experiments on a wide range of tasks -- including time series, generative modeling, and physical control -- demonstrate a median improvement of 40% fewer function evaluations. On some problems we see as much as 62% fewer function evaluations, so that the overall training time is roughly halved.","url_abs":"https://arxiv.org/abs/2009.09457v2","url_pdf":"https://arxiv.org/pdf/2009.09457v2.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":"hey-that-s-not-an-ode-faster-ode-adjoints","repo_url":"https://github.com/patrick-kidger/FasterNeuralDiffEq","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"hey-that-s-not-an-ode-faster-ode-adjoints","repo_url":"https://github.com/rtqichen/torchdiffeq","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"hey-that-s-not-an-ode-faster-ode-adjoints","repo_url":"https://github.com/2024-MindSpore-1/Code3/tree/main/ODE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"mindspore","reach":null}],"tasks":[{"task_slug":"time-series-1","task_name":"Time Series"},{"task_slug":"time-series","task_name":"Time Series Analysis"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2009.09457","atlas_url":"https://app.syntology.ai/?focus=2009.09457","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2009.09457"}},"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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