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This is achieved by defining the\nmeasurement sequence to consist of the observations of the difference between\nthe derivative of the GP and the vector field evaluated at the GP---which are\nall identically zero at the solution of the ODE. When the GP has a state-space\nrepresentation, the problem can be reduced to a non-linear Bayesian filtering\nproblem and all widely-used approximations to the Bayesian filtering and\nsmoothing problems become applicable. Furthermore, all previous GP-based ODE\nsolvers that are formulated in terms of generating synthetic measurements of\nthe gradient field come out as specific approximations. Based on the non-linear\nBayesian filtering problem posed in this paper, we develop novel Gaussian\nsolvers for which we establish favourable stability properties. Additionally,\nnon-Gaussian approximations to the filtering problem are derived by the\nparticle filter approach. The resulting solvers are compared with other\nprobabilistic solvers in illustrative experiments.","url_abs":"http://arxiv.org/abs/1810.03440v4","url_pdf":"http://arxiv.org/pdf/1810.03440v4.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":"probabilistic-solutions-to-ordinary","repo_url":"https://github.com/mlysy/rodeo","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[{"method_slug":"gaussian-process","method_name":"Gaussian Process"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1810.03440","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1810.03440"}},"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. 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