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Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.","url_abs":"https://arxiv.org/abs/2110.11847v2","url_pdf":"https://arxiv.org/pdf/2110.11847v2.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-numerical-method-of-lines-for","repo_url":"https://github.com/schmidtjonathan/pnmol-experiments","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":null},{"paper_slug":"probabilistic-numerical-method-of-lines-for","repo_url":"https://github.com/pnkraemer/probfindiff","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"bayesian-inference","task_name":"Bayesian Inference"}],"methods":[{"method_slug":"gaussian-process","method_name":"Gaussian Process"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2110.11847","atlas_url":"https://app.syntology.ai/?focus=2110.11847","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2110.11847"}},"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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