Probabilistic Symbolic Regression for Equation Discovery via Operator-induced and Regularized Symbolic Forests

[2509.19710] Probabilistic Symbolic Regression for Equation Discovery via Operator-induced and Regularized Symbolic Forests Skip to main content Search Submit Donate Log in Statistics > Methodology arXiv:2509.19710 (stat) [Submitted on 24 Sep 2025 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Probabilistic Symbolic Regression for Equation Discovery via Operator-induced and Regularized Symbolic Forests Authors: Somjit Roy , Pritam Dey , Bani K. Mallick , Debdeep Pati View PDF HTML (experimental) Abstract: Symbolic regression has emerged as a powerful tool for artificial intelligence-driven scientific discovery by learning interpretable analytical expressions that reveal governing relationships directly from data. Existing methods, however, often rely on heuristic search, struggle to balance predictive accuracy with expression complexity in noisy settings, and offer limited characterization of symbolic uncertainty. Probabilistic approaches that address these challenges in a unified manner remain underexplored. We introduce a probabilistic symbolic regression framework that represents mathematical expressions as ensembles of symbolic trees. A regularizing prior over tree topology controls expression complexity, while an Occam's window-based posterior summary captures uncertainty across multiple plausible symbolic models. Given the limited existing theoretical treatment of symbolic regression, we develop posterior concentration guarantees when symbolic expressions approximate the underlying relationship arbitrarily well, with a near-parametric rate when an exact finite formula exists. Additionally, we establish a sharp oracle concentration result under symbolic misspecification. Comparisons of our proposed framework with state-of-the-art competitors demonstrate superior predictive accuracy, optimal symbolic complexity, and stable structural recovery when learning benchmark scientific equations, together with the identification of scientifically interpretable descriptor formulas in a challenging materials discovery application. Comments: 152 pages, 37 figures, 24 tables, Submitted Subjects: Methodology (stat.ME) ; Machine Learning (cs.LG); Symbolic Computation (cs.SC); Statistics Theory (math.ST); Machine Learning (stat.ML) Cite as: arXiv:2509.19710 [stat.ME] (or arXiv:2509.19710v3 [stat.ME] for this version) https://doi.org/10.48550/arXiv.2509.19710 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Somjit Roy [ view email ] [v1] Wed, 24 Sep 2025 02:42:25 UTC (3,523 KB) [v2] Fri, 24 Jul 2026 23:31:07 UTC (6,171 KB) [v3] Fri, 28 Aug 2026 00:29:58 UTC (2,010 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: stat.ME < prev | next > new | recent | 2025-09 Change to browse by: cs cs.LG cs.SC math math.ST stat stat.ML stat.TH References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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