aiqbench analysis
This research advances AI-driven scientific discovery by introducing a probabilistic framework for symbolic regression that balances predictive accuracy with expression complexity using regularized tree ensembles. The work provides rigorous theoretical guarantees on posterior concentration and uncertainty quantification, addressing key limitations in existing heuristic search methods for learning interpretable governing equations from noisy data. By enabling stable structural recovery of benchmark scientific laws, this approach strengthens the intersection between machine learning methodology and fundamental physics or materials science discovery.
Probabilistic Symbolic Regression for Equation Discovery via Operator-induced and Regularized Symbolic Forests
arXiv:2509.19710v2 Announce Type: cross 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 under approximate symbolic realizability, yielding a near-parametric rate for exact symbolic representability. 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 descriptors in a challenging materials discovery problem.
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 under approximate symbolic realizability, yielding a near-parametric rate for exact symbolic representability. 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 descriptors in a challenging materials discovery problem.
Methodology (stat.ME); Machine Learning (cs.LG); Symbolic Computation (cs.SC); Statistics Theory (math.ST); Machine Learning (stat.ML)
arXiv:2509.19710 [stat.ME]
(or arXiv:2509.19710v2 [stat.ME] for this version)
https://doi.org/10.48550/arXiv.2509.19710 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 279 words
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