Language models guide symbolic equation discovery by controlling search

Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.

Merge Kernel for Bayesian Optimization on Permutation Space

Bayesian Optimization (BO) algorithm is a standard tool for black-box optimization problems. The current state-of-the-art BO approach for permutation spaces relies on the Mallows kernel-an $\Omega(n^2)$ representation that explicitly enumerates every pairwise comparison. Inspired by the close relationship between the Mallows kernel and pairwise comparison, we propose a novel framework for generating kernel functions on permutation space based on sorting algorithms. Within this framework, the Mallows kernel can be viewed as a special instance derived from bubble sort. Further, we introduce the \textbf{Merge Kernel} constructed from merge sort, which replaces the quadratic complexity with $\Theta(n\log n)$ to achieve the lowest possible complexity. The resulting feature vector is significantly shorter, can be computed in linearithmic time, yet still efficiently captures meaningful permutation distances. To boost robustness and right-invariance without sacrificing compactness, we further incorporate three lightweight, task-agnostic descriptors: (1) a shift histogram, which aggregates absolute element displacements and supplies a global misplacement signal; (2) a split-pair line, which encodes selected long-range comparisons by aligning elements across the two halves of the whole permutation; and (3) sliding-window motifs, which summarize local order patterns that influence near-neighbor objectives. Our empirical evaluation demonstrates that the proposed kernel consistently outperforms the state-of-the-art Mallows kernel across various permutation optimization benchmarks. Results confirm that the Merge Kernel provides a more compact yet more effective solution for Bayesian optimization in permutation space.