ERBench: A Benchmark and Testsuite for Equation Discovery Algorithms
Authors: Paul Kahlmeyer, Henrik Voigt, Michael Habeck, Joachim Giesen
Organizations: Department of Computer Science University of Jena Inselplatz 5, 07743 Jena, Germany
Abstract
Equation discovery aims to automate the discovery of scientific models in the form of mathematical equations from data. Technically, equation discovery is implemented by symbolic regression algorithms. Performance of symbolic regression for equation discovery is measured along two dimensions: Prediction accuracy on test data, and recovery of known groundtruth formulas. For standard regression, accuracy is typically measured on in-domain test data, for instance, by splitting a data set randomly into training and test data. While this makes sense for in-domain interpolation, which is the common goal in ordinary regression, it can be a misleading proxy for true model discovery and generalization. The obvious alternative is to measure out-of-domain accuracy. However, obtaining challenging out-of-domain test data is a non-trivial problem. Therefore, we focus on equation recovery for evaluating symbolic regression algorithms for equation discovery. The rationale is that symbolic regression algorithms that perform well in recovering known groundtruth formulas are good candidates to perform well in unknown equation discovery. Existing benchmarks for symbolic regression include equation recovery tasks, however, with only a small number of groundtruth formulas that are publicly known. Moreover, these benchmarks place less emphasis on evaluating the robustness of algorithms in terms of their behavior under changing dimensionality, sampling size, sampling distribution and sampling domain. This, however, is of central importance to practitioners wanting to discover equations for modeling natural phenomena, since data is almost certainly noisy and comes from diverse domains, distributions, and sample sizes. To fill this gap, we introduce the Equation Recovery Benchmark (ERBench), a new evaluation framework designed to rigorously assess algorithms explicitly targeting the task of equation discovery.
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.
LLM-based equation discovery offers a promising route to recovering symbolic laws from data, but many systems still rely on generation-centered loops that propose candidates, fit parameters, score results, and reuse selected examples. Such loops can misjudge useful skeletons under unreliable fitting, discard near-correct equations that require repair, and accumulate redundant memories that provide limited guidance. We propose STRIDE, a self-reflective agent framework that improves reliability by coordinating data-aware generation, mixed-fitting evaluation, critic--executor repair, and diversity-preserving semantic memory. By turning fitted scores and candidate behavior into shared feedback, STRIDE enables equations to be proposed, assessed, refined, and reused within a closed-loop discovery process. Experiments on representative symbolic-regression benchmarks and LSR-Synth suites show that STRIDE improves accuracy, OOD robustness, and structural recovery across multiple LLM backbones, with ablations and analyses confirming the contribution of its core components.
Many equations arising in science currently cannot be solved by available analytical techniques and are therefore solved numerically, without yielding explicit symbolic expressions. Existing symbolic regression approaches can recover symbolic expressions, but require training data obtained from the underlying process, rather than the governing equation alone. We propose the Symbolic Equation Solver (SES), a framework that formulates equation solving as an optimization problem over differentiable symbolic models. SES constructs its objective from the equation together with initial or boundary conditions, eliminating the need for paired input-output data. The learned model is expressed in explicit symbolic form, enabling further analysis. We evaluate SES on representative algebraic and differential equations, including a system of algebraic equations, an equation with transcendental terms, an ordinary differential equation, and partial differential equations with different initial or boundary conditions. Across these settings, SES recovers compact symbolic expressions that match the corresponding analytical solutions.