ERBench: A Benchmark and Testsuite for Equation Discovery Algorithms

2026-06-08

AI Summary

Existing symbolic regression benchmarks lack systematic evaluation of algorithmic robustness and true formula recovery across varying data dimensions, sample sizes, distributions, and noise conditions. To address this gap, this work proposes ERBench—the first dedicated benchmark for equation discovery—integrating a diverse collection of test problems with known ground-truth equations. ERBench encompasses a wide range of data configurations, sampling strategies, and functional complexities, thereby substantially enhancing the reliability and challenge of empirical assessments. The framework establishes a unified and rigorous standard for evaluating the effectiveness and generalization capability of symbolic regression algorithms in scientific model discovery.

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.

Problem

• equation discovery
• symbolic regression
• benchmark
• equation recovery
• robustness

Innovation

• equation discovery
• symbolic regression
• benchmark
• equation recovery
• robustness evaluation