Contemporary Symbolic Regression Methods and their Relative Performance

TL;DR

The paper tackles the lack of standardized benchmarking in symbolic regression by introducing SRBench, an open-source, reproducible platform. It benchmarks 14 symbolic regression methods and 7 machine learning methods across 252 regression problems, including real-world datasets and ground-truth physics/ODE systems, to map state-of-the-art performance. Key findings show that real-world performance is strongest for GA-based SR with parameter estimation and semantic search, while noiseless ground-truth recovery is led by AIFeynman; under noisy conditions, DSR and AFP-family methods show robustness. The work also provides a detailed, reproducible workflow and invites community contributions to maintain a living benchmark for symbolic regression research.

Abstract

Many promising approaches to symbolic regression have been presented in recent years, yet progress in the field continues to suffer from a lack of uniform, robust, and transparent benchmarking standards. In this paper, we address this shortcoming by introducing an open-source, reproducible benchmarking platform for symbolic regression. We assess 14 symbolic regression methods and 7 machine learning methods on a set of 252 diverse regression problems. Our assessment includes both real-world datasets with no known model form as well as ground-truth benchmark problems, including physics equations and systems of ordinary differential equations. For the real-world datasets, we benchmark the ability of each method to learn models with low error and low complexity relative to state-of-the-art machine learning methods. For the synthetic problems, we assess each method's ability to find exact solutions in the presence of varying levels of noise. Under these controlled experiments, we conclude that the best performing methods for real-world regression combine genetic algorithms with parameter estimation and/or semantic search drivers. When tasked with recovering exact equations in the presence of noise, we find that deep learning and genetic algorithm-based approaches perform similarly. We provide a detailed guide to reproducing this experiment and contributing new methods, and encourage other researchers to collaborate with us on a common and living symbolic regression benchmark.

Figures (11)

• Figure 1: Results on the black-box regression problems. Points indicate the mean of the median test set performance on all problems, and bars show the 95% confidence interval. Methods marked with an asterisk are SR methods.
• Figure 2: Pareto plot comparing the rankings of SR methods in terms of model size and $R^2$ score on the black-box problems. Points denote median rankings and the bars denote 95% confidence intervals. Connecting lines and color denote Pareto dominance rankings.
• Figure 3: Solution rates for the ground-truth regression problems. Color/shape indicates level of noise added to the target variable.
• Figure 4: An example code contribution, defining the estimator, its hyperparameters, and functions to return the complexity and symbolic model.
• Figure 5: Distribution of dataset sizes in PMLB.

Theorems & Definitions (1)

• Definition 4.1: Symbolic Solution