Bayesian symbolic regression via posterior sampling

Symbolic regression (SR) is a powerful tool for discovering governing equations directly from data, but its sensitivity to noise hinders its broader application. This article introduces a sequential Monte Carlo (SMC) framework for Bayesian SR that approximates the posterior distribution over symbolic expressions, enhancing robustness and enabling uncertainty quantification for SR in the presence of noise. Differing from traditional genetic programming approaches, the SMC-based algorithm combines probabilistic selection, adaptive tempering and the use of normalized marginal likelihood (NML) to efficiently explore the search space of symbolic expressions, yielding parsimonious expressions with improved generalization. When compared with standard genetic programming baselines, the proposed method better deals with challenging, noisy benchmark datasets. The reduced tendency to overfit and enhanced ability to discover accurate and interpretable equations paves the way for more robust SR in scientific discovery and engineering design applications. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.