Incorporating Background Knowledge in Symbolic Regression using a Computer Algebra System

TL;DR

This paper explores how incorporating symbolic mathematical constraints as background knowledge influences symbolic regression, demonstrating that soft constraints improve search effectiveness and model meaningfulness, especially within Bayesian SR frameworks.

Contribution

It introduces methods for integrating background knowledge into symbolic regression using constraints, comparing their effects on genetic algorithms and Bayesian approaches.

Findings

Soft constraints improve search effectiveness and model interpretability.

Bayesian SR handles constraints better than genetic algorithms.

Hard constraints can hinder the search process.

Abstract

Symbolic Regression (SR) can generate interpretable, concise expressions that fit a given dataset, allowing for more human understanding of the structure than black-box approaches. The addition of background knowledge (in the form of symbolic mathematical constraints) allows for the generation of expressions that are meaningful with respect to theory while also being consistent with data. We specifically examine the addition of constraints to traditional genetic algorithm (GA) based SR (PySR) as well as a Markov-chain Monte Carlo (MCMC) based Bayesian SR architecture (Bayesian Machine Scientist), and apply these to rediscovering adsorption equations from experimental, historical datasets. We find that, while hard constraints prevent GA and MCMC SR from searching, soft constraints can lead to improved performance both in terms of search effectiveness and model meaningfulness, with computational costs increasing by about an order-of-magnitude. If the constraints do not correlate well with the dataset or expected models, they can hinder the search of expressions. We find Bayesian SR is better these constraints (as the Bayesian prior) than by modifying the fitness function in the GA