Comparison of Single- and Multi- Objective Optimization Quality for Evolutionary Equation Discovery

TL;DR

This paper compares single- and multi-objective evolutionary algorithms for discovering differential equations, demonstrating their effectiveness on classical models like Burgers, wave, and Korteweg-de Vries equations.

Contribution

It provides a systematic comparison of single- and multi-objective optimization approaches in evolutionary differential equation discovery, highlighting their relative advantages.

Findings

Multi-objective optimization balances accuracy and complexity effectively.

Both approaches successfully recover classical equations from data.

Multi-objective methods tend to produce simpler, more interpretable models.

Abstract

Evolutionary differential equation discovery proved to be a tool to obtain equations with less a priori assumptions than conventional approaches, such as sparse symbolic regression over the complete possible terms library. The equation discovery field contains two independent directions. The first one is purely mathematical and concerns differentiation, the object of optimization and its relation to the functional spaces and others. The second one is dedicated purely to the optimizational problem statement. Both topics are worth investigating to improve the algorithm's ability to handle experimental data a more artificial intelligence way, without significant pre-processing and a priori knowledge of their nature. In the paper, we consider the prevalence of either single-objective optimization, which considers only the discrepancy between selected terms in the equation, or multi-objective optimization, which additionally takes into account the complexity of the obtained equation. The proposed comparison approach is shown on classical model examples -- Burgers equation, wave equation, and Korteweg - de Vries equation.