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

arXiv: https://arxiv.org/abs/2306.17038 · HTML: https://mlsanity.com/paper/2306.17038/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 17 references that are in the map.
- First posted on arXiv 29 Jun 2023.

## Builds on (most cited first, 9 of 9)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery](https://mlsanity.com/paper/2301.12649/) (2023)
- [Noise-aware Physics-informed Machine Learning for Robust PDE Discovery](https://mlsanity.com/paper/2206.12901/) (2022)
- [Bayesian Bootstrap Spike-and-Slab LASSO](https://mlsanity.com/paper/2011.14279/) (2020)
- [Multi-objective discovery of PDE systems using evolutionary approach](https://mlsanity.com/paper/2103.06739/) (2021)

## Abstract (arXiv)

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.